Stochastic Analysis with Financial Applications: Hong Kong 2009

Progress in Probability Volume 65

Series Editors Charles Newman Sidney I. Resnick

For other volumes published in this series, go to www.springer.com/series/4839

Stochastic Analysis with Financial Applications Hong Kong 2009

Arturo Kohatsu-Higa Nicolas Privault Shuenn-Jyi Sheu Editors

Editors Arturo Kohatsu-Higa Department of Mathematical Sciences Japan Science and Technology Agency Ritsumeikan University 1-1-1 Nojihigashi, Kusatsu Shiga, 525-8577 Japan [email protected]

Nicolas Privault Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University SPMS-MAS-05-43, 21 Nanyang Link Singapore 637371 [email protected]

Shuenn-Jyi Sheu Department of Mathematics National Central University No. 300, Zhongda Rd., Zhongli City Taoyuan County 320 Taiwan (R.O.C.) [email protected]

2010 Mathematical Subject Classification 60H, 65C, 91B, 91G, 93E ISBN 978-3-0348-0096-9 DOI 10.1007/978-3-0348-0097-6

e-ISBN 978-3-0348-0097-6

Library of Congress Control Number: 2011933482 © Springer Basel AG 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use, permission of the copyright owner must be obtained. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

List of Speakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Part I: Stochastic Analysis N. Bouleau and L. Denis Dirichlet Forms for Poisson Measures and L´evy Processes: The Lent Particle Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

S. Chakraborty, E.T. Kolkovska and J.A. L´ opez-Mimbela Stability of a Nonlinear Equation Related to a Spatially-inhomogeneous Branching Process . . . . . . . . . . . . . . . . . . . . . . . . .

21

S.N. Cohen and R.J. Elliott Backward Stochastic Difference Equations with Finite States . . . . . . . .

33

A.B. Cruzeiro and E. Shamarova On a Forward-backward Stochastic System Associated to the Burgers Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

S. Fang and H. Lee On the Estimate for Commutators in DiPerna–Lions Theory . . . . . . . .

61

F. Gao and H. Jiang Approximation Theorem for Stochastic Differential Equations Driven by G-Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

B. Goldys and X. Zhang Stochastic Flows for Nonlinear SPDEs Driven by Linear Multiplicative Space-time White Noises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

Y. Ishikawa Optimal Stopping Problem Associated with Jump-diffusion Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

B. Jourdain and A. Kohatsu-Higa A Review of Recent Results on Approximation of Solutions of Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121

vi

Contents

A. Kohatsu-Higa, N. Vayatis and K. Yasuda Strong Consistency of Bayesian Estimator Under Discrete Observations and Unknown Transition Density . . . . . . . . . . . . . . . . . . . . . . 145 J. Luo Exponentially Stable Stationary Solutions for Delay Stochastic Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169

B. Øksendal and A. Sulem Robust Stochastic Control and Equivalent Martingale Measures . . . . . 179 J. Ren and J. Wu Multi-valued Stochastic Differential Equations Driven by Poisson Point Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 A. Takeuchi Sensitivity Analysis for Jump Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 J. Yang and J. Duan Quantifying Model Uncertainties in Complex Systems . . . . . . . . . . . . . . . 221 Part II: Financial Applications T.R. Bielecki, S. Cr´epey, M. Jeanblanc and M. Rutkowski Convertible Bonds in a Defaultable Diffusion Model . . . . . . . . . . . . . . . . . 255 T.-S. Chiang and S.-J. Sheu A Convexity Approach to Option Pricing with Transaction Costs in Discrete Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 J.M. Corcuera Completeness and Hedging in a L´evy Bond Market . . . . . . . . . . . . . . . . . . 317 M. Fukasawa Asymptotically Efficient Discrete Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 C.-H. Han Efficient Importance Sampling Estimation for Joint Default Probability: The First Passage Time Problem . . . . . . . . . . . . . . . . . . . . . . . 347 L. Li and M. Rutkowski Market Models of Forward CDS Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 J. Wei, H. Yang and R. Wang Optimal Threshold Dividend Strategies under the Compound Poisson Model with Regime Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

413

Preface The Workshop on Stochastic Analysis and Finance took place at City University of Hong Kong from June 29 to July 03, 2009. The goal of this workshop was to present a broad overview of the range of applications of stochastic analysis and its recent theoretical developments, while giving some weight to the research being carried out in the East Asia region. The topics of the talks given in the conference ranged from mathematical aspects of the theory of stochastic processes, to their applications to finance. This is reflected in the organization of the volume which is split into two sections on stochastic analysis and on financial applications. In recent times the applications of stochastic analysis to finance and insurance have bloomed exponentially, and for this reason we have devoted to them a significant attention. Stochastic analysis has also a variety of other applications to biological systems, physical and engineering problems, requiring the development of advanced techniques, a representative sample of which is also included here. A large number of articles in this volume deal with stochastic equations, and in particular stochastic (partial) differential equations and stochastic delay equations which arise naturally in physical systems depending on time and space. Contributions dealing with the numerical simulation and error analysis of these stochastic systems, which are an obligatory step before carrying out the actual applications, are also included and can also be of crucial importance to finance. The important and difficult topics of statistical estimation of parameters in these models, as well as their control and robustness, are also treated in this volume. The conference has also covered two areas that are growing rapidly. First the area of backward stochastic differential equations and all its variants that have deep connections with non-linear partial differential equations. Secondly, the recent developments of (non-linear) G-Brownian motion and its potential uses in risk analysis, which are also opening a new venue of development for stochastic analysis. From a technical point of view, the existence of densities of random variables associated with stochastic differential equations is an important matter, for which a quite complete basic theory is already available for continuous diffusion processes. The case of jump processes, which has already been the object of many important advances, is still in need of many developments that are motivated by applications, as shown in this volume. Concerning the applications to finance, many of the articles deal with a topic that has taken by storm our current society, which is how to deal with the valuation and hedging of credit risk in various forms. The

viii

Preface

results presented in the financial applications section cover in particular pricing and hedging in credit risk and jump models, including recent results on markets with frictions such as transaction costs, and L´evy driven market models. The articles contained in these proceedings are survey articles and original research papers which have been peer reviewed, and we take this opportunity to thank the colleagues who have largely contributed with their time as referees. We also thank the contributors for answering our requests to improve the presentation and results in order to produce a high quality volume, and all workshop participants for lively discussions. The participants and organizers are also grateful to the Lee Hysan Foundation, the Hong Kong Pei Hua Education Foundation, and the Department of Mathematics at City University of Hong Kong, for their generous financial support and for providing the conference venue. Last but not least, we acknowledge Ms Lonn Chan of the Mathematics General Office, whose highly efficient organizational skills ensured the complete success of the event. October 2010

Arturo Kohatsu-Higa Nicolas Privault Shuenn-jyi Sheu

List of Speakers Bouleau, N. Corcuera, J.M. Cruzeiro, A.B. Da Fonseca, J. Dai, M. Denis, L. Di Nunno, G. Dong, Z. Duan, J. El Khatib, Y. Elliott, R.J. Fang, S. Fukasawa, M. Gao, F.Q. Han C.H. Hayashi, M. Hwang, C.R. Ishikawa, Y. Jeanblanc, M. Kohatsu Higa, A. Li, X.D. L´ opez Mimbela, J.A. Øksendal, B. Ren, J. Rutkowski, M. Sheu, S.J. Shin, Y.H. Siu, T.K. Takeuchi, A. Wu, C.T. Yang, H. Yasuda, K. Ying, J. Zambrini, J.-C. Zhang, X. Zheng, H.

ENPC Paris, France University of Barcelona, Spain IST Lisbon, Portugal Auckland University of Technology, New Zealand National University of Singapore University of Evry, France University of Oslo, Norway Chinese Academy of Sciences, China Illinois Institute of Technology, USA United Arab Emirates University University of Calgary, Canada University of Bourgogne, France Osaka University, Japan Wuhan University, China National Tsing-Hua University, Taiwan Kyoto University, Japan Academia Sinica, Taiwan Ehime University, Japan University of Evry, France Ritsumeikan University, Japan Fudan University, China CIMAT Guanajuato, Mexico University of Oslo, Norway Sun Yat-sen University, China University of Sydney, Australia National Central University, Taiwan KAIST Daejeon, Korea Macquarie University, Australia Osaka City University, Japan National Chiao Tung University, Taiwan University of Hong Kong, China Hosei University, Japan Fudan University, China University of Lisbon, Portugal Wuhan University, China Imperial College London, U.K.

Part I Stochastic Analysis

Progress in Probability, Vol. 65, 3–20 c 2011 Springer Basel AG


Dirichlet Forms for Poisson Measures and L´evy Processes: The Lent Particle Method Nicolas Bouleau and Laurent Denis Abstract. We present a new approach to absolute continuity of laws of Poisson functionals. The theoretical framework is that of local Dirichlet forms as a tool for studying probability spaces. The argument gives rise to a new explicit calculus that we present first on some simple examples: it consists in adding a particle and taking it back after computing the gradient. Then we apply the method to SDE’s driven by Poisson measure. Mathematics Subject Classification (2000). Primary 60G57, 60H05; secondary 60J45, 60G51. Keywords. Stochastic differential equation, Poisson functional, Dirichlet form, energy image density, L´evy processes, gradient, carr´e du champ.

1. Introduction In order to situate the method it is worth to emphasize some features of the Dirichlet forms approach with comparison to the Malliavin calculus which is generally better known among probabilists. First the arguments hold under only Lipschitz hypotheses: for example the method applies to a stochastic differential equation with Lipschitz coefficients. Second a general criterion exists, (EID) the Energy Image Density property, (proved on the Wiener space for the Ornstein-Uhlenbeck form, still a conjecture in general cf. Bouleau-Hirsch [7] but established in the case of random Poisson measures with natural hypotheses) which ensures the existence of a density for a Rd -valued random variable. Third, Dirichlet forms are easy to construct in the infinite-dimensional frameworks encountered in probability theory and this yields a theory of errors propagation through the stochastic calculus, especially for finance and physics cf. Bouleau [2], but also for numerical analysis of PDE’s and SPDE’s cf. Scotti [18]. Our aim is to extend, thanks to Dirichlet forms, the Malliavin calculus applied to the case of Poisson measures and SDE’s with jumps. Let us recall that in the

4

N. Bouleau and L. Denis

case of jumps, there are several ways for applying the ideas of Malliavin calculus. The works are based either on the chaos decomposition (Nualart-Vives [14]) and provide tools in analogy with the Malliavin calculus on Wiener space, but nonlocal (Picard [15], Ishikawa-Kunita [12]) or dealing with local operators acting on the size of the jumps using the expression of the generator on a sufficiently rich class and closing the structure, for instance by Friedrichs’ argument (cf. especially Bichteler-Gravereaux-Jacod [1], Coquio [8] and Ma-R¨ ockner [13]). We follow a way close to this last one. We will first expose the method from a practical point of view, in order to show how it acts on concrete cases. Then in a separate part we shall give the main elements of the proof of the main theorem on the lent particle formula. Eventually we will display several examples where the method improves known results. Then, in the last section, we shall apply the lent particle method to SDE’s driven by a Poisson measure or a L´evy process. Complete details of the proofs and hypotheses for getting (EID) are published in [3] and [4].

2. The lent particle method Consider a random Poisson measure as a distribution of points, and let us see a L´evy process as defined by a Poisson random measure, that is let us think on the configuration space. We suppose the particles live in a space (called the bottom space) equipped with a local Dirichlet form with carr´e du champ and gradient. This makes it possible to construct a local Dirichlet form with carr´e du champ on the configuration space (called the upper space). To calculate for some functional the Malliavin matrix – which in the framework of Dirichlet forms becomes the carr´e du champ matrix – the method consists first in adding a particle to the system. The functional then possesses a new argument which is due to this new particle. We can compute the bottom-gradient of the functional with respect to this argument and as well its bottom carr´e du champ. Then taking back the particle we have added does not erase the new argument of the obtained functional. We can integrate the new argument with respect to the Poisson measure and this gives the upper carr´e du champ matrix – that is the Malliavin matrix. This is the exact summary of the method. 2.1. Let us give more details and notation Let (X, X , ν, d, γ) be a local symmetric Dirichlet structure which admits a carr´e du champ operator. This means
that (X, X , ν) is a measured space, ν is σ-finite and the bilinear form e[f, g] = 12 γ[f, g] dν is a local Dirichlet form with domain d ⊂ L2 (ν) and carr´e du champ γ (cf. Fukushima-Oshima-Takeda [10] in the locally compact case and Bouleau-Hirsch [7] in a general setting). (X, X , ν, d, γ) is called the bottom space. Consider a Poisson random measure N on [0, +∞[×X with intensity measure dt × ν. A Dirichlet structure may be constructed canonically on the probability

Dirichlet Forms for Poisson Measures and L´evy Processes

5

space of this Poisson measure that we denote (Ω1 , A1 , P1 , D, Γ). We call this space the upper space. D is a set of functions in the domain of Γ, in other words a set of random variables which are functionals of the random distribution of points. The main result is the following formula: For all F ∈ D   +∞

ε− (γ[ε+ F ]) dN

Γ[F ] = 0

X

in which ε+ and ε− are the creation and annihilation operators. Let us explain the meaning and the use of this formula on an example. 2.2. First example Let Yt be a centered L´evy process with L´evy measure ν integrating x2 . We assume that ν is such that a local Dirichlet structure may be constructed on R\{0} with carr´e du champ γ[f ] = x2 f 2 (x). The notion of gradient in the sense of Dirichlet forms is explained in [7] Chapter V. It is a linear operator with values in an auxiliary Hilbert space giving the carr´e du champ by taking the square of the Hilbert norm. It is convenient to choose as Hilbert space a space L2 of a probability space. Here we define a gradient  associated with γ by choosing ξ such that  1  1 ξ(r)dr = 0 and ξ 2 (r)dr = 1 0

0

and putting f
= xf  (x)ξ(r). Practically  acts as a derivation with the chain rule (ϕ(f ))
= ϕ (f ).f
(for ϕ ∈ C 1 ∩ Lip or even only Lipschitz). N is
the Poisson random measure associated with Y with intensity dt × σ
t ˜ (dsdx) for h ∈ L2 (R+ ). such that 0 h(s) dYs = 1[0,t] (s)h(s)xN loc We study the regularity of  t V = ϕ(Ys− )dYs 0

where ϕ is Lipschitz and C . 1) We add a particle (α, x), i.e., a jump to Y at time α with size x what gives  t ε+ V = V + ϕ(Yα− )x + (ϕ(Ys− + x) − ϕ(Ys− ))dYs . 1

]α


2) V = 0 since V does not depend on x, and    t +
 (ε V ) = ϕ(Yα− )x + ϕ (Ys− + x)xdYs ξ(r) ]α


because x = xξ(r).

6

N. Bouleau and L. Denis 3) We compute



γ[ε+ V ] =



t

(ε+ V )
2 dr = (ϕ(Yα− )x +

ϕ (Ys− + x)xdYs )2 .

]α

4) We take back the particle what gives ε− γ[ε+ V ] = (ϕ(Yα− )x +



t

ϕ (Ys− )xdYs )2

]α


and compute Γ[V ] = ε− γ[ε+V ]dN (lent particle formula) 2    t  ϕ (Ys− )dYs x2 N (dαdx) Γ[V ] = ϕ(Yα− ) + =

 α
t



]α t

∆Yα2 (

ϕ (Ys− )dYs + ϕ(Yα− ))2

]α

where ∆Yα = Yα − Yα− . For real functional, the condition (EID) is always fulfilled: V possesses a density as soon as Γ[V ] > 0. Then the above expression may be used to discuss the strict positivity of Γ[V ] depending on the finite or infinite mass of ν cf. [4] Example 5.2. Before giving a typical set of assumptions that the L´evy measure ν has to fulfill, let us explicit the (EID) property. 2.3. Energy Image Density property (EID) A Dirichlet form on L2 (Λ) (Λ σ-finite) with carr´e du champ γ satisfies (EID) if, for any d and all U with values in Rd whose components are in the domain of the form, the image by U of the measure with density with respect to Λ the determinant of the carr´e du champ matrix is absolutely continuous with respect to the Lebesgue measure, i.e., U∗ [(detγ[U, U t]) · Λ]  λd . This property is true for the Ornstein-Uhlenbeck form on the Wiener space, and in several other cases cf. Bouleau-Hirsch [7]. It was conjectured in 1986 that it were always true. It is still a conjecture. It is therefore necessary to prove this property in the context of Poisson random measures. With natural hypotheses, cf. [4] Parts 2 and 4, as soon as (EID) is true for the bottom space, (EID) is true for the upper space. Our proof uses a result of Shiqi Song [19]. 2.4. Main example of bottom structure in Rd Let (Yt )t  0 be a centered d-dimensional L´evy process without gaussian part, with L´evy measure ν = kdx. Under standard hypotheses, we have the following representation:  t ˜ (ds, dx), Yt = xN 0

Rd

Dirichlet Forms for Poisson Measures and L´evy Processes

7

˜ is a compensated Poisson measure with intensity dt × kdx. In this case, where N the idea is to introduce an ad-hoc Dirichlet structure on Rd . The following example gives a case of such a structure (d, e) which satisfies all the required hypotheses and which is flexible enough to encompass many cases: Lemma 1. Let r ∈ N∗ , (X, X ) = (Rr , B(Rr )) and ν = kdx where k is non-negative and Borelian. We are given ξ = (ξi,j )1
i,j
r an Rr×r -valued and symmetric Borel function. We assume that there exist an open set O ⊂ Rr and a function ψ continuous on O and null on Rr \ O such that 1) k > 0 on O ν-a.e. and is locally bounded on O. 2) ξ is locally bounded and locally elliptic on O. 3) k
ψ > 0 ν-a.e. on O. 1 4) for all i, j ∈ {1, . . . , r}, ξi,j ψ belongs to Hloc (O). 2 We denote by H the subspace of functions f ∈ L (ν) ∩ L1 (ν) such that the restriction of f to O belongs to Cc∞ (O). Then, the bilinear form defined by r   ∀f, g ∈ H, e(f, g) = ξi,j (x)∂i f (x)∂j g(x)ψ(x) dx i,j=1

O

is closable in L2 (ν). Its closure, (d, e), is a local Dirichlet form on L2 (ν) which admits a carr´e du champ γ: ∀f ∈ d, γ(f )(x) =

r 

ξi,j (x)∂i f (x)∂j f (x)

i,j=1

ψ(x) . k(x)

Moreover, it satisfies property (EID). Remark. In the case of a L´evy process, we will often apply this lemma with ξ the identity mapping. We shall often consider an open domain of the form O = {x ∈ Rd ; |x| < ε} which means that we “differentiate” only w.r.t. small jumps and hypothesis 3. means that we do not need to assume regularity on k but only that k dominates a regular function. 2.5. Multivariate example Consider as in the previous section, a centered L´evy process without gaussian part Y such that its L´evy measure ν satisfies assumptions of Lemma 1 (which imply 1 + ∆Ys = 0 a.s.) with d = 1 and ξ(x) = x2 . We want to study the existence of density for the pair (Yt , Exp(Y )t ) where Exp(Y ) is the Dol´eans exponential of Y .  Exp(Y )t = eYt (1 + ∆Ys )e−∆Ys . s
t

1) We add a particle (α, y), i.e., a jump to Y at time α  t with size y:  Yt +y ε+ (1 + ∆Ys )e−∆Ys (1 + y)e−y = Exp(Y )t (1 + y). (α,y) (Exp(Y )t ) = e s
t

8

N. Bouleau and L. Denis 2) We compute γ[ε+ Exp(Y )t ](y) = (Exp(Y )t )2 y 2 ψ(y) . k(y) 3) We take back the particle:  2 ψ(y) ε− γ[ε+Exp(Y )t ] = Exp(Y )t (1 + y)−1 y 2 k(y) we integrate w.r.t. N and that gives the upper carr´e du champ operator (lent particle formula):   2 ψ(y) Γ[Exp(Y )t ] = Exp(Y )t (1 + y)−1 y 2 N (dα, dy) k(y) [0,t]×R   2 ψ(∆Yα ) = Exp(Y )t (1 + ∆Yα )−1 ∆Yα2 . k(∆Yα ) α
t

By a similar computation the matrix Γ of the pair (Yt , Exp(Yt )) is given by

 1 Exp(Y )t (1 + ∆Yα )−1 ψ(∆Yα )   Γ= ∆Yα2 . 2 k(∆Yα ) Exp(Y )t (1 + ∆Yα )−1 Exp(Y )t (1 + ∆Yα )−1 α
t

Hence under hypotheses implying (EID), such as those of Lemma 1, the density of the pair (Yt , Exp(Yt )) is yielded by the condition



1 dim L α ∈ JT =2 Exp(Y )t (1 + ∆Yα )−1 where JT denotes the jump times of Y between 0 and t. Making this in details we obtain Let Y be a real L´evy process with infinite L´evy measure with density dominating near 0 a positive continuous function, then the pair (Yt , Exp(Y )t ) possesses a density on R2 .

3. Demonstration of the lent particle formula 3.1. The construction Let us recall that (X, X , ν, d, γ) is a local Dirichlet structure with carr´e
du champ called the bottom space, ν is σ-finite and the bilinear form e[f, g] = 12 γ[f, g] dν is a local Dirichlet form with domain d ⊂ L2 (ν) and with carr´e du champ γ. We assume {x} ∈ X for all x ∈ X and ν is diffuse. The associated generator is denoted a, its domain is D(a) ⊂ d. We consider a random Poisson measure N , on [0, +∞[×X with intensity dt×ν. It is defined on (Ω1 , A1 , P1 ) where Ω1 is the configuration space of countable sums of Dirac masses on [0, +∞[×X, A1 is the σ-field generated by N and P1 is the law of N . (Ω1 , A1 , P1 ) is called the upper space. The question is to construct a Dirichlet structure on the upper space, induced “canonically” by the Dirichlet structure of the bottom space.

Dirichlet Forms for Poisson Measures and L´evy Processes

9

This question is natural by the following interpretation. The bottom structure may be thought as the elements for the description of a single particle moving according to a symmetric Markov process associated with the bottom Dirichlet form. Then considering an infinite family of independent such particles with initial law given by (Ω1 , A1 , P1 ) shows that a Dirichlet structure can be canonically considered on the upper space (cf. the introduction of [4] for different ways of tackling this question). Because of typical formulas on functions of the form eiN (f ) related to the Laplace functional, we consider the space of test functions D0 to be the set of ˜ are the linear of variables of the form eiN (f ) elements in L2 (P1 ) which    combinations 2 1 2 where f belongs to D(a) ⊗ L (dt) L (dt×ν) and is such that γ[f ] ∈ L (dt×ν), ˜ = N − dt × ν. recall that N Remark 1. As we need D0 to be a dense subset, we make what we call aBottom core hypothesis. Namely we assume that there exists a subspace H of D(a) L1 (ν), dense in L1 (ν)∩L2 (ν) and such that ∀f ∈ H, γ[f ] ∈ L2 (ν) (see [4] for more details on the technical hypotheses we adopt). ˜ If U = p λp eiN(fp ) belongs to D0 , we put  ˜ ˜ (a[fp ]) − 1 N (γ[fp ])), A0 [U ] = λp eiN(fp ) (iN (1) 2 p where, in a natural way, if f (x, t) = l ul (x)ϕl (t) ∈ D(a) ⊗ L2 (dt)   a[f ] = a[ul ]ϕl and γ[f ] = γ[ul ]ϕl . l

l

In order to show that A0 is uniquely defined and is the generator of a Dirichlet form satisfying the required properties, starting from a gradient of the bottom structure we construct a gradient for the upper structure defined first on the test functions. Then we show that this gradient does not depend on the form of the test function and this allows to extend the operators thanks to Friedrichs’ property yielding the closedness of the upper structure. 3.2. The bottom gradient We suppose the space d separable, then there exists a gradient for the bottom space, i.e., there is a separable Hilbert space and a linear map D from d into L2 (X, ν; H) such that ∀u ∈ d, D[u] 2H = γ[u], then necessarily – If F : R → R is Lipschitz then ∀u ∈ d, D[F ◦ u] = (F  ◦ u)Du, – If F is C 1 and Lipschitz from Rd into R then D[F ◦ u] = di=1 (Fi ◦ u)D[ui ] ∀u = (u1 , . . . , ud ) ∈ dd . We take for H a space L2 (R, R, ρ) where (R, R, ρ) is a probability space s.t. 2 L (R, R, ρ) is infinite dimensional. The gradient D is denoted by : ∀u ∈ d, Du = u
∈ L2 (X × R, X ⊗ R, ν × ρ).

10

N. Bouleau and L. Denis

Without loss of generality, we assume moreover that the operator  takes its values in the orthogonal space of 1 in L2 (R, R, ρ). So that we have  ∀u ∈ d, u
dρ = 0 ν-a.e. 3.3. Candidate gradient for the upper space Now, we introduce the creation operator (resp. annihilation operator) which consists in adding (resp. removing if necessary) a jump at time t with size u: ε+ / w1 } (t,u) (w1 ) = w1 1{(t,u)∈supp w1 } + (w1 + ε(t,u) })1{(t,u)∈supp ε− / w1 } + (w1 − ε(t,u) })1{(t,u)∈supp w1 } . (t,u) (w1 ) = w1 1{(t,u)∈supp In a natural way, we extend these operators to the functionals by − − ε+ H(w1 , t, u) = H(ε+ (t,u) w1 , t, u) ε H(w1 , t, u) = H(ε(t,u) w1 , t, u).

Definition. For F ∈ D0 , we define the pre-gradient  +∞  F = ε− ((ε+ F )
) dN ρ, 0

X×R

where N ρ is the point process N “marked” by ρ, i.e., if N is the family of marked points (Ti , Xi ), N ρ is the family (Ti , Xi , ri ) where the ri are new independent random variables mutually independent and identically distributed with law ρ, ˆ A, ˆ P). ˆ So N ρ is a Poisson random defined on an auxiliary probability space (Ω, measure on [0, +∞[×X × R. 3.4. Main result The above candidate may be shown to extend in a true gradient for the upper structure. The argument is based on the extension of the pregenerator A0 thanks to Friedrichs’ property (cf. for instance [7] p. 4): A0 is shown to be well defined on D0 which is dense, A0 is non-positive and symmetric and therefore possesses a selfadjoint extension. Before stating the main theorem, let us introduce some notation. We denote by D the completion of D0 ⊗ L2 ([0, +∞[, dt) ⊗ d with respect to the norm  +∞ 

12 − H D = E ε (γ[H])(w, t, u)N (dt, du) 0

X



+∞



(ε− |H|)(w, t, u)η(t, u)N (dt, du)

+E  = E

0

X

12 γ[H](w, t, u)ν(du)dt

+∞ 

0



X

+∞



+E

|H|(w, t, u)η(t, u)ν(du)dt, 0

X

where η is a fixed positive function in L2 (R+ × X, dt × dν).

Dirichlet Forms for Poisson Measures and L´evy Processes

11

As we shall see below, a peculiarity of the method comes from the fact that it involves, in the computation, successively mutually singular measures, such as measures PN = P1 (dω)N (ω, dt, dx) and P1 × dt × ν. This imposes some care in the applications. Main theorem. The formula



+∞



∀F ∈ D, F  = 0

ε− ((ε+ F )
) dN ρ,

X×R

extends from D0 to D, it is justified by the following decomposition: ε+ −I

F ∈ D → ε+ F − F ∈ D

ε− ((.)
)

→

ε− ((ε+ F )
) ∈ L20 (PN × ρ)

d(N ρ)

→

ˆ F  ∈ L2 (P1 × P)

where each operator is continuous on the range of the preceding one
and where L20 (PN × ρ) is the closed set of elements G in L2 (PN × ρ) such that R Gdρ = 0 PN -a.s. Furthermore for all F ∈ D  +∞   2 ˆ Γ[F ] = E(F ) = ε− γ[ε+ F ] dN. (2) 0

X

Let us explain the steps of a typical calculation applying this theorem. Let H = Φ(F1 , . . . , Fn ) with Φ ∈ C 1 ∩ Lip(Rn ) and F = (F1 , . . . , Fn ) with Fi ∈ D, we have:  a) γ[ε+ H] = Φi (ε+ F )Φj (ε+ F )γ[ε+ Fi , ε+ Fj ] P × ν-a.e. ij −

+

b) ε γ[ε H] =  c) Γ[H] =



Φi (F )Φj (F )ε− γ[ε+ Fi , ε+ Fj ] PN -a.e.

ij −

+

ε γ[ε H]dN =



Φi (F )Φj (F )



ε− γ[ε+ Fi , ε+ Fj ]dN P-a.e.

ij

Let us eventually remark that the lent particle formula (2) has been encountered previously by some authors for test functions (see, e.g., [17] before Prop. 8). Here, it is proved on the whole domain D, this is essential to apply the method to SDE’s and to exploit the full strength of the functional calculus of Dirichlet forms.

4. Applications 4.1. Sup of a stochastic process on [0, t] The fact that the operation of taking the maximum is typically a Lipschitz operation makes it easy to apply the method. Let Y be a centered L´evy process as in §2.2. Let K be a c` adl` ag process independent of Y . We put Hs = Ys + Ks . Proposition. If ν(R\{0}) = +∞ and if P1 [sups
t Hs = H0 ] = 0, the random variable sups
t Hs has a density.

12

N. Bouleau and L. Denis

As a consequence, any L´evy process starting from zero and immediately entering R∗+ , whose L´evy measure dominates a measure ν satisfying Hamza condition and infinite, is such that sups
t Xs has a density. Let us recall that the Hamza condition (cf. Fukushima et al. [10] Chapter 3) gives a necessary and sufficient condition of existence of a Dirichlet structure on L2 (ν). Such a necessary and sufficient condition is only known in dimension one. 4.2. Regularity without H¨ormander Consider the following SDE driven by a two-dimensional Brownian motion  1
t  Xt = z1 + 0 dBs1  
t
t Xt2 = z2 + 0 2Xs1 dBs1 + 0 dBs2  
t
t  3 Xt = z3 + 0 Xs1 dBs1 + 2 0 dBs2 .

(3)

This diffusion is degenerate and the H¨ormander conditions are not fulfilled. The generator is A = 12 (U12 + U22 ) + V and its adjoint A∗ = 12 (U12 + U22 ) − V with ∂ ∂ ∂ ∂ ∂ U1 = ∂x + 2x1 ∂x + x1 ∂x , U2 = ∂x + 2 ∂x and V = − ∂z∂ 2 − 12 ∂z∂ 3 . The Lie 1 2 3 2 3 brackets of these vectors vanish and the Lie algebra is of dimension 2: the diffusion remains on the quadric of equation 34 x21 − x2 + 12 x3 − 34 t = C. Consider now the same equation driven by a L´evy process:  1
t Zt = z1 + 0 dYs1   
t
t Zt2 = z2 + 0 2Zs1− dYs1 + 0 dYs2  
t
t  3 Zt = z3 + 0 Zs1− dYs1 + 2 0 dYs2 under hypotheses on the L´evy measure such that the bottom space may be equipped with the carr´e du champ operator γ[f ] = y12 f12 + y22 f22 satisfying the hypotheses yielding (EID). Let us apply in full details the lent particle method. For α  t     y1 y1
t  2Y 1 y + 2 y dYs1 + y2  ε+ α− 1  = Zt +  2y1 Yt1 + y2  , ]α 1 (α,y1 ,y2 ) Zt = Zt + 
t 1 y1 Yt1 + 2y2 Yα− y1 + ]α y1 dYs1 + 2y2 1 where we have used Yα− = Yα1 because ε+ send into P1 × dt × ν classes. That gives  2  y1 y12 2Yt1 y12 Yt1 γ[ε+ Zt ] =  id y12 4(Yt1 )2 + y22 y12 2(Yt1 )2 + 2y22  id id y12 (Yt1 )2 + 4y22

and



y12 − + ε γ[ε Zt ] =  id id

y12 2(Yt1 − ∆Yα1 ) 2 y1 4(Yt1 − ∆Yα1 )2 + y22 id

 y12 (Yt1 − ∆Yα1 ) y12 2(Yt1 − ∆Yα1 )2 + 2y22  , y12 (Yt1 − ∆Yα1 )2 + 4y22

Dirichlet Forms for Poisson Measures and L´evy Processes

13

where id denotes the symmetry of the matrices. Hence 

1 2(Yt1 − ∆Yα1 ) Γ[Zt ] = (∆Yα1 )2  id 4(Yt1 − ∆Yα1 )2 id id α
t 

 (Yt1 − ∆Yα1 ) 2(Yt1 − ∆Yα1 )2  (Yt1 − ∆Yα1 )2 

0 + (∆Yα2 )2  0 0

 0 0 1 2 . 2 4

With this formula we can reason, trying to find conditions for the determinant of Γ[Z] to be positive. For instance if the L´evy measures of Y 1 and Y 2 are infinite, it follows that Zt has a density as soon as      1 0   dim L  2(Yt1 − ∆Yα1 )  ,  1  α ∈ JT = 3.   (Yt1 − ∆Yα1 ) 2 But Y 1 possesses necessarily jumps of different sizes, hence Zt has a density on R3 . It follows that the integro-differential operator  ˜ (z) = Af





 z 1 + y1 f (z) − f  z2 + 2z1 y1 + y2  z3 + z1 y1 + 2y2

 y1 −(f1 (z) f2 (z) f3 (z))  2z1 y1 + y2  σ(dy1 dy2 ) z1 y1 + 2y2 

is hypoelliptic at order zero, in the sense that its semigroup Pt has a density. No minoration is supposed on the growth of the L´evy measure near 0 as assumed by some authors. This result implies that for any L´evy process Y satisfying the above hypotheses, even a subordinated one in the sense of Bochner, the process Z is never subordinated of the Markov process X solution of equation (3) (otherwise it would live on the same manifold as the initial diffusion).

5. Application to SDE’s driven by a Poisson measure 5.1. The equation we study We consider another probability space (Ω2 , A2 , P2 ) on which an Rn -valued semimartingale Z = (Z 1 , . . . , Z n ) is defined, n ∈ N∗ . We adopt the following assumption on the bracket of Z and on the total variation of its finite variation part. It is satisfied if both are dominated by the Lebesgue measure uniformly:

14

N. Bouleau and L. Denis

Assumption on Z: There exists a positive constant C such that for any square integrable Rn -valued predictable process h:  t  t hs dZs )2 ]  C 2 E[ |hs |2 ds]. (4) ∀t
0, E[( 0

0

We shall work on the product probability space: (Ω, A, P) = (Ω1 ×Ω2 , A1 ⊗A2 , P1 × P2 ). For simplicity, we fix a finite terminal time T > 0. Let d ∈ N∗ , we consider the following SDE:  t  t ˜ (ds, du) + Xt = x + c(s, Xs− , u)N σ(s, Xs− )dZs (5) 0

X

0

where x ∈ R , c : R × R × X → R and σ : R × Rd → Rd×n satisfy the set of hypotheses below denoted (R). d

+

d

d

+

Hypotheses (R): 1. There exists η ∈ L2 (X, ν) such that: a) for all t ∈ [0, T ] and u ∈ X, c(t, ·, u) is differentiable with continuous derivative and ∀u ∈ X,

|Dx c(t, x, u)|  η(u),

sup t∈[0,T ],x∈Rd

b) ∀(t, u) ∈ [0, T ] × X, |c(t, 0, u)|  η(u), c) for all t ∈ [0, T ] and x ∈ Rd , c(t, x, ·) ∈ d and sup

γ[c(t, x, ·)](u)  η 2 (u),

t∈[0,T ],x∈Rd

d) for all t ∈ [0, T ], all x ∈ Rd and u ∈ X, the matrix I + Dx c(t, x, u) is invertible and     −1 sup (I + Dx c(t, x, u)) × c(t, x, u)  η(u). t∈[0,T ],x∈Rd

2. For all t ∈ [0, T ], σ(t, ·) is differentiable with continuous derivative and sup

|Dx σ(t, x)| < +∞.

t∈[0,T ],x∈Rd

3. As a consequence of hypotheses 1. and 2. above, it is well known that equation (5) admits a unique solution X such that E[supt∈[0,T ] |Xt |2 ] < +∞. We suppose that for all t ∈ [0, T ], the matrix (I + nj=1 Dx σ·,j (t, Xt− )∆Ztj ) is invertible and its inverse is bounded by a deterministic constant uniformly with respect to t ∈ [0, T ]. Remark. We have defined a Dirichlet structure (D, E) on L2 (Ω1 , P1 ). Now, we work on the product space, Ω1 × Ω2 . Using natural notations, we consider from now on that (D, E) is a Dirichlet structure on L2 (Ω, P). In fact, it is the product structure of (D, E) with the trivial one on L2 (Ω2 , P2 ) (see [7] ). Of course, all the properties

Dirichlet Forms for Poisson Measures and L´evy Processes

15

remain true. In other words, we only differentiate w.r.t. the Poisson noise and not w.r.t. the one introduced by Z. 5.2. Spaces of processes and functional calculus We denote by P the predictable sigma-field on [0, T ]×Ω and we define the following sets of processes: – HD : the set of real-valued processes (Ht )t∈[0,T ] , which belong to L2 ([0, T ]; D), i.e., such that  T  T H 2HD = E[ |Ht |2 dt] + E(Ht )dt < +∞. 0

0

– HD,P : the subvector space of predictable processes in HD . – HD⊗d,P : the set of real-valued processes H defined on [0, T ] × Ω × X which are predictable and belong to L2 ([0, T ]; D ⊗ d), i.e., such that   T 2 H HD⊗d,P = E |Ht |2 ν(du)dt 0



T

X





T

E(Ht (·, u))ν(du)dt + E

+ 0

X

e(Ht )dt < +∞. 0

The main idea is to differentiate equation (5). To do that, we need some functional calculus. It is given by the next proposition that we prove by approximation: n Proposition 2. Let H ∈ HD⊗d,P and G ∈ HD,P , then:

1) The process

 t ∀t ∈ [0, T ], Xt =

˜ (ds, du) H(s, w, u)N 0

X

is a square integrable martingale which belongs to HD and such that the process X − = (Xt− )t∈[0,T ] belongs to HD,P . The gradient operator satisfies for all t ∈ [0, T ]:  t ˜ (ds, du) Xt (w, w) ˆ = H  (s, w, u, w)d ˆ N 0 X (6)  t
+ H (s, w, u, r)N ρ(ds, du, dr). 0

X×R

2) The process



t

∀t ∈ [0, T ], Yt =

G(s, w)dZs 0

is a square integrable semimartingale which belongs to HD , Y − = (Yt− )t∈[0,T ] belongs to HD,P and  t  ˆ = G (s, w, w)dZ ˆ (7) ∀t ∈ [0, T ], Yt (w, w) s. 0

16

N. Bouleau and L. Denis

5.3. Computation of the Carr´e du champ matrix of the solution Applying the standard functional calculus related to Dirichlet forms, the previous proposition and a Picard iteration argument, we obtain: Proposition 3. The equation (5) admits a unique solution X in HDd . Moreover, the gradient of X satisfies:  t Xt =

 ˜ Dx c(s, Xs− , u) · Xs− N (ds, du) 0 X  t + c
(s, Xs− , u, r)N ρ(ds, du, dr)



0

X×R t  Dx σ(s, Xs− ) · Xs− dZs .

+ 0

Let us define the Rd×d -valued processes U by dUs =

n 

Dx σ.,j (s, Xs− )dZsj ,

j=1

and the derivative of the flow generated by X:  t



t

˜ (ds, du) + Dx c(s, Xs− , u)Ks− N

Kt = I + 0

X

dUs Ks− . 0

Proposition 4. Under our hypotheses, for all t
0, the matrix Kt is invertible and ¯ t = (Kt )−1 satisfies: its inverse K  t ¯t = I − K 0



¯ s− (I + Dx c(s, Xs− , u))−1 Dx c(s, Xs− , u)N ˜ (ds, du) K

X



t

¯ s− dUs + K

− 0



¯ s− (∆Us )2 (I + ∆Us )−1 K

s
t t

¯ s d < U c , U c >s . K

+ 0

We are now able to calculate the carr´e du champ matrix. This is done in the next theorem whose proof is sketched to show how simple is the lent particle method. Theorem 5. For all t ∈ [0, T ],  t Γ[Xt ] = Kt 0

X

¯ s γ[c(s, Xs− , ·)]K ¯ ∗ N (ds, du)K ∗ . K s t

Dirichlet Forms for Poisson Measures and L´evy Processes (α,u)

Proof. Let (α, u) ∈ [0, T ] × X. We put Xt  (α,u)

Xt

α

 (α,u)

= x+ 

0



0

c(s, Xs−

α

+ +

X

= ε+ (α,u) Xt .

˜ (ds, du ) , u )N

(α,u)

(α,u)

σ(s, Xs− )dZs + c(α, Xα− , u)   (α,u)  ˜  c(s, Xs− , u )N (ds, du ) +

]α,t]

17

X

]α,t]

(α,u)

σ(s, Xs−

)dZs .

(α,u)

= Xt if t < α so that, taking the gradient with respect Let us remark that Xt to the variable u, we obtain:   (α,u)
(α,u) (α,u) (α,u) ˜ (ds, du ) (Xt ) = (c(α, Xα− , u))
+ Dx c(s, Xs− , u ) · (Xs− )
N ]α,t] X  (α,u) (α,u) + Dx σ(s, Xs− ) · (Xs− )
dZs . ]α,t]

(α,u)

Let us now introduce the process Kt = ε+ (α,u) (Kt ) which satisfies the following SDE:  t  t (α,u) (α,u) (α,u) ˜ (α,u) Kt =I+ Dx c(s, Xs− , u )Ks− N (ds, du ) + dUs(α,u) Ks− 0

X

0

¯ (α,u) = (K (α,u) )−1 . Then, using the flow property, we have: and its inverse K t t (α,u)


∀t
0, (Xt

(α,u)

) = Kt

¯ α(α,u) (c(α, Xα− , u))
. K

Now, we calculate the carr´e du champ and then we take back the particle: ∀t
0, ε− (α,u) γ[(Xt

(α,u)

¯ α γ[c(α, Xα− , ·)]K ¯ α∗ Kt∗ . )] = Kt K

Finally integrating with respect to N we get  t ¯ s γ[c(s, Xs− , ·)](u)K ¯ s∗ N (ds, du)Kt∗ . ∀t
0, Γ[Xt ] = Kt K 0



X

5.4. First application: the regular case An immediate consequence of the previous theorem is: Proposition 6. Assume that X is a topological space, that the intensity measure ds × ν of N is such that ν has an infinite mass near some point u0 in X. If the matrix (s, y, u) → γ[c(s, y, ·)](u) is continuous on a neighborhood of (0, x, u0 ) and invertible at (0, x, u0 ), then the solution Xt of (5) has a density for all t ∈]0, T ].

18

N. Bouleau and L. Denis

5.5. Application to SDE’s driven by a L´evy process Let Y be a L´evy process with values in Rd , independent of another variable X0 . We consider the following equation  t Xt = X0 + a(Xs− , s) dYs , t
0 0

where a : R × R → R k

+

k×d

is a given map.

Proposition 7. We assume that: 1) The L´evy measure, ν, of Y satisfies hypotheses of the example given in Section 2.4 with ν(O) = +∞ and ξi,j (x) = xi δi,j . Then we may choose the operator γ to be ψ(x)  2  xi (∂i f )2 k(x) d

γ[f ] =

d

i=1

for f ∈ C0∞ (Rd ).

i=1

2) a is C ∩ Lip with respect to the first variable uniformly in s and 1

sup |(I + Dx a · u)−1 (x, t)|  η(u), t,x

where η ∈ L (ν). 3) a is continuous with respect to the second variable at 0, and such that the matrix aa∗ (X0 , 0) is invertible; then for all t > 0 the law of Xt is absolutely continuous w.r.t. the Lebesgue measure. 2

Proof. We just give an idea of the proof in the case d = 1: ψ(u) 2 2 Let us recall that γ[f ](u) = u f (u). k(u)
t
˜ (ds, du), so that We have the representation: Yt = 0 R uN  t ˜ (ds, du). Xt = X0 + a(s, Xs− )u N 0

R

The lent particle method yields:  t 2 ¯ 2 a2 (s, Xs− )γ[j](u)N (ds, du) Γ[Xt ] = Kt K s 0

X

where j is the identity application: γ[j](u) = So

ψ(u) 2 u . k(u)

 t

¯ 2 a2 (s, Xs− ) ψ(u) u2 N (ds, du) K s k(u) 0 X  ¯ 2 a2 (s, Xs− ) ψ(∆Ys ) ∆Y 2 , = Kt2 K s s k(∆Ys ) α
Γ[Xt ] =

Kt2

and it is easy to conclude.



Dirichlet Forms for Poisson Measures and L´evy Processes

19

Remarks (i) We refer to [5] for other examples and applications. (ii) Let us finally remark that, as easily seen, the gradient we have introduced may be naturally iterated. This yields a criteria of regularity for the density of Poisson functionals such as solutions of SDE’s, this is the object of a forthcoming paper.

References [1] Bichteler K., Gravereaux J.-B., Jacod J. Malliavin Calculus for Processes with Jumps (1987). [2] Bouleau N. Error Calculus for Finance and Physics, the Language of Dirichlet Forms, De Gruyter (2003). [3] Bouleau N. “Error calculus and regularity of Poisson functionals: the lent particle method” C. R. Acad. Sc. Paris, Math´ematiques, Vol 346, n. 13-14, pp. 779–782, (2008). [4] Bouleau N. and Denis L. “Energy image density property and the lent particle method for Poisson measures” Jour. of Functional Analysis 257, pp. 1144–1174, (2009). [5] Bouleau N. and Denis L. “Application of the lent particle method to Poisson driven SDE’s”, to appear in Probability Theory and Related Fields, available online DOI: 10.1007/s00440-010-0303-x. [6] Bouleau N. and Hirsch F. “Formes de Dirichlet g´en´erales et densit´e des variables al´eatoires r´eelles sur l’espace de Wiener” J. Funct. Analysis 69, 2, pp. 229–259, (1986). [7] Bouleau N. and Hirsch F. Dirichlet Forms and Analysis on Wiener Space De Gruyter (1991). [8] Coquio A. “Formes de Dirichlet sur l’espace canonique de Poisson et application aux ´equations diff´erentielles stochastiques” Ann. Inst. Henri Poincar´ e vol. 19, n. 1, pp. 1–36, (1993). [9] Denis L. “A criterion of density for solutions of Poisson-driven SDEs” Probab. Theory Relat. Fields 118, pp. 406–426, (2000). [10] Fukushima M., Oshima Y. and Takeda M. Dirichlet Forms and Symmetric Markov Processes De Gruyter (1994). [11] Ikeda N., Watanabe S. Stochastic Differential Equation and Diffusion Processes, North-Holland, Koshanda (1981). [12] Ishikawa Y. and Kunita H. “Malliavin calculus on the Wiener-Poisson space and its application to canonical SDE with jumps” Stoch. Processes and their App. 116, pp. 1743–1769, (2006). ¨ ckner M. “Construction of diffusion on configuration spaces” Osaka J. [13] Ma and Ro Math. 37, pp. 273–314, (2000). [14] Nualart D. and Vives J. “Anticipative calculus for the Poisson process based on the Fock space”, S´em. Prob. XXIV, Lect. Notes in M. 1426, Springer (1990).

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[15] Picard J. “On the existence of smooth densities for jump processes” Probab. Theorie Relat. Fields 105, pp. 481–511, (1996). [16] Privault N. “A pointwise equivalence of gradients on configuration spaces”, C. Rendus Acad. Sc. Paris, 327, 7, pp. 677–682, (1998). [17] Privault N. “Equivalence of gradients on configuration spaces” Random Oper. and Stoch. Equ. Vol. 7, No. 3, pp. 241–262, (1999). [18] Scotti S. Applications de la Th´ eorie des Erreurs par Formes de Dirichlet, Thesis Univ. Paris-Est, Scuola Normale Pisa, 2008. (http://pastel.paristech.org/4501/) [19] Song Sh. “Admissible vectors and their associated Dirichlet forms” Potential Analysis 1, 4, pp. 319–336, (1992). Nicolas Bouleau Ecole des Ponts ParisTech, Paris-Est 6 Avenue Blaise Pascal F-77455 Marne-La-Vall´ee Cedex 2, France e-mail: [email protected] Laurent Denis Equipe Analyse et Probabilit´es Universit´e d’Evry-Val-d’Essonne Boulevard Fran¸cois Mitterrand F-91025 Evry Cedex, France e-mail: [email protected]

Progress in Probability, Vol. 65, 21–32 c 2011 Springer Basel AG


Stability of a Nonlinear Equation Related to a Spatially-inhomogeneous Branching Process S. Chakraborty, E.T. Kolkovska and J.A. L´opez-Mimbela Abstract. Consider the nonlinear equation ∞  ∂ u(x, t) = ∆α u(x, t) + a(x) pk uk (x, t) + (p0 + p1 u(x, t))φ(x), x ∈ Rd , ∂t k=2

where α ∈ (0, 2], u(x, 0) is nonnegative, {pk , k = 0, 1, . . .} is a probability distribution on Z+ , and a and φ are positive functions satisfying certain growth conditions. We prove existence of non-trivial positive global solutions when p0 , p1 and u(x, 0) are small. Mathematics Subject Classification (2000). Primary: 35K58, 35B44. Secondary: 35B09, 35B51. Keywords. Semi-linear partial differential equation, positive solutions, global solutions.

1. Introduction and main result In this paper we investigate existence of non-negative global solutions to the semilinear partial differential equation ∞  ∂u = ∆α u + a(x) pk uk + (p0 + p1 u)φ(x), u(x, 0) = f (x), x ∈ Rd , (1) ∂t k=2

where ∆α , 0 < α ≤ 2, is the generator of the spherically symmetric α-stable process in Rd , φ and a are non-negative functions satisfying certain growth conditions, and {pk , k = 0, 1, . . .} is a probability distribution with p1 > 0 and p0 > 0. Pinsky [10] and Zhang [13] studied both, existence and non-existence, of positive global solutions to the above equation when the sum in the reaction term is of the form a(x)up with p > 1, p0 does not depend on x and p1 = 0. Using a probabilistic approach, in [2] we investigated existence of positive global solutions to semilinear equations of the form ∞  ∂u = ∆α u + pk uk , (2) ∂t k=2

22

S. Chakraborty, E.T. Kolkovska and J.A. L´ opez-Mimbela

where pk ≥ 0 for all k ≥ 2 and k pk = 1. Both equations (1) and (2) admit the probabilistic representation (see [8], [9] or [2])    u(t, x) = E eLt u(0, z) , z∈suppXtx

where Xtx is a branching α-stable motion in Rd (with exponential individual lifetimes and offspring number Z such that P [Z = k] = pk ) starting from an ancestor at x, and Lt is the total length of the family tree up to time t. The approach developed in [2] works not only for the α-Laplacian but also for a wider class of Markov generators. In the framework of the above representation, for suitably decaying initial values u(0) := f , it is the exponential of the tree length that makes the solution u to grow. Hence, in order to get a global solution it is necessary that the motion process be mobile enough as to quickly populate the regions where f is near to 0. Therefore, in order to know for which values of the equation parameters (2) admits global solutions, a careful analysis of the underlying branching model is in order. For simpler semi-linear equations, this was successfully achieved in a number of previous papers (see, e.g., [11, 9, 2]). At the level of the branching model, (1) corresponds to a branching α-stable motion in which the branching rate depends on the space variable in such a way that it can grow and needs not to be away from 01 . These circumstances posse serious technical problems to directly apply the probabilistic approach, and therefore we have to proceed in a more analytical way. In Section 2 we prove existence and uniqueness of mild solutions to Equation (1) on Rd × [0, tmax ), where [0, tmax ) is a maximal existence interval (see Proposition 2). Typically, the “life span” tmax of a positive solution u depends on its initial value u(·, 0); when tmax = +∞ the corresponding solution is called global. Our main result, Theorem 4 below, gives conditions on the equation parameters, and on the decay of the initial value of (1), which are sufficient for existence of a nontrivial positive global solution to (1). Our proof of Theorem 4 relies on a comparison procedure (which is based on Proposition 3; see Section 2), and is carried out in Section 3. The final section is devoted to prove a technical lemma which is necessary to implement the comparison method.

2. Preliminary results We denote by | · | the usual Euclidean norm in Rd , and by · ∞ the norm in L∞ (Rd ). For any T > 0, let CB (Rd × [0, T ]) be the space of bounded continuous functions on Rd × [0, T ], endowed with the supremum norm. Let us write {St , t ≥ 0} for the semigroup of the spherically symmetric α-stable process in Rd , 1 Other relevant aspects of related branching models featuring this form of spatially-inhomogeneous branching have been investigated, e.g., in [5] and [6]; see also [3].

Stability of a Nonlinear Equation namely,

23

 St g(x) = E[g(x + Wt )] =

(α)

Rd

g(y)pt (y − x) dy,

where g : Rd → R is bounded and measurable, and {Wt , t ≥ 0} is the standard (α) d-dimensional symmetric stable process with transition densities pt (·), t > 0. When α = 2, the infinitesimal generator ∆α of {St , t ≥ 0} is the d-dimensional Laplacian. When 0 < α < 2, the generator is given by  f (x + y) − f (x) ∆α f (x) = γ−α,d P V dy f ∈ Cb2 (Rd ), d+α |y| d R where γα,d is given in (11) and P V stands for Principal Value, see [1]. Let ω(x) = (1 + |x|)m , x ∈ Rd , where 0 ≤ m < α, and d ∞ d L∞ ω (R ) = {f ∈ L (R ) : f ω ∞ < +∞}. d We put f ω ≡ f ω ∞ when f ∈ L∞ ω (R ). We also denote ! " d ∞ d L∞ ([0, T ], L∞ < +∞ , ω (R )) = F : [0, T ] → Lω (R ) : F L∞ [0,T ]

with F L∞ := ess sup0≤t≤T F(t) ω . [0,T ] Lemma 1. Let T > 0 and α ∈ (0, 2]. There exists a constant C1 (T ) > 0 such that d for all t ∈ [0, T ] and all f ∈ L∞ ω (R ), (St |f |) (x) ≤ (2m + C1 (T )) f ω ω −1 (x). (α)

(α)

Proof. Recall that td/α pt (t1/α x) = p1 (x) for all t > 0 and x ∈ Rd . Therefore  (α) St |f |(x) = pt (x − y)|f (y)| dy Rd  (α) ≤ f ω pt (x − y)ω −1 (y) dy d R (α) = f ω pt (t1/α η)(1 + |x − t1/α η|)−m td/α dη Rd #  $ 

(α) 1/α −m = f ω + p1 (η)(1 + |x − t η|) dη R1 R2 %

  |x| &   R1 := η ∈ Rd : x − t1/α η ≤ ; R2 := Rd \ R1 2 

−m  |x| (α) (α) ≤ f ω p1 (η) dη + p1 (η) dη 1 + 2 R1 Rd 

(α) ≤ f ω p1 (η) dη + 2m ω −1 (x) . (3) R1

24

S. Chakraborty, E.T. Kolkovska and J.A. L´ opez-Mimbela

If |x| > 0, using that |x| − t1/α |η| ≤ |x − t1/α η| ≤ |x|/2 on R1 , we obtain    (α) (α) (α) I1 := p1 (η) dη ≤ p1 (η) dη ≤ p1 (η) dη, |η|≥

R1

|x| −1/α 2 t

|η|≥

|x| −1/α 2 T

hence, % & % & |x| −1/α |x|m 2m T m/α E|W1 |m I1 ≤ Pr |W1 | ≥ T = Pr |W1 |m ≥ m T −m/α ≤ , 2 2 |x|m where E|W1 |m < ∞ due to α > m. Since I1 ≤ 22m T m/α E|W1 |m ω −1 (x) for |x| ≥ 1, and I1 ≤ 1 ≤ 2m ω −1 (x) if |x| < 1, it follows that I1 ≤ 4m T m/α E|W1 |m ω −1 (x)1{|x|≥1} + 2m ω −1 (x)1{|x|<1} ≤ C1 (T )w−1 (x), with C1 (T ) = max{2m , 4m T m/αE|W1 |m }.

(4) 

Plugging the above inequality into (3) proves the lemma.

Solutions u = u(x, t) to equation (1) are going to be understood in the mild sense, meaning that, for certain T > 0, u ∈ CB (Rd × [0, T  ]) for any 0 < T  < T , and   t ∞  u(x, t) = St f (x) + St−s a(·) pk uk (·, s) + (p0 + p1 u(·, s))φ(·) (x) ds 0

k=2

for each (x, t) ∈ Rd × [0, T ). We define a mild upper (lower) solution to (1) by replacing “=” above by “≥” (“≤”). In the proofs of Propositions 2 and 3 below we follow a method of [12]. Proposition 2. Let d > α. Consider the inhomogeneous equation (1), where 1. 0 ≤ a(x) ≤ c1 (1+|x|)m , x ∈ Rd , for some constants c1 > 0 and m ∈ (−∞, α), and 2. 0 ≤ φ(x) ≤ c2 (1 + |x|)−q , x ∈ Rd , for certain constants c2 > 0 and q ∈ (α, d]. d If f ∈ L∞ ω (R ) is continuous, nonnegative and satisfies ∞  k=2 ∞ 

' (k pk f ω (3 + 2m+2 ) < ∞,

(5)

' (k−1 kpk f ω (3 + 2m+2 ) < ∞,

(6)

k=2

then there exists a constant Tf > 0 such that Equation (1) has a unique mild solution on Rd × [0, Tf ). Moreover, if Tf < ∞, then limT →T − u(·, T ) ∞ = +∞. f

Stability of a Nonlinear Equation

25

Proof. We deal only with the case of m ≥ 0; if m < 0 the function a is bounded by a constant and the result can be obtained in the same way as for m = 0. Let us define the operator  ∞  t  F (u) = St f + St−s a pk uk + (p0 + p1 u)ϕ ds. (7) 0

We denote

k=2

% & ∞ ∞ d = u ∈ L ([0, T ], Lω (R )) : sup u(·, t) − f ω ≤ δ ,

BδT (f )

0≤t≤T

where δ > 0 and T > 0 are going to be specified afterwards. For any u ∈ BδT (f ), |F (u) − f |(x, t) ≤ |f (x)| + St |f |(x) #  t ∞  (α) + pt−s (x − y) c1 (1 + |y|)m pk |u(y, s)|k Rd

0

k=2

$ + c2 (p0 + p1 |u(y, s)|)(1 + |y|)−m dy ds

≤ f ω ω(x)−1 (1 + C1 (T ) + 2m ) + I, where we have used Lemma 1, and    t

(α)

I := 0

Rd

0

Rd

≤ 0

(α) pt−s (x

Rd

pk |u(y, s)|k

( +c2 (p0 + p1 |u(y, s)|)(1 + |y|)−m dy ds

− y) c1 (1 + |y|)m

∞ 

pk u(·, s) kω ω −k (y)

k=2



 t

∞  k=2



 t ≤

pt−s (x − y) c1 (1 + |y|)m

(α) pt−s (x

( +c2 (p0 + p1 u(·, s) ω ω −1 (y))(1 + |y|)−m dy ds

− y) c1

∞ 

pk u(·, s) kω + c2 (p0 + p1 u(·, s) ω ) ω −1 (y) dy ds.

k=2

Noting that u(·, t) ω ≤ f ω + δ for 0 ≤ t ≤ T , from Lemma 1 we get  t (α) pt−s (x − y)ω −1 (y) dy ds ≤ C2 (2m + C1 (T ))T ω −1(x), I ≤ C2 0

where C2 = c1

∞

Rd

k=2

pk ( f ω + δ)k + c2 (p0 + p1 ( f ω + δ)). Therefore,

F (u) − f ω ≤ f ω (1 + 2m + C1 (T )) + C2 T (2m + C1 (T )) ,

(8)

26

S. Chakraborty, E.T. Kolkovska and J.A. L´ opez-Mimbela

where C1 (T ) is given by (4). In a similar way, for any u, v ∈ BδT (f ), |F (u) − F (v)|(x, t)   t ∞  (α) ≤ pt−s (x − y) c1 (1 + |y|)m pk |uk − vk |(y, s) 0

Rd



 t ≤ 0

Rd

0

(α) pt−s (x

− y) c1 (1 + |y|)m

Rd

∞ 

  kpk |u|k−1 + |v|k−1 (y, s)

k=2



 t ≤

k=2

( +c2 p1 |u − v|(y, s)(1 + |y|)−m dy ds

(α) pt−s (x

× u − v ω ω

( +c2 p1 ω −1 (y) |u − v|(y, s) dy ds

− y) c1

∞ 

  kpk u(·, s) k−1 + v(·, s) k−1 + c2 p1 ω ω

k=2 −1

(y) dy ds.

Since max{ u(·, t) ω , v(·, t) ω } ≤ f ω + δ for 0 ≤ t ≤ T , using again Lemma 1 gives |F (u) − F (v)|(x, z) ≤ C3 (2m + C1 (T )) T ω −1 (x) u − v ω , k−1 where C3 = 2c1 ∞ + c2 p1 . Hence, k=2 kpk ( f ω + δ) F (u) − F (v) ω ≤ C3 (2m + C1 (T ))T u − v ω . Let us now choose δ = 2(2m+1 + 1) f ω and ! ) * " −α/m T < min 2−α (E|W1 |m ) , f ω 2−(m+1) + 1 C2−1 , 2−(m+1) C3−1 := Tf . Then C2 < ∞ and C3 < ∞ due to assumptions (5) and (6), whereas C1 (T ) = 2m . It follows that F (u) ∈ BδT (f ) and that F is a contraction on BδT (f ). Therefore F possesses a unique fixed point in BδT (f ), which is the unique mild solution to Equation (1) on Rd × [0, Tf ). The fact that limT →T − u(·, T ) ∞ = +∞ when f Tf < ∞, follows from a ladder argument.  Remark. Notice that, due to k pk = 1, assumptions (5) and (6) are trivially fulfilled when f ω < (3 + 2m+2 )−1 . Proposition 3. Under the assumptions in Proposition 2, suppose that u and u are, respectively, upper and lower mild solutions of (1) on Rd × [0, T ). Then the unique mild solution of (1) on Rd × [0, Tf ) satisfies the inequalities u ≤ u ≤ u on Rd × [0, T ) for each T ≤ Tf . Proof. Let us take T+ = min{T, Tf }, where Tf is given in Proposition 2. From the proof of Proposition 2 we know that there exists a T1 ≤ T+ such that the T1 T1 operator F is a contraction !on BδT1 (f ) and maps " Bδ (f ) into Bδ (f ), where δ =  m+2  f ω 2 + 2 . Let B = v ∈ BδT1 (f ) : v ≤ u . Since F is increasing, F (v) ≤ F (u) for any v ∈ BδT1 (f ). In addition, F (u) ≤ u because u is a supersolution

Stability of a Nonlinear Equation

27

to (1). Hence F : B → B, and therefore, F has a unique fixed point in B. Since u is the unique fixed point of F in BδT1 (f ), it follows that u ∈ B and u ≤ u on Rd × [0, T1 ]. By a ladder argument, one can prove that u ≤ u on Rd × [T1 , T2 ] for some T+ ≥ T2 > T1 , and so on. The assertion regarding the inequality u ≤ u can be proved similarly. 

3. Main theorem Theorem 4. Assume that d > α, and let pk , k = 0, 1, . . . , be nonnegative numbers such that ∞ k=0 pk = 1 and 0 < p0 , p1 < 1. Consider the inhomogeneous equation (1), where 1. 0 ≤ a(x) ≤ c1 (1+|x|)m , x ∈ Rd , for some constants c1 > 0 and m ∈ (−∞, α), 2. 0 ≤ φ(x) ≤ c2 (1 + |x|)−q , x ∈ Rd , for certain constants c2 > 0 and q ∈ (α, d], 3. q − α > (α + m)+ .   Let δ ∈ (0, q − α),  > q − α − (α + m)+ and µ ∈ 0, (2c1 γα,d C 2 (δ) + C(δ))−1 , where γα,d > 0 is given) by (11) and C(δ) > 0 ) is given in Lemma*5 below. * −1

−1

Assume also that p0 ∈ 0, µ (4γα,d c2 ) , p1 ∈ 0, (4γα,d C(δ)c2 ) and let  , - m+2 −1 c ∈ 0, min µC∗ , (3 + 2 ) , where C∗ is given in Lemma 5 below. Then, for all continuous initial values f satisfying 0 ≤ f (x) ≤ c (1 + |x|)

−(α+m)+ −

,

x ∈ Rd ,

the mild solution to (1) is global. Our method of proof is an adaptation of a technique used in [7] and [10]. We use the following lemma (which is proved in the final section of the present paper). Lemma 5. There exists a C∗ > 0 and for all δ > 0, a C(δ) > 0 such that,  (1 + |y|)−q C∗ (1 + |x|)α−q ≤ dy ≤ C(δ)(1 + |x|)α−q+δ , q ∈ (α, d]. d−α Rd |x − y| For a given constant µ > 0 we define  (1 + |y|)−q v(x) = µ dy. d−α Rd |x − y|

(10)

Recall [1] that the Green’s function G of the linear operator −∆α satisfies γα,d G(x, y) = where γα,d = Γ((d − α)/2)/(2α π d/2 |Γ(α/2)|). |x − y|d−α Lemma 6. Let v be given by (10). There exists a δ > 0 such that ∞  ∆α v + a(x) pk vk + (p0 + p1 v)φ(x)  ≤

k=2 ∞

 −µ + c1 pk C(δ)k µk + (p0 + p1 C(δ)µ)c2 γα,d k=2

(9)

 (1 + |x|)−q .

(11)

(12)

28

S. Chakraborty, E.T. Kolkovska and J.A. L´ opez-Mimbela

Proof. This follows from Inequality (9). Indeed, from (11) we get that µ ∆α v(x) = − (1 + |x|)−q , γα,d (see, e.g., [4], Chapter 2), and by assumption p0 φ(x) ≤ p0 c2 (1 + |x|)−q . Using Lemma 5 we obtain that, for any δ > 0, a(x)

∞ 

pk v k ≤ c1

k=2

∞ 

pk C(δ)k µk (1 + |x|)m+k(α−q+δ) .

k=2

Let δ ∈ (0, q − α). Since the inequality m + k(α − q + δ) ≤ m + 2(α − q + δ) is valid for all k ≥ 2, to prove (12) it suffices to show that m + 2(α − q + δ) ≤ −q.

(13)

If m + α < 0, then we choose δ > 0 so small that m + α + δ < 0, and thus m + 2(α − q + δ) ≤ −q. In case of m + α ≥ 0, since q − α > (m + α)+ = m + α, we can choose δ > 0 so small that m + α + (α − q) + 2δ is negative. This proves (13). In addition, because of α − q + δ < 0, p1 vφ(x) ≤ p1 c2 C(δ)µ(1 + |x|)α−q+δ−q ≤ p1 c2 C(δ)µ(1 + |x|)−q . Thus (12) is satisfied.  Proof of Theorem 4. Since by assumption δ ∈ (0, q−α), Lemma 5 implies c2 v(x) ≤ c2 µC(δ) for all x ∈ Rd . Therefore p1 c2 C(δ)µ < µ/(4γα,d ) because ) * −1 p1 ∈ 0, (4γα,d C(δ)c2 ) . Hence, ∞

 −µ −µ µ2 C(δ)2 2µ + c1 pk C(δ)k µk + (p0 + p1 C(δ)µ)c2 < + c1 + <0 γα,d γα,d 1 − µC(δ) 4γα,d k=2

and ∆α v(x) + a(x)

∞ 

pk v k (x) + (p0 + p1 v(x))φ(x) < 0

(14)

k=2

due to Lemma 6. Using again Lemma 5 we deduce that +

v(x) ≥ µC∗ (1 + |x|)α−q ≥ c(1 + |x|)−(α+m)

−

≥ f (x),

x ∈ Rd .

(15)

We now put g(s, x) = St−s v(x), 0 ≤ s ≤ t. Using that {St , t ≥ 0} is a positive semigroup and (14), we get that ∂ g(s) = −∆α St−s v ∂s



≥ −∆α St−s v + St−s ∆α v + a  = St−s

a

∞ 

 pk v + (p0 + p1 v)φ

k=2 ∞  k=2



k

k

pk v + (p0 + p1 v)φ ,

Stability of a Nonlinear Equation

29

which after integration from 0 to t renders  ∞   t  v(x) ≥ St v(x) + St−s a pk v k + (p0 + p1 v)φ ds, 0

k=2

and due to (15),  v(x) ≥ St f (x) +



t

St−s 0

a

∞ 

 k

pk v + (p0 + p1 v)φ

ds = F (v),

k=2

where F is given by (7). Thus, v is a mild upper solution to (1), and it follows from propositions 2 and 3 that u(x, t) ≤ v(x) on Rd × [0, Tf ) with Tf = +∞, i.e., the solution to Equation (1) is global. 

4. Proof of Lemma 5 We prove first the RHS inequality in Lemma 5. For fixed x ∈ Rd let split Rd into three regions D1 = {y ∈ Rd : |y − x| ≥ 2|x|}, D2 = {y ∈ Rd : |y − x| ≤ 2|x|, |y| ≥ |x|/2}, D3 = {y ∈ Rd : |y − x| ≤ 2|x|, |y| ≤ |x|/2}. In the following, b1 , b2 , . . . , denote positive constants. For y ∈ D1 we have |y| ≥ |y − x| − |x| ≥ |y − x| − |y − x|/2 = |y − x|/2. In the integral in (9) we pass to polar coordinates (y1 −x1 , y2 −x2 , . . . , yd −xd ) → (r cos θ1 , r sin θ1 cos θ2 , . . . , r sin θ1 sin θ2 . . . sin θd ), the Jacobian of the transformation being equal to rd−1 sind−2 θ1 · · · sin θd−2 . After the change of variables we see that the integral over D1 is bounded from above by  ∞ b1 r−(d−α) (1 + r)−q rd−1 dr = b2 (1 + |x|)α−q . (16) 2|x|

If y ∈ D2 , we bound (1 + |y|)−q from above by b3 (1 + |x|)−q and pass to polar coordinates as before. The corresponding integral is bounded from above by  2|x| −q b4 (1 + |x|) r−(d−α) rd−1 dr = b5 (1 + |x|)−q |x|α ≤ b5 (1 + |x|)α−q . (17) 0

When y ∈ D3 we have 2|x| ≥ |y − x| ≥ |x| − |y| ≥ |x|/2 ≥ |y|, and therefore |y − x|−(d−α) ≤ b6 |x|−(d−α) . Expressing y in polar coordinates we see that the

30

S. Chakraborty, E.T. Kolkovska and J.A. L´ opez-Mimbela

integral in (9) over D3 is bounded from above by  |x|/2 −(d−α) b7 |x| (1 + r)−q rd−1 dr 0   |x|/2    b7 |x|−(d−α) (1 + r)d−1−q dr ≤ b8 |x|−(d−α) (1 + |x|)d−q , d > q, 0 ≤ (18)    −(d−α) log(1 + |x|), d = q. b9 |x| If in addition |x| ≤ 1,  |x|/2  −(d−α) −q d−1 −(d−α) |x| (1 + r) r dr ≤ |x| 0

|x|/2

rd−1 dr = b10 |x|α ≤ b10 . (19)

0

Let κ = sup|x|≥1 ( 1+|x| )d−α . When x ∈ D3 and |x| ≥ 1 we get from (18) that for |x| d > q,  |x|/2 b7 |x|−(d−α) (1 + r)−q rd−1 dr ≤ b8 κ(1 + |x|)α−q+δ , (20) 0

whereas for d = q,  b7 |x|−(d−α)

|x|/2

(1 + r)−q rd−1 dr ≤ b9 κ(1 + |x|)−(d−α) log(1 + |x|)

0

< b11 (1 + |x|)

α−q+δ

(21)

.

Putting together (16), (17) and (19)–(21) gives the RHS inequality in (9). To prove the LHS inequality in (9) we notice that   (1 + |y|)−q dy −q ≥ b12 (1 + |x|) |x − y|α−d dy, (22) d−α |x − y| d + R D + = {y ∈ Rd : |x − y| < |x|/2}. This is because |x − y| < |x|/2 implies where D |y| < 3|x|/2, hence (1 + |x|)q (1 + |x|)q ≥ ≥ b13 , q (1 + |y|) (1 + 3|x|/2)q yielding (1 + |y|)−q ≥ b13 (1 + |x|)−q . We use once again polar coordinates to show that the RHS of (22) equals  b12 (1 + |x|)−q |x − y|α−d dy {y: 0<|x−y|<|x|/2}

= b12 (1 + |x|)−q



|x|/2 

0 −q

= b13 |x|α (1 + |x|) We have proved that 

Rd

 ···

θ1

rα−1 sind−2 θ1 · · · sin θd−2 dθd−2 · · · dθ1 dr θd−1

.

(1 + |y|)−q dy ≥ b13 (1 + |x|)−q · |x|α . |x − y|d−α

Stability of a Nonlinear Equation

31

From here the LHS inequality in (9) follows easily if |x| ≥ 1. In the case |x| < 1
one has to verify that Rd (1 + |y|)−q /|x − y|d−α dy is bounded away from zero. To prove this, notice that, as |x| < 1,   (1 + |y|)−q dy (1 + |y|)−q dy ≥ d−α d−α |x − y| Rd Rd (|x| + |y|)  (1 + |y|)−q ≥ dy d−α Rd (1 + |y|)  dy = , d−α+q Rd (1 + |y|) which is a non-zero constant, and hence is no smaller than C∗ (1 + |x|)α−q for a suitable constant C∗ > 0. This finishes the proof of Lemma 5. Acknowledgement The authors are grateful to two anonymous referees for their useful comments and suggestions, and specially for providing a substantially shorter proof of Lemma 5.

References [1] K. Bogdan, T. Byczkowski, Potential theory of Schr¨ odinger operator based on fractional Laplacian. Probab. Math. Statist. 20 (2000), no. 2, 293–335. [2] Chakraborty, S. and L´ opez-Mimbela, J.A. Nonexplossion of a class of semilinear equations via branching particle representations. Adv. Appl. Prob. 40 (2008), 250– 272. [3] Erickson, K.B. Rate of expansion of an inhomogeneous branching process of Brownian particles. Z. Wahrsch. verw. Gebiete, 66 (1984), 129–140. [4] Folland, G.B. Introduction to partial differential equations 2nd Edition. Princeton University Press. Princeton, New Jersey 1995. [5] Lalley, S. and Sellke, T. Limit theorems for the frontier of a one-dimensional branching diffusion. Ann. Probab. 20 (1992), 1310–1340. [6] Lalley, S. and Sellke, T. Traveling waves in inhomogeneous branching Brownian motions. I and II. Ann. Probab. 16 (1988), 1051–1062, and 17 (1989), 116–127. [7] Lee T.Y., Some limit theorems for super-Brownian motion and semilinear differential equations, Annals of Probab. 21 (1993), 979–995. [8] L´ opez-Mimbela, J.A. (1996). A probabilistic approach to existence of global solutions of a system of nonlinear differential equations, Aportaciones Matem´ aticas Serie Notas de Investigaci´ on 12, 147–155. [9] L´ opez-Mimbela, J.A. and Wakolbinger, A. (1998). Length of Galton-Watson trees and blow up of semilinear systems. Journal of Applied Probability 35, 802–811. [10] Pinsky, Ross G. Finite time blow-up for the inhomogeneous equation ut = ∆u + a(x)up + λφ in Rd . Proc. Amer. Math. Soc. 127 (1999), no. 11, 3319–3327. [11] Nagasawa, M. and Sirao, T. (1969). Probabilistic treatment of the blowing up of solutions for a nonlinear integral equation. Trans. Amer. Math. Soc., 139, 301–310.

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[12] Wang, X. On the cauchy problem for reaction-difusion equations. Trans. Amer. Math. Soc., 337 (1993), 549–590. [13] Zhang, Qi S. A new critical phenomenon for semilinear parabolic problems. J. Math. Anal. Appl. 219 (1998), no. 1, 125–139. S. Chakraborty Department of Mathematics University of Texas Pan American Edinburg, Texas, USA e-mail: [email protected] E.T. Kolkovska and J.A. L´ opez-Mimbela Department of Probability and Statistics Centro de Investigaci´ on en Matem´ aticas Guanajuato, Mexico e-mail: [email protected] [email protected]

Progress in Probability, Vol. 65, 33–42 c 2011 Springer Basel AG


Backward Stochastic Difference Equations with Finite States Samuel N. Cohen and Robert J. Elliott Abstract. We define Backward Stochastic Difference Equations on spaces related to discrete time, finite state processes. Solutions exist and are unique under weaker assumptions than are needed in the continuous time setting. A comparison theorem for these solutions is also given. Applications to the theory of nonlinear expectations are explored, including a representation result. Mathematics Subject Classification (2000). 60H10, 60G42, 65C30. Keywords. BSDE, comparison theorem, nonlinear expectation, dynamic risk measures.

1. Introduction A Backward Stochastic Differential Equation (BSDE) defines processes (Y, Z) through an equation of the form   Yt − F (ω, u, Yu− , Zu )du + Zu dMu = Q. ]t,T ]

]t,T ]

Here Q is a square-integrable terminal condition, F a progressively measurable ‘driver’ function, and M an N -dimensional Brownian Motion, all defined on an appropriate filtered probability space. The ‘solutions’ (Y, Z) are required to be adapted to the forward filtration, and Z is required to be predictable. In this paper, we shall describe results for an analogous situation in discrete time. A fuller treatment with proofs can be found in [5]. Such processes have been considered previously in [12] and other works, particularly as numerical approximations to continuous time processes. In contrast to previous work, we consider discrete time BSDEs as entities in their own right, and do not significantly address their use as approximations and the related numerical methods. Consequently, we obtain considerably more general conditions under which solutions exist, and also establish fundamental results, including, for example, a comparison theorem. This work was done while Samuel Cohen was at the University of Adelaide. Robert Elliott wishes to thank the Australian Research Council for support.

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We begin by defining a backward stochastic difference equation and giving conditions for existence and uniqueness. We then state a comparison theorem. We apply these results to obtain a theory of nonlinear expectations. Every nonlinear expectation obeying certain assumptions in this context is the solution to a discrete BSDE.

2. Dynamics Consider an underlying discrete-time, finite state process X. Without loss of generality, this can be assumed to take values in the standard basis vectors of RN , where N is the number of states of the process. That is, for each t ∈ {0, 1, . . . }, Xt ∈ {e1 , e2 , . . . , eN }, where ei = (0, 0, . . . , 0, 1, 0, . . . , 0)∗ are the basis vectors of appropriate dimension (in this case, in RN ), and [·]∗ denotes vector transposition. Let (Ω, F, {Ft }0≤t≤T , P) be a filtered probability space, where Ft is the completion of the sigma algebra generated by the process X up to time t. Unless otherwise noted, we index all quantities by the first time t such that they are Ft -measurable. X can then be represented as: Xt = E[Xt |Ft−1 ] + Mt ∈ RN . By definition, M is the martingale difference process Mt = Xt − E[Xt |Ft−1 ]. The central process considered in this paper is the solution (Y, Z) of a BSDE based on M , that is an equation of the form   Yt − F (ω, u, Yu , Zu ) + Zu Mu+1 = Q, (1) t≤u
t≤u
where T is a deterministic terminal time, F an adapted map F : Ω × {0, . . . , T } × RK × RK×N → RK , and Q an RK -valued, FT -measurable terminal condition. We consider only solutions (Y, Z) which are adapted to the filtration {Ft }, that is, (Yt , Zt ) ∈ RK × RK×N is Ft -measurable for all t. For simplicity, we shall also assume Yt , Zt ∈ L∞ (Ft ) for all t, Q ∈ L∞ (FT ) and F (ω, t, Yt , Zt ) ∈ L∞ (Ft ) for all t and all Yt , Zt ∈ L∞ (Ft ). Note, however, that as there are only finitely many possible paths for X on {0, . . . , t}, and all our quantities are finite dimensional, it is clear that L1 (Ft ) = L2 (Ft ) = L∞ (Ft ). Definition 2.1. For any integer K, we shall denote by · M the seminorm on the space of adapted processes Z in RK×N , given by #  $ ∗ Z 2M : = E Tr Zu · E[Mu+1 Mu+1 |Fu ] · Zu∗ =



0≤u
0≤u
Tr E[(Zu Mu+1 )(Zu Mu+1 )∗ ].

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Lemma 2.1. The following statements are equivalent: (i) Z 1 − Z 2 2M = 0. ∗ (ii) E Tr[(Zu1 − Zu2 )Mu+1 Mu+1 (Zu1 − Zu2 )∗ ] = 0 for all u ∈ {0, . . . , T − 1}. 1 2 (iii) Zu Mu+1 = Zu Mu+1 P-a.s. for all u ∈ {0, . . . , T − 1}. (iv) For all t ∈ {1, . . . , T }, P-a.s.,   Zu1 Mu+1 = Zu2 Mu+1 0≤u
0≤u
In this case we shall write Z ∼M Z . 1

2



Proof. See [5].

1 2 Definition 2.2. For two Ft−1 -measurable random variables Zt−1 and Zt−1 , we shall 1 2 1 2 write Zt−1 ∼Mt Zt−1 if Zt−1 Mt = Zt−1 Mt , P-a.s. 1 2 By Lemma 2.1, Z 1 ∼M Z 2 if and only if Zt−1 ∼Mt Zt−1 for all t ∈ {1, . . . , T }. It is straightforward to show that ∼M and ∼Mt are both equivalence relations.

3. Existence and uniqueness We now state under what conditions solutions of the BSDE (1) exist and are unique. Recall the following Martingale Representation Theorem (from [8]). Theorem 3.1. For any {Ft }-adapted, RK -valued martingale L, there exists an adapted RK×N process Z such that  Lt+1 = L0 + Zu Mu+1 . 0≤u
This Z process is unique up to equivalence ∼M . Corollary 3.1.1. For any RK -valued, Ft -measurable random variable W with E[W |Ft−1 ] = 0, there exists an Ft−1 -measurable random variable Zt−1 such that, P-a.s., W = Zt−1 Mt. This variable is unique up to equivalence ∼Mt . Theorem 3.2. Suppose F is such that the following two assumptions hold: (i) For any Y , if Z 1 ∼M Z 2 , then F (ω, t, Yt , Zt1 ) = F (ω, t, Yt , Zt2 ) P-a.s. for all t. (ii) For any Z, for all t, the map Yt → Yt − F (ω, t, Yt , Zu ) is P-a.s. a bijection from RK → RK , up to equality P-a.s. Then for any terminal condition Q essentially bounded, FT -measurable, and with values in RK , the BSDE (1) has an adapted solution (Y, Z). Moreover, this solution is unique up to indistinguishability for Y and equivalence ∼M for Z.

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Proof. Clearly we can find an (adapted) solution YT = Q at time T . We shall construct the solution for all t using backward induction. Suppose we have a solution at time t + 1. In this case, equation (1) can be simplified to Yt − F (ω, t, Yt , Zt ) + Zt Mt+1 = Yt+1 . (2) Taking a conditional expectation gives Yt − F (ω, t, Yt , Zt ) = E[Yt+1 |Ft ],

(3)

and so the martingale difference term must be Yt+1 − E[Yt+1 |Ft ]. From Corollary 3.1.1, there is a Zt such that Zt Mt+1 = Yt+1 − E[Yt+1 |Ft ]. This Zt is unique up to equivalence ∼Mt+1 . Using this Zt , consider the equation Yt − F (ω, t, Yt , Zt ) = E[Yt+1 |Ft ]. This is now uniquely determined as an equation in Yt . By assumption (ii), this equation has a unique solution Yt , up to equality P-a.s. The pair (Yt , Zt ) will solve the desired BSDE (i) at time t. Backward induction starting with T = t gives the desired result for all t ∈ {0, 1, . . . , T }. The solution Yt is unique up to equality P-a.s. for all t, and hence Y is unique up to indistinguishability (as we are working in discrete time). The solution Zt is unique up to equivalence ∼Mt+1 for all t, and hence Z is unique up to equivalence ∼M .  Remark 3.1. Unlike in the continuous time case, we have not assumed any continuity conditions on F , Lipschitz or otherwise. In particular, the assumptions on F as a function of Z are very weak, essentially demanding only that F does not distinguish “equivalent” strategies Z. We can then prove Corollary 3.2.1. In general, both assumptions of Theorem 3.2 are necessary for a unique solution to exist. That is, if either assumption fails for some Y or Z, then there will exist terminal conditions Q such that either no solutions or multiple solutions of the BSDE (1) exist (on some set t ∈ {s, . . . , T }). 

Proof. See [5]. The following lemma is of interest.

Lemma 3.1. Let (Y, Z) be the solution to a discrete BSDE satisfying Theorem 3.2. Then, for each t, there exists a bijection, (up to equality P-a.s. and equivalence ∼Mt+1 ), between the pair (Yt , Zt ) and the value Yt+1 . Furthermore, this implies that there exists a bijection, (up to equality P-a.s. and equivalence ∼M ), between (Y0 , {Zs}s
(4)

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For any Yt+1 ∈ L∞ (Ft+1 ), we wish to show there is a unique pair (Yt , Zt ) such that (4) is true. It follows from (4), and the fact F (ω, t, Yt , Zt ) is Ft-measurable, that Zt Mt+1 = Yt+1 − E[Yt+1 |Ft ]. By Corollary 3.1.1, there is a unique such Zt , up to equivalence ∼Mt+1 . Finally, it follows from (4) that Yt − F (ω, t, Yt , Zt ) = E[Yt+1 |Ft ]. We have assumed the left-hand side of this equation is bijective as a function of Yt and therefore there is a unique (Yt , Zt ) which satisfies (4), for a given Yt+1 . Therefore, there is a bijective relationship between the pair (Yt , Zt) and Yt+1 . As there exists a bijection between the pair (Yt , Zt) and Yt+1 for all t, the remaining statements clearly follow by induction. 

4. A comparison theorem A basic result in the study of BSDEs is the ‘Comparison Theorem’, first obtained in [13]. As is shown in [4], for Markov chain driven BSDEs, there are conditions under which a comparison theorem remains true when the underlying process is not a Brownian motion. We now state a comparison theorem for discrete time BSDEs. We make the following definition: Definition 4.1. Let Jt denote the Ft -measurable set of (indices of) possible values of Xt+1 , given Ft . That is, Jt := {i : P(Xt+1 = ei |Ft ) > 0} = {i : e∗i E[Xt+1 |Ft ] > 0}. Note, if all paths of X have positive probability, then Jt ≡ {1, 2, . . . , N }. In the following, an inequality on a vector quantity is to hold componentwise. Theorem 4.1 (Comparison Theorem). Consider two discrete time BSDEs as in (1), corresponding to coefficients F 1 , F 2 and terminal values Q1 , Q2 . Suppose the conditions of Theorem 3.2 are satisfied for both equations, let (Y 1 , Z 1 ) and (Y 2 , Z 2 ) be the associated solutions. Suppose the following conditions hold, all vector inequalities being taken componentwise: (i) Q1 ≥ Q2 P-a.s. (ii) P-a.s., for all times t, F 1 (ω, t, Yt2 , Zt2 ) ≥ F 2 (ω, t, Yt2 , Zt2 ). (iii) P-a.s., for all t, for all i, the ith component of F 1 , given by e∗i F 1 , satisfies e∗i F 1 (ω, t, Yt2 , Zt1 ) − e∗i F 1 (ω, t, Yt2 , Zt2 ) ≥ min{e∗i (Zt1 − Zt2 )(ej − E[Xt+1 |Ft ])} j∈Jt

(iv) P-a.s., for all t, if Yt1 − F 1 (ω, t, Yt1 , Zt1 ) ≥ Yt2 − F 1 (ω, t, Yt2 , Zt1 ) then Yt1 ≥ Yt2 . It is then true that Y 1 ≥ Y 2 P-a.s.

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Proof. See [5].

Corollary 4.1.1. Suppose Theorem 4.1 holds and the inequality in Assumption (iii) is strict, that is, for all i, e∗i F 1 (ω, t, Yt2 , Zt1 ) − e∗i F 1 (ω, t, Yt2 , Zt2 ) > min{e∗i (Zt1 − Zt2 )(ej − E[Xt+1 |Ft ])} j∈Jt

e∗i Zt1

e∗i Zt2 .

unless ∼Mt+1 Then this comparison is strict, that is, if on some A ∈ Ft we have Yt1 = Yt2 P-a.s. on A, then Q1 = Q2 P-a.s. on A, and for all s ∈ {t, . . . , T }, P-a.s. on A, F 1 (ω, s, Ys2 , Zs2 ) = F 2 (ω, s, Ys2 , Zs2 ), Zs1 ∼Ms+1 Zs2 and Ys1 = Ys2 . 

Proof. See [5].

Corollary 4.1.2. Theorem 4.1 remains true if we replace Assumptions (iii) and (iv) by (iii ) P-a.s., for all t, for all i, e∗i F 1 satisfies e∗i F 1 (ω, t, Yt1 , Zt1 ) − e∗i F 1 (ω, t, Yt1 , Zt2 ) ≥ min{e∗i (Zt1 − Zt2 )(ej − E[Xt+1 |Ft ])}. j∈Jt



(iv ) P-a.s., for all t, if Yt1 − F 1 (ω, t, Yt1 , Zt2 ) ≥ Yt2 − F 1 (ω, t, Yt2 , Zt2 ) then Yt1 ≥ Yt2 . or with a mixture of (iii–iv) and (iii –iv ) for different times t, provided one of these pairs holds for each t. Similarly for the strict comparison of Corollary 4.1.1. Remark 4.1. Assumption (iii) of Theorem 4.1 is closely related to the assumption used in [4, Thm 4.2] for continuous time BSDEs driven by Markov Chains. It essentially ensures that there will always be a value of Mt+1 , that occurs with positive probability, such that −F 1 (ω, t, Yt2 , Zt1 ) + F 1 (ω, t, Yt2 , Zt2 ) + (Zt1 − Zt2 )Mt+1 is negative.

5. Applications to nonlinear expectations We now focus our attention on the theory of risk measures, as in [9], and nonlinear expectations, as in [14]. The connection between these is present in [1], and more generally in [15]. Many of the results in this section parallel those in [6], which discusses continuous time processes related to Brownian motion. As in [4], we follow [14] by giving the following definition. Definition 5.1. Let {Qt } be a family of subsets {Qt ⊂ L2 (FT )}. A system of operators E(·|Ft ) : L2 (FT ) → L2 (Ft ), 0 ≤ t ≤ T

Backward Stochastic Difference Equations with Finite States

39

is called an Ft -consistent nonlinear expectation for {Qt } if it satisfies the following properties: 1. For Q, Q ∈ Qt , if Q ≥ Q P-a.s. componentwise, then E(Q|Ft ) ≥ E(Q |Ft ) P-a.s. componentwise, with, for each i, e∗i E(Q|Ft ) = e∗i E(Q |Ft ) only if e∗i Q = e∗i Q P-a.s. 2. E(Q|Ft ) = Q P-a.s. for any Ft -measurable Q. 3. E(E(Q|Ft )|Fs ) = E(Q|Fs ) P-a.s. for any s ≤ t 4. For any A ∈ Ft , IA E(Q|Ft ) = E(IA Q|Ft ) P-a.s. We know from [15] that the theory of dynamic risk measures is closely related to the theory of nonlinear expectations. In particular, we can define a dynamic risk measure as the function ρt (Q) := −E(Q|Ft ), where E(·|Ft ) is an Ft -consistent nonlinear expectation. Similar ideas are used in [11], where E is assumed to be concave, and is referred to as a ‘monetary concave utility functional’. In the same discrete time, finite state context as considered here, [10] considers this structure under the name of a ‘concave valuation operator’. As shown in [6], in continuous time with Brownian motions, the theory of Backward Stochastic Differential Equations is an appropriate context to study these operators in general. It is straightforward to prove various properties of risk measures from their definitions in terms of nonlinear expectations. The interested reader is referred to Section 9 of [4] – the proofs in this discrete time context follow without changes. An important contribution to the theory of risk measures, which follows from our work, is that the quantities may be vector-valued. Our proofs all work in a multi-dimensional context, which may be significant in applications related to multiobjective optimisation. Definition 5.2. A family of maps E(·|Ft ) : L∞ (FT ) → L∞ (Ft ) will be called dynamically monotone for {Qt } if, for all s ≤ t, (i) E is an Ft -consistent nonlinear expecation for {Qt }, (ii) Qs ⊆ Qt ({Qt } is nondecreasing in t) (iii) E(Q|Ft ) ∈ Qs for all Q ∈ Qs . Remark 5.1. In many applications, it may be that Qt = L∞ (FT ) for all t. In this case, a Ft -consistent nonlinear expectation for {Qt } will automatically be dynamically monotone for {Qt }. Definition 5.3. An Ft -consistent nonlinear expectation E(.|Ft ) is said to be (dynamically) translation invariant if for any Q ∈ L2 (FT ), any q ∈ L2 (Ft ), E(Q + q|Ft ) = E(Q|Ft ) + q.

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As in [4], we make the following definition, which ensures a comparison theorem will hold on [t, T ] under certain circumstances. Definition 5.4. Consider some nondecreasing family of sets {Qt ⊂ L2 (FT )} and some driver F , satisfying the assumptions of Theorem 3.2. Suppose that, for each t, for any Q1 , Q2 ∈ Qt , the corresponding BSDE solutions (Y 1 , Z 1 ), (Y 2 , Z 2 ) satisfy (iii) P-a.s., for all i, the ith component of F , given by e∗i F , satisfies e∗i F (ω, t, Yt2 , Zt1 ) − e∗i F (ω, t, Yt2 , Zt2 ) ≥ min{e∗i (Zt1 − Zt2 )(ej − E[Xt+1 |Ft ])}, j∈Jt

with equality only if e∗i Zt1 ∼Mt+1 e∗i Zt2 . (iv) P-a.s., if Yt1 − F (ω, t, Yt1 , Zt1 ) ≥ Yt2 − F (ω, t, Yt2 , Zt1 ) componentwise, then Yt1 ≥ Yt2 componentwise, (cf. assumptions (iii) and (iv) of Theorem 4.1 and the assumption of Corollary 4.1.1). Then we shall call F a balanced driver on {Qt }. Remark 5.2. Note that as {Qt } is nondecreasing, this implies that, for any Q1 , Q2 ∈ Qt , for all i, the corresponding BSDE solutions (Y 1 , Z 1 ), (Y 2 , Z 2 ) satisfy e∗i F (ω, s, Ys2 , Zs1 ) − e∗i F (ω, s, Ys2 , Zs2 ) ≥ min{e∗i (Zs1 − Zs2 )(ej − E[Xs+1 |Fs ])}, j∈Js

with equality only if e∗i Zs1 ∼Ms+1 e∗i Zs2 , for all s > t. Similarly for Assumption (iv). Remark 5.3. As in Corollary 4.1.2, these assumptions can be generalised slightly by changing which of Y 1 and Y 2 , and which of Z 1 and Z 2 , appears in each place. In this situation of a discrete time space generated by a finite state system, we establish the following theorem, which directly relates BSDEs, (and their drivers), to nonlinear expectations. This theorem applies in both the scalar and vector cases. Theorem 5.1. For some family of operators E(·|Ft ), let {Qt ⊂ L2 (FT )}, be such that if Q ∈ Qt then Q+q ∈ Qt for all q ∈ L∞ (Ft ), and E is dynamically monotone for {Qt}. Then the following two statements are equivalent. 1. E(·|Ft ) is an Ft -consistent, dynamically translation invariant, nonlinear expectation for {Qt }. 2. There exists a driver F , which is balanced on {Qt}, is independent of Y , and satisfies the normalisation condition F (ω, t, 0) = 0, such that, for all Q, Yt = E(Q|Ft ) is the solution to a BSDE with terminal condition Q and driver F . Furthermore, these two statements are related by the equation F (ω, t, Zt) = E(Zt Mt+1 |Ft ). Proof. See [5].



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41

6. From static to dynamic behaviour A final question that remains to be answered is whether, given a particular map Q → Y0 , it is possible to find an Ft -consistent nonlinear expectation which agrees with it. The following theorem addresses this problem in the scalar case, under the assumption of monotonicity. Theorem 6.1. Consider a measurable, scalar-valued map E : L2 (FT ) → L2 (F0 ). Suppose this map satisfies: (i) (Ft -consistency) For any Q ∈ L2 (FT ) and any t ≤ T , there exists an Ft measurable random variable Yt such that E(IA Q) = E(IA Yt )

(5)

for any A ∈ Ft . (ii) (F0 -triviality) E(Q) = Q for all F0 -measurable Q. (iii) (Monotonicity) For any Q, Q ∈ L2 (Ft ), if Q ≥ Q P-a.s., then E(Q) ≥ E(Q ), with equality if and only if Q = Q P-a.s. Then there exists a unique dynamic nonlinear expectation E(.|Ft ) (for Q = L2 (FT )) such that E(Q) = E(Q|F0 ) 2 for all Q ∈ L (FT ). This dynamic nonlinear expectation is given by E(Q|Ft ) = Yt with Yt as in (i). Proof. See [5].



References [1] Pauline Barrieu and Nicole El Karoui. Inf-convolution of risk measures and optimal risk transfer. Finance and Stochastics, 9(2):269–298, 2005. [2] Philippe Briand, Francois Coquet, Ying Hu, Jean M´emin, and Shige Peng. A converse comparison theorem for bsdes and related properties of g-expectation. Electronic Communications in Probability, 5:101–117, 2000. [3] Samuel N. Cohen and Robert J. Elliott. Solutions of backward stochastic differential equations on Markov chains. Communications on Stochastic Analysis, 2(2):251–262, August 2008. [4] Samuel N. Cohen and Robert J. Elliott. Comparisons for Backward Stochastic Differential Equations on Markov Chains and related no-arbitrage conditions, The Annals of Applied Probability, 20(1):267–311. [5] Samuel N. Cohen and Robert J. Elliott. A General Theory of Finite State Backward Stochastic Difference Equations, Stochastic Processes and their Applications, 120(4):442-466. [6] Fran¸cois Coquet, Ying Hu, Jean Memin, and Shige Peng. Filtration consistent nonlinear expectations and related g-expectations. Probability Theory and Related Fields, 123(1):1–27, May 2002.

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[7] Nicole El Karoui and S-J Huang. Backward Stochastic Differential Equations, Chapter 2: A general result of existence and uniqueness of backward stochastic differential equations, pages 27–36. Pitman Research Notes in Mathematics. Longman, 1997. [8] Robert J. Elliott and Hailang Yang. How to count and guess well: Discrete adaptive filters. Applied Mathematics and Optimization, 30(1):51–78, 1994. [9] Hans F¨ ollmer and Alexander Schied. Stochastic Finance: An introduction in discrete time. Studies in Mathematics 27. de Gruyter, 2002. [10] Arnaud Jobert and L. C. G. Rogers. Valuations and dynamic convex risk measures. Mathematical Finance, 18(1):1–22, January 2008. [11] Susanne Kl¨ oppel and Martin Schweizer. Dynamic indifference valuation via convex risk measures. Mathematical Finance, 17(4):599–627, October 2007. [12] Jin Ma, Philip Protter, Jaime San Martin, and Soledad Torres. Numerical method for backward stochastic differential equations. The Annals of Applied Probability, 12(1):302–316, Febuary 2002. [13] Shige Peng. A generalized dynamic programming principle and Hamilton-JacobiBellman equation. Stochastics and Stochastics Reports, 38:119–134, 1992. [14] Shige Peng. Stochastic Methods in Finance, Chapter 4: Nonlinear Expectations, Nonlinear Evaluations and Risk Measures, 165–254. Springer, Berlin – Heidelberg – New York, 2004. [15] Emanuela Rosazza Gianin. Risk measures via g-expectations. Insurance Mathematics and Economics, 39:19–34, 2006. [16] Albert N. Shiryaev. Probability. Springer, 2nd edition, 2000. Samuel N. Cohen University of Oxford e-mail: [email protected] Robert J. Elliott University of Calgary and University of Adelaide e-mail: [email protected]

Progress in Probability, Vol. 65, 43–59 c 2011 Springer Basel AG


On a Forward-backward Stochastic System Associated to the Burgers Equation Ana Bela Cruzeiro and Evelina Shamarova Abstract. We describe a probabilistic construction of H α -regular solutions for the spatially periodic forced Burgers equation by using a characterization of this solution through a forward-backward stochastic system. Mathematics Subject Classification (2000). 65C30, 35Q53. Keywords. Burgers equation, forward-backward SDE, diffeomorphism group.

1. Introduction Burgers equation, given by ∂s y(s, θ) + (y, ∇)y(s, θ) − ν∆y(s, θ) + F (s, θ) = 0 is sometimes presented as a simplified model for turbulence and describes the motion of a compressible fluid with viscosity ν under the influence of a force F . In this paper we establish a connection between a time-changed spatially periodic Burgers equation and a forward-backward stochastic system on the group of diffeomorphisms of a torus, similar to the characterization we have studied in [C-S] for the incompressible (Navier-Stokes equation) case. It is well known that forward-backward systems are closely related to partial differential equations. For references in the subject one can use for example those in [D] or in the more recent work [C-S-T-V]. One difference in our approach is that the stochastic processes are defined in the group of diffeomorphisms of the underlying configuration space of the fluid (in our case, the torus) and not in the configuration space itself. This group is endowed with a Sobolev topology. The importance of working with infinite-dimensional geometry, in the line of thought introduced by V. Arnold ([A]) for the Euler equation is, partly, that it allows to construct solutions which are “automatically” Sobolev regular in the space variable. Also, which is more important, the geometric objects defined in the (infinite-dimensional) path

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spaces may allow to prove several properties of the dynamics, such as stability for the corresponding flows. Generally speaking, our approach can be regarded as a stochastic deformation of the Lagrangian picture keeping the “mean velocity” (the Eulerian picture) of the motion unchanged. This means that the mean velocity, given by the drift, is still the relevant deterministic solution of the (velocity) equations of motion. This approach finds its roots in the works [N-Y-Z], [Y]. It is the point of view described in [G], but is completely different from the approaches that consist in perturbing the Eulerian (velocity) dynamics by a random noise. In [C-S] we have developed in this spirit a construction for the Navier-Stokes equations. We derived a solution of the stochastic system from a solution of NavierStokes equation and, “reciprocally”, defined Navier-Stokes solutions from the stochastic forward-backward flows. The incompressibility condition there makes the geometry much more delicate to study then in the present Burgers case and, in this respect, the Burgers equation is a simplification of the framework of [C-S]. On the other hand here we prove an existence result of the stochastic forward-backward equation (without assuming the existence of the p.d.e. solution), a result which is not proved for the Navier-Stokes case. This is therefore the main result of this paper and the one that really distinguishes it from [C-S]. The methods we use to prove this result are close to those of Delarue in [D]. Therefore we obtain here a completely probabilistic construction for the Burgers solutions. We treat the torus case since it is one of the simplest compact manifolds; the results, with the necessary adjustments, should extend to other manifolds. We refer to [C-S] and references therein for the general framework of the stochastic approach to partial differential equations such as Burgers and NavierStokes that we are dealing with.

2. The main result Let us consider the spatially periodic backward Burgers equation in Rn : .

∂s y(s, θ) + (y, ∇)y(s, θ) + ν∆y(s, θ) + F (s, θ) = 0 y(T, θ) = h(θ).

(1)

It is obtained from the classical Burgers equation by means of the substitution u(t, θ) ↔ −u(T − t, θ), where T > 0 is fixed arbitrary. We assume that h and F belong to the Sobolev space of order α, H α (Tn , Rn ), where α is an integer bigger than n2 + 2. The symbol Tn denotes the n-dimensional torus, namely Tn = S 1 × · · · × S 1 , and S 1 is a unit circle. We extend the functions F and h to Rn / 01 2 n

periodically, and use the same symbols for the extended functions. Consider the

Forward-backward Stochastic System

45

following system of forward-backward SDEs in H α (Tn , Rn ):  √ t,e t,e   dZs = Ys ds + 2ν dWs ,  √ dYst,e = −F (s, · ) ◦ Zst,e ds + 2ν Xst,e dWs ,    Ztt,e = e; YTt,e = h ◦ ZTt,e

(2)

where e : Tn → Tn is the identical map, Ws is an n-dimensional Brownian motion and ν > 0. Let Fs = σ(Wr , r ∈ [0, s]). We would like to find an Fs -adapted triple of stochastic processes (Zst,e , Yst,e , Xst,e ) with values in H α (Tn , Rn ) × H α (Tn , Rn ) × H α (Tn , Rn×n ) which solves (2). Note that the process Xst,e takes values in the space of linear operators L(Rn , H α (Tn , Rn )) (≈ H α (Tn , Rn×n )), i.e., Xst,e =

n 

Xsi ⊗ ei

(3)

i=1

where the processes Xsi take values in H α (Tn , Rn ), and {ei }ni=1 is an orthonormal basis of Rn . Define K = sup |∇h| + sup |∇F | (4) Tn

[0,T ]×Tn

where T is the arbitrary fixed number that we used to obtain the backward Burgers equation. Our main result is the following. Theorem 1. Let h ∈ H α (Tn , Rn ), F (s, · ) ∈ H α (Tn , Rn ), s ∈ [0, T ], be such that ∇l h and ∇l F (s, · ) are bounded for all l  α and for all s ∈ [0, T ]. Then, there exists a T0 > 0 that depends only on α and K defined by (4) and such that for every T < T0 and for every t ∈ [0, T ], there exists a unique Fs -adapted solution (Zst,e , Yst,e , Xst,e) to (2) on [t, T ] with values in H α (Tn , Rn ) × H α (Tn , Rn ) × H α (Tn , Rn×n ). Moreover, the function y : [0, T ] × Tn → Rn , (t, θ) → Ytt,e (θ) is deterministic, and solves the problem (1). First we prove the existence and uniqueness of an Fs -adapted H α (Tn , Rn ) × H α (Tn , Rn ) × H α (Tn , Rn×n )-valued solution (Zst,ξ , Yst,ξ , Xst,ξ ) to the following problem:  s √ t,ξ Zs = ξ + Yrt,ξ dr + 2ν (Ws − Wt ), (5) t

Yst,ξ = h ◦ ZTt,ξ +



T

F (r, · ) ◦ Zrt,ξ dr −

√

s



T

Xrt,ξ dWr ,

2ν

(6)

s

where ξ is an Ft -measurable H α (Tn , Rn )-valued random variable. Without loss of generality we will assume that the derivatives ∇l h and ∇l F (t, · ), t ∈ [0, T ], are everywhere defined. In the following, we will identify H 0 (Tn , Rn ) and L2 (Tn , Rn ). The proof of Theorem 1 will be devided in several lemmas. Lemma 1. Let E ξ pLp (Tn ,Rn ) and E ∇i ξ pL

p (T

n ,Rni )

be bounded for all integers

i  l and p
2. Further suppose that the FBSDEs (5), (6) have a solution in H l (Tn , Rn ). Then for any integer p
2 there exists a T0 > 0 that depends only

46

A.B. Cruzeiro and E. Shamarova

on p and K defined by (4) and such that for all positive T < T0 , for all i  l, E Zst,ξ pLp (Tn ,Rn ) and E ∇i Zst,ξ pL (Tn ,Rni ) are bounded on [t, T ]. p

Proof. Everywhere below, for convenience, we use the same symbols γ and γi , i = 1, 2, 3 . . ., for (different) constants in different formulas. All γ and γi below are positive and do not depend on s ∈ [t, T ] and θ ∈ Tn . Note that, for any Hilbert norm, 

 g p h = p g p−2 (g, h),   g p h1 h2 = p(p − 2) g p−4 (g, h1 )(g, h2 ) + p g p−2 (h1 , h2 ). Fix a θ ∈ Tn , and let zs = Zst,ξ (θ), ys = Yst,ξ (θ), xs = Xst,ξ (θ). BSDE (6) and Itˆ o’s formula imply: E |ys |p −

 √ 2ν p(p − 2)

T

# $ n  E |yr |p−4 |(xir , yr )|2 dr

s

 T √ ' ( − 2ν p E |yr |p−2 |xr |2 dr s



= E |h(zT )| + 2p p

T

i=1

' ( E |yr |p−2 (F (r, zr ), yr ) dr

s

where xir = Xri (θ) and the processes Xri were introduced in representation (3). Taking into account that h and F are bounded on Tn and resp. [0, T ] × Tn , and applying Young’s inequality we obtain the existence of constants γ1 and γ2 such that  T p E |ys |  γ1 + γ2 E |yr |p dr. s

Applying Gronwall’s lemma and then integrating over Tn we obtain that there exists a constant γ such that E Yst,ξ pLp (Tn ,Rn )  γ. From SDE (5), we deduce the existence of constants γ1 and γ2 such that 

s

E |zs |p  γ1 E |ξ|p + γ2

E |yr |p dr. t

Integrating over Tn and modifying γ we obtain: E Zst,ξ pLp (Tn ,Rn )  γ.

Forward-backward Stochastic System

47

Let us prove now that E ∇Zst,ξ pLp (Tn ,Rn ) is bounded, where ∇ stands, as usual, for the space derivative. The triple (∇Zst,ξ , ∇Yst,ξ , ∇Xst,ξ ) solves the FBSDEs:  s ∇Zst,ξ = ∇ξ + ∇Yrt,ξ dr (7) t



 ∇Yst,ξ = ∇h (ZTt,ξ ( · ) ∇ZTt,ξ +



√   ∇F r, Zrt,ξ ( · ) ∇Zrt,ξ dr − 2ν

T



T

∇Xrt,ξ dWr .

s

s

(8) For simplicity of the notation, let us introduce the processes zs = Zst,ξ (θ), ys = Yst,ξ (θ) and xs = Xst,ξ (θ). Itˆo’s formula together with the BSDE (8) imply:  T n  √ p p−4 |∇ys | + 2ν p(p − 2) |∇yr | |(∇xir , ∇yr )|2 dr +

√

s



i=1

T

|∇ys |p−2 |∇xr |2 dr

2ν p s



T

= |∇h(zT )∇zT |p + 2p 

|∇yr |p−2 (∇F (r, zr )∇zr , ∇yr ) dr s

T

|∇yr |p−2 (∇yr , ∇xr dWr ).

+ 2p s

This implies that there exist constants γ1 , γ2 , and γ3 such that for all s ∈ [t, T ]  T  T p p p−2 2 |∇yr | |∇zr | dr + γ3 E |∇yr |p dr. E|∇ys |  γ1 E |∇ξ| + γ2 E t

t

From Young’s inequality, p−2 2 |∇yr |p + |∇zr |p . p p

|∇yr |p−2 |∇zr |2 

(9)

Therefore, we can find constants γ1 and γ2 such that  T p p E|∇ys |  γ1 E |∇ξ| + γ2 E|∇yr |p dr. t

Choosing T0 smaller than

1 γ2

we deduce that there exists a constant γ such that E|∇ys |p  γ E |∇ξ|p .

Integrating over Tn gives: E ∇Yst,ξ pL

p (T

n ,Rn2 )

 γ E ∇ξ pL

p (T

n ,Rn2 )

.

(10)

Next, the SDE (7) implies that there exist positive constants γ3 and γ4 such that  s p t,ξ p E ∇Zs L (Tn ,Rn2 )  γ3 E ∇ξ L (Tn ,Rn2 ) + γ4 E ∇Yrt,ξ pL (Tn ,Rn2 ) dr. p

p

t

p

48

A.B. Cruzeiro and E. Shamarova

Combining this and (10) and modifying γ and γ1 we obtain: E ∇Zst,ξ p

2

Lp (Tn ,Rn )

Let us assume E ∇i Zst,ξ pL E ∇

l

Zst,ξ p n n2 Lp (T ,R )

n n2 ) p (T ,R

 γ1 E ∇ξ pL

p (T

n ,Rn2 )

 γ.

 γ for all integers i  l − 1, and prove that

 γ. The triple (∇l Zst,ξ , ∇l Yst,ξ , ∇l Xst,ξ ) solves the FBSDEs

(11), (12) below which are obtain from (5), (6) by differentiating both parts l times:  s l t,ξ l ∇ Zs = ∇ ξ + ∇l Yrt,ξ dr (11) t

∇l Yst,ξ

  = ∇h ZTt,ξ ( · ) ∇l ZTt,ξ + +

l 

 ∇j h ZTt,ξ ( · )

s

√

  ∇F r, Zrt,ξ ( · ) ∇l Zrt,ξ dr

s



∇i1 ZTt,ξ . . . ∇ij ZTt,ξ

4

i1 +···+ij =l−j+1

T

+ −

T

3

j=2





l 

 3 ∇j F r, Zrt,ξ ( · )

j=2



4 ∇i1 Zrt,ξ . . . ∇ij Zrt,ξ dr



i1 +···+ij =l−j+1

T

∇l Xrt,ξ dWr .

2ν

(12)

s

The argument below is similar to the one we have used for the first-order derivatives. Itˆ o’s formula and the BSDE (12) imply: |∇ ys | + l

p

√



T

|∇l ys |p−2 |∇l xr |2 dr

2ν p

(13)

s

+

√



T

2ν p(p − 2)

|∇l yr |p−4 s

n  i=1

l  3   = ∇h(zT )∇l zT + ∇j h(zT ) j=2



|∇ yr | l

i1 +···+ij =l−j+1

p−2



(∇F (r, zr )∇ zr , ∇ yr ) dr + 2p l

T

|∇l yr |p−2 (∇l yr , ∇l xr dWr )

l

s

T

|∇ yr | s

4p  ∇i1 zT . . . ∇ij zT 

s l

+



T

+ 2p 

|(∇l xir , ∇l yr )|2 dr

p−2

l ) j=2

∇j F (r, zr )

3

4 * ∇i1 zr . . . ∇ij zr , ∇l yr dr.

 i1 +···+ij =l−j+1

Note that by (11), there exist constants γ1 and γ2 so that  s l p l p |∇ zs |  γ1 |∇ ξ| + γ2 |∇l yr |p dr. t

Forward-backward Stochastic System

49

This and (13) imply for all s ∈ [t, T ] there exist constants γ3 , γ4 , γ5 , γ6 , and γ7 such that  T  T E|∇l ys |p  γ3 E |∇l ξ|p + γ4 E |∇l yr |p−2 |∇l zr |2 dr + γ5 E |∇l yr |p dr t t  + γ6 E |∇i1 zT . . . ∇ij zT |p i1 +···+ij
l−1



T

+ γ7 E

l  3  |∇l yr |p−2  ∇j F (r, zr )

t

j=2



42  ∇i1 zr . . . ∇ij zr  dr

i1 +···+ij =l−j+1

Note that we can apply inequality (9) where ∇yr is replaced by ∇l yr and ∇zr is replaced by ∇l zr , r ∈ [t, T ]. Also, Young’s inequality implies that there exists a constant γ8 such that l  3  |∇l yr |p−2  ∇j F (r, zr ) j=2

42  ∇i1 zr . . . ∇ij zr 

 i1 +···+ij =l−j+1

l 3 p − 2 l p 2  j  |∇ yr | +  ∇ F (r, zr ) p p j=2

p−2 l p  |∇ yr | + γ8 p

4p  ∇i1 zr . . . ∇ij zr 



i1 +···+ij =l−j+1



|∇i1 zr . . . ∇ij zr |p .

i1 +···+ij
l−1

Finally we obtain that there exist constants γ1 , γ2 , γ3 , and γ4 such that  E |∇l ys |p  γ1 E |∇l ξ|p + γ2 E |∇i1 zT . . . ∇ij zT |p i1 +···+ij
l−1





T

+ γ3 t

 ij

E |∇l yr |p dr.

p

t

i1 +···+ij
l−1

Choosing T0 smaller than

T

E |∇ zr . . . ∇ zr | dr + γ4 i1

1 γ4

and modifying γ1 , γ2 and γ3 we obtain that  E |∇l ys |p  γ1 E |∇l ξ|p + γ2 E |∇i1 zT . . . ∇ij zT |p 

i1 +···+ij
l−1 T

+ γ3 t



E |∇i1 zr . . . ∇ij zr |p dr

(14)

i1 +···+ij
l−1

and moreover, modifying γ1 , γ2 and γ3 we obtain that  E|∇l zs |p  γ1 E |∇l ξ|p + γ2 E |∇i1 zT . . . ∇ij zT |p  + γ3 t

i1 +···+ij
l−1 T

 i1 +···+ij
l−1

E |∇i1 zr . . . ∇ij zr |p dr.

(15)

50

A.B. Cruzeiro and E. Shamarova

Integrating over Tn and taking into account that E ∇l ξ p assumption,

E ∇i1 Zrt,ξ . . . ∇ij Zrt,ξ p n nl , Lp (T ,R )

l

Lp (Tn ,Rn )

is bounded by

r ∈ [t, T ], are bounded by H¨older’s

inequality and the induction hypothesis, we obtain that E ∇l Yst,ξ p E ∇

l

Zst,ξ p n nl Lp (T ,R )

l

Lp (Tn ,Rn )

and 

are bounded.

Lemma 2. There exists a T0 > 0 such that for every positive T < T0 and for every t ∈ [0, T ], FBSDEs (5), (6) has a unique Fs -adapted solution on [t, T ] with values in H α (Tn , Rn ) × H α (Tn , Rn ) × H α (Tn , Rn×n ). Proof. First we prove the existence of solution in L2 (Tn , Rn ). Let us consider the map Γ : L2 (Ω, L2 (Tn , Rn )) → L2 (Ω, L2 (Tn , Rn )), Yst,ξ → Y¯st,ξ which is defined by the FBSDEs below:  s √ Z¯st,ξ = ξ + Yrt,ξ dr + 2ν (Ws − Wt ), t

  Y¯st,ξ = h Z¯Tt,ξ ( · ) +



T

√   F r, Z¯rt,ξ ( · ) dr − 2ν

s

(16) 

T

¯ t,ξ dWr . X r

(17)

s

First we find Z¯st,ξ from the SDE (16), and substitute it into BSDE (17). Then we ¯ t,ξ ) of BSDE (17). Namely, find the unique Fs -adapted solution (Y¯st,ξ , X s  T     Y¯st,ξ = E [h Z¯Tt,ξ ( · ) + F r, Z¯rt,ξ ( · ) dr | Fs ], s

¯ t,ξ exists by the martingale representation theorem (see [C-S]). Note that and X s since h and F are bounded on Tn and resp. on [0, T ] × Tn , Y¯st,ξ takes values in ¯ t,ξ is actually not needed for the definition of the map Γ. L2 (Tn , Rn ). The process X s Let us prove that the map Γ is a contraction. Let Vst,ξ , Yst,ξ ∈ L2 (Ω, L2 (Tn , Rn )), t,ξ ¯st,ξ be obtained from (16), i.e., and V¯st,ξ =
Γ(Vst,ξ ), Y¯s√ = Γ(Yst,ξ ). Further let U s t,ξ t,ξ t,ξ ¯ ¯ Us = ξ + t Vr dr + 2ν (Ws − Wt ), and Hs be obtained from the BSDE (17), √
T t,ξ  t,ξ 
T   ¯ (·) + ¯ t,ξ ¯ dWr . The SDE (16) i.e., V¯st,ξ = h U 2ν s H r T s F r, Ur ( · ) dr − implies that for all s ∈ [t, T ]  s t,ξ t,ξ 2 ¯ ¯ Zs − Us L2 (Tn ,Rn )  (s − t) Yrt,ξ − Vrt,ξ 2L2 (Tn ,Rn ) dr. (18) t

The SDE (17) and Itˆ o’s formula imply that  T ¯ t,ξ − H ¯ t,ξ 2 n n dr E Y¯st,ξ − V¯st,ξ 2L2 (Tn ,Rn ) + 2ν E X r r L2 (T ,R ) s  t,ξ   t,ξ  2 ¯ (·) = E h Z¯T ( · ) − h U L2 (Tn ,Rn ) T  T     ¯ t,ξ ( · ) , Y¯ t,ξ − V¯ t,ξ )L (Tn ,Rn ) dr. +2 E (F r, Z¯rt,ξ ( · ) − F r, U r r 2 T s

Forward-backward Stochastic System

51

Hence,    t,ξ  2 ¯ (·) E Y¯st,ξ − V¯st,ξ 2L2 (Tn ,Rn )  E h Z¯Tt,ξ ( · ) − h U L2 (Tn ,Rn ) T  T     ¯ t,ξ ( · ) 2 n n dr + E F r, Z¯rt,ξ ( · ) − F r, U r L2 (T ,R ) 

t

T

+ s

E Y¯rt,ξ − V¯rt,ξ 2L2 (Tn ,Rn ) dr.

Gronwall’s lemma and inequality (18) imply that  T E Y¯st,ξ − V¯st,ξ 2L2 (Tn ,Rn )  γ˜ (T ) E Yrt,ξ − Vrt,ξ 2L2 (Tn ,Rn ) dr t

where γ˜ (T ) = eT K and K is defined by (4). Let us pick T0 so that γ(T0 ) = T0 γ˜ (T0 ) < 1. Then, if T < T0 , sup E Y¯st,ξ − V¯st,ξ L2 (Tn ,Rn )  γ(T ) sup E Yst,ξ − Vst,ξ L2 (Tn ,Rn ) .

s∈[t,T ]

(19)

s∈[t,T ]

This proves that for T < T0 there is a unique fixed point Yst,ξ of the map Γ. The processes Zst,ξ and Xst,ξ can be determined from (6) as described above. Let us consider now the FBSDEs which is obtained from (5), (6) by differentiation with respect to θ ∈ Tn : 
s ∇Zst,ξ = ∇ξ + t ∇Yrt,ξ dr     
T   (20) ∇Yst,ξ = ∇h ZTt,ξ ( · ) ∇ZTt,ξ + s ∇F r, Zrt,ξ ( · ) ∇Zrt,ξ dr   √
T  t,ξ − 2ν s ∇Xr dWr . Now we assume that the solution (Zst,ξ , Yst,ξ , Xst,ξ ) is known, and therefore the FBSDEs (20) are regarded as a system of SDEs with random coefficients. Clearly, if we prove the existence of solution (∇Zst,ξ , ∇Yst,ξ , ∇Xst,ξ ) to (20) it would imply that the solution (Zst,ξ , Yst,ξ , Xst,ξ ) is differentiable in θ, and solves (5), (6) in H 1 (Tn , Rn ). The proof of this fact uses standard approaches described for example in [D] or [B]. The same argument as before applied to the triple (∇Zst,ξ , ∇Yst,ξ , ∇Xst,ξ ) as well as the boundedness of ∇h and ∇F on Tn and resp. on [0, T ] × Tn imply the existence and uniqueness of a solution to (20), and therefore the existence and uniqueness of a solution to (5), (6) with respect to the H 1 (Tn , Rn )-norm. Indeed, consider the map 2

2

Γ(1) : L2 (Ω, L2 (Tn , Rn )) → L2 (Ω, L2 (Tn , Rn )), ∇Yst,ξ → ∇Yst,ξ which is defined by the FBSDEs: ∇Zst,ξ

 = ∇ξ +

s

∇Yrt,ξ dr, t

(21)

52

A.B. Cruzeiro and E. Shamarova

  ∇Yst,ξ = ∇h ZTt,ξ ( · ) ∇ZTt,ξ +



T

√   ∇F r, Zrt,ξ ( · ) ∇Zrt,ξ dr − 2ν

s



T

∇Xrt,ξ dWr . s

(22) The process ∇Zst,ξ is obtained from (21), and the processes ∇Yst,ξ and ∇Xst,ξ are obtained from the BSDE (22) as its unique Fs -adapted solution. Let ∇Vst,ξ , ∇Ust,ξ , ∇Vst,ξ , and ∇Hst,ξ be associated with the map Γ(1) and correspond to the processes ¯st,ξ , V¯st,ξ , and H ¯ st,ξ in the fixed point argument for the map Γ. The SDE Vst,ξ , U (21) implies the estimate:  ∇Zst,ξ − ∇Ust,ξ 2L

n2

n 2 (T ,R

)

 (s − t) t

s

∇Yrt,ξ − ∇Vrt,ξ 2L

Application of Itˆ o’s formula to ∇Yrt,ξ − ∇Vrt,ξ 2L  E ∇Yst,ξ − ∇Vst,ξ 2L

2

+ 2ν (Tn ,Rn2 )

s

2 (T

n ,Rn2 )

n n2 ) 2 (T ,R

dr.

gives

T

E ∇Xrt,ξ − ∇Hrt,ξ 2L

2 (T

n ,Rn3 )

  = E ∇h ZTt,ξ ( · ) (∇ZTt,ξ − ∇UTt,ξ ) 2L (Tn ,Rn2 ) 2  T     +2 E ∇F r, Zrt,ξ ( · ) (∇Zrt,ξ − ∇Urt,ξ ), ∇Yrt,ξ − ∇Vrt,ξ L2 (Tn ,Rn2 ) . s

The same argument as for the map Γ implies that sup E ∇Yst,ξ − ∇Vst,ξ 2L

s∈[t,T ]

2 (T

n ,Rn2 )

 γ(T ) sup E ∇Yst,ξ − ∇Vst,ξ 2L s∈[t,T ]

n n2 ) 2 (T ,R

where γ(T ) and T can be choosen exactly the same as in (19). Now let us assume that we proved the existence of solution to (5), (6) in H l−1 (Tn , Rn ). Namely, we formally differentiate (5), (6) l − 1 times with respect to θ, and assume that we have proved the existence of a solution (∇l−1 Zst,ξ , ∇l−1 Yst,ξ , ∇l−1 Xst,ξ ) in the l−1 space L2 (Tn , Rn ). Let us differentiate the FBSDE (5), (6) once again. We obtain l the FBSDEs (11), (12) which we consider as the FBSDEs in L2 (Tn , Rn ) with random coefficients with respect to three unknown processes (∇l Zst,ξ , ∇l Yst,ξ , ∇l Xst,ξ ). Consider the map l

l

Γ(l) : L2 (Ω, L2 (Tn , Rn )) → L2 (Ω, L2 (Tn , Rn )), ∇l Yst,ξ → ∇l Yst,ξ which is defined as follows: first we determine ∇l Zst,ξ from the SDE  s t,ξ l l ∇ Zs = ∇ ξ + ∇l Yrt,ξ dr. t

Forward-backward Stochastic System

53

Then we substitute ∇l Zst,ξ in the SDE  T     ∇l Yst,ξ = ∇h ZTt,ξ ( · ) ∇l ZTt,ξ + ∇F r, Zrt,ξ ( · ) ∇l Zrt,ξ dr +

√ 2ν

s



T

∇l Xrt,ξ dWr + s



l 

T

+ s

l 

 3 ∇j h ZTt,ξ ( · )

j=2

∇i1 ZTt,ξ . . . ∇ij ZTt,ξ

4

i1 +···+ij =l−j+1

 3 ∇j F r, Zrt,ξ ( · )

j=2



4 ∇i1 Zrt,ξ . . . ∇ij Zrt,ξ dr



(23)

i1 +···+ij =l−j+1

  and find a couple ∇l Yst,ξ , ∇l Xst,ξ as the unique Fs -adapted solution to (23). Namely, we have the following expression for ∇l Yst,ξ :  T 3     t,ξ t,ξ t,ξ l l ∇ Ys = E ∇h ZT ( · ) ∇ ZT + ∇F r, Zrt,ξ ( · ) ∇l Zrt,ξ dr s

+

l 

 ∇j h ZTt,ξ ( · )

j=2

 + s

T

3



∇i1 ZTt,ξ . . . ∇ij ZTt,ξ

4

i1 +···+ij =l−j+1 l  j=2

 3 ∇j F r, Zrt,ξ ( · )



4 4 ∇i1 Zrt,ξ . . . ∇ij Zrt,ξ dr | Fs .

(24)

i1 +···+ij =l−j+1 l

By Lemma 1, the last two terms of (24) belong to L2 (Tn , Rn ), and therefore l

∇l Yst,ξ takes values in L2 (Tn , Rn ). As before, we find ∇l Xrt,ξ by the martingale representation theorem. Since the coefficients of ∇l ZTt,ξ and of ∇l Yrt,ξ under the integral sign are the same as in the case l = 1, the fixed point argument will be also the same. In particular, T0 will be the same as for the FBSDEs (20) and (5), (6). By induction, we conclude that (11), (12) has Fs -adapted solutions (∇l Zst,ξ , ∇l Yst,ξ , ∇l Xst,ξ ) for every l = 1, . . . , α. This implies that there exists an Fs -adapted H α (Tn , Rn ) × H α (Tn , Rn ) × H α (Tn , Rn×n )-solution to (5), (6). Uniqueness of solution can be shown as in the proof of Lemma 15 of [C-S].



We have now shown the existence of solution for the forward-backward system of stochastic equations (2). From here we proceed to obtain the deterministic function y which actually determines the drift of the process Zst,e . This procedure is the same that we have followed in [C-S] to derive the solution of Navier-Stokes equations from the solution of the corresponding stochastic system. The difference is that, since now we are dealing with Burgers equation the incompressibility condition (div y = 0) is not present and, accordingly, the process Zst,e here belongs to the group Gα of H α -diffeomorphisms Tn → Tn whereas in [C-S] the corresponding relevant space is the subgroup of the volume-preserving diffeomorphisms. Still, for sake of completeness, we present here the rest of the proof. Everywhere below we assume that T < T0 , where T0 is defined in Lemma 2.

54

A.B. Cruzeiro and E. Shamarova

Lemma 3. If ξ is an Ft-measurable H α (Tn , Rn )-valued random variable, then the solution (Zst,ξ , Yst,ξ , Xst,ξ ) to (5), (6) takes the form: (Zst,ξ , Yst,ξ , Xst,ξ ) = (Zst,e ◦ ξ, Yst,e ◦ ξ, Xst,e ◦ ξ).

(25)

Proof. It suffices to prove the statement of the lemma in the space L2 (Tn , Rn ). Indeed, by uniqueness of solution, (Zst,ξ , Yst,ξ , Xst,ξ ) is the unique solution to (5), (6) in L2 (Tn , Rn ), and therefore if we prove (25) in L2 (Tn , Rn ), then the triple (Zst,e ◦ ξ, Yst,e ◦ ξ, Xst,e ◦ ξ) is the unique solution to (5), (6) also in H α (Tn , Rn ). Let us prove the statement in L2 (Tn , Rn ). We apply the operator Rξ of the composition with ξ to both parts of the SDEs (26) and (27):  s √ t,e Zs = e + Yrt,e dr + 2ν (Ws − Wt ), (26) t



Yst,e = h ◦ ZTt,e +

T

F (r, · ) ◦ Zrt,e dr − s

√



T

Xrt,e dWr .

2ν

(27)

s

Let us observe that we can write Rξ under the integral signs of the Bochner integrals. Indeed, since these integrals converge in H α (Tn , Rn ), they also converge with respect to at least the C(Tn , Rn )-topology. Due to the periodicity of the functions under the (Bochner) integrals signs, the composition of the integrands with ξ will preserve the convergence with respect to the C(Tn , Rn )-topology. For the integrals converging in the C(Tn , Rn )-topology, we can easily see that we are allowed to write Rξ under the integral signs. This implies that if we consider convergence of the Bochner integrals with respect to the L2 (Tn , Rn )-topology, we can also write Rξ under the integral signs. Let us prove now that we are allowed to write Rξ under the integral signs of the stochastic integrals in (26), (27). First we prove this for the case when ξ = g is deterministic. Let s and S be such that t  s < S  T , Φr be an
S Fr -adapted stochastically integrable process, and I(Φr ) denote s Φr dWr . Let (m) (m) Φr be a sequence of simple stochastic processes such that I(Φr ) converge to I(Φr ) with respect to the L2 (Ω)× H α (Tn , Rn )-norm. Note that if F ∈ H α (Tn , Rn ) extended to Rn periodically, then there exist constants γ1 and γ2 not depending on F and such that F ◦ g L2 (Tn ,Rn )  γ1 F ◦ g C(Tn ,Rn )  γ1 F C(Tn ,Rn )  γ2 F H α (Tn ,Rn ) . Therefore, since E


S

(m)

− Φr 2H α (Tn ,Rn ) dr → 0, then E


S

(m)

◦ g − Φr ◦ (m) 2 → 0. By Itˆo’s isometry, E I(Φr ◦ g) − I(Φr ◦ g) L2 (Tn ,Rn ) → 0. (m) (28), we conclude that E I(Φr )◦g −I(Φr )◦g 2L2(Tn ,Rn ) → 0 because s

Φr

(28)

s

Φr

g 2L2 (Tn ,Rn ) dr Again using (m)

E I(Φr

) − I(Φr ) 2H α (Tn ,Rn ) → 0. Clearly, for simple stochastic processes it holds

(m)

that I(Φr

(m)

) ◦ g = I(Φr

◦ g), and therefore I(Φr ) ◦ g = I(Φr ◦ g).

Forward-backward Stochastic System

55

Now let us take an Ft -measurable stepwise function ξ = ∞ i=1 gi IAi , where gi ∈ H α (Tn , Rn ) and the sets Ai are Ft -measurable. We obtain:  S  S ∞ ∞ ∞  S    Φr dWr ◦ g i I Ai = IAi Φr ◦ gi dWr = IAi Φr ◦ gi dWr s

i=1

s

i=1

i=1 s S



Φr ◦

= s

∞ 

gi IAi dWr .

i=1

Next, we find a sequence of Ft-measurable stepwise functions converging to ξ in the space of continuous functions C(Tn , Rn ). This is possible due to the separability of C(Tn , Rn ). Indeed, let us consider a countable number of disjoint Borel sets Oim covering C(Tn , Rn ), and such that their diameter in the norm of C(Tn , Rn ) ∞ 1 m −1 m m m is smaller than m . Let Ai = ξ (Oi ) and gi ∈ Oi . Define ξm = i=1 gim IAm . i 1 Then for all ω ∈ Ω, we have ξ − ξm C(Tn ,R2 ) < m . As before, I(Φ) and I(Φ ◦ ξ)
S
S denote s Φr dWr and s Φr ◦ ξ dWr resp. We have to prove that a.s. I(Φ) ◦ ξ = I(Φ ◦ ξ). For this it suffices to prove that lim E I(Φ) ◦ ξm − I(Φ) ◦ ξ 2L2 (Tn ,Rn ) = 0,

(29)

lim E I(Φ ◦ ξm ) − I(Φ ◦ ξ) 2L2 (Tn ,Rn ) = 0.

(30)

m→∞ m→∞

By (28), I(Φ) ◦ ξn L2 (Tn ,Rn )  γ2 I(Φ) H α (Tn ,Rn ) , and I(Φ) ◦ ξ L2 (Tn ,Rn )  γ2 I(Φ) H α (Tn ,Rn ) . By Lebesgue’s theorem, in (29) we can pass to the limit under the expectation sign. Relation (29) holds then by the continuity of I(Φ) in θ ∈ o’s isometry, the limit in (30) equals to Tn . To prove (30) we observe that by Itˆ
S limn→∞ E s Φr ◦ ξn − Φr ◦ ξ 2L2 (Tn ,Rn ) dr. The same argument that we used to prove (29) implies that we can pass to the limit under the expectation and the integral signs. Relation (30) follows from the continuity of Φr in θ ∈ Tn . Hence, (Zst,e ◦ ξ, Yst,e ◦ ξ, Xst,e ◦ ξ) is a solution to (5), (6) in L2 (Tn , Rn ).  Lemma 4. The processes Zst,e and Yst,e have continuous path modifications. Namely, for these modifications it holds that the trajectories [t, T ] → L2 (Tn , Rn ), s → Zst,e and [t, T ] → L2 (Tn , Rn ), s → Yst,e are continuous with probability 1. Proof. Let s > s. Application of Itˆo’s formula together with the BSDE (27) imply:  s  s 2 t,e 2 2 Yst,e −Yst,e F (r, Z ) dr+ Yrt,e −Yst,e n n  L (Tn ,Rn )   L (Tn ,Rn ) dr. r L2 (T ,R ) 2 2 s

s

Gronwall’s lemma implies that there exist constants γ˜ > 0 and γ > 0 such that  s 2 Yst,e − Yst,e  γ ˜ F (r, Zrt,e ) 2L2 (Tn ,Rn ) dr  γ(s − s).  L2 (Tn ,Rn ) s

This implies that if p > 1 then 2p  p Yst,e − Yst,e  L (Tn ,Rn )  γ|s − s | . 2

56

A.B. Cruzeiro and E. Shamarova

By Kolmogorov’s continuity criteria, Yst,e has a continuous path modification with respect to the L2 (Tn , Rn )-topology. The SDE (26) implies that Zst,e has a continuous path modification in the L2 (Tn , Rn )-topology as well.  Lemmas 5 and 6 below characterize the deterministic nature of the process Yst,e and describe its continuity properties. Lemma 5. The map [0, T ] × Tn → Rn , (t, θ) → Ytt,e (θ) is deterministic and the function [0, T ] → H α (Tn , Rn ), t → Ytt,e is continuous. Proof. The first statement is a consequence of Blumenthal’s zero-one law and the fact that the random variable Ytt,e is F0 -measurable (as in Lemma 13 of [C-S] or Corollary 1.5. of [D]). The proof of the continuity of the map [0, T ] → L2 (Tn , Rn ), t → Ytt,e follows as in Lemma 14 of [C-S]. Consider the FBSDEs below on the interval [0, T ] with respect to (∇l Zst,ξ , ∇l Yst,ξ , ∇l Xst,ξ ):  s ∇l Zst,ξ = ∇l ξ + I[t,T ] ∇l Yrt,ξ dr (31) 0

∇

l

Yst,ξ

  = ∇h ZTt,ξ ( · ) ∇l ZTt,ξ + +

l 

 ∇j h ZTt,ξ ( · )

j=2



I[t,T ] s

−

√

T

  I[t,T ] ∇F r, Zrt,ξ ( · ) ∇l Zrt,ξ dr

s

3



∇i1 ZTt,ξ . . . ∇ij ZTt,ξ

4

i1 +···+ij =l−j+1

T

+





l 

 3 ∇j F r, Zrt,ξ ( · )

j=2



4 ∇i1 Zrt,ξ . . . ∇ij Zrt,ξ dr

i1 +···+ij =l−j+1

T

∇l Xrt,ξ dWr

2ν

(32)

s

and note that its solution (∇l Zst,ξ , ∇l Yst,ξ , ∇l Xst,ξ ) can be obtained from the solution to (11), (12) by extending it to [0, t] as follows: ∇l Zst,ξ = ∇l ξ, ∇l Yst,ξ = ∇l Ytt,ξ , ∇l Xst,ξ = 0, s ∈ [0, t]. The extended triple solves the FBSDEs (31), (32) on [0, T ]. The same argument as in the proof of Lemma 14 of [C-S] implies that there exists a constant γ > 0 such that 

∇l Ytt,ξ − ∇l Ytt ,ξ L2 (Tn ,Rnl )  γ|t − t |. Therefore the map t → Ytt,ξ is continuous with respect to the H α (Tn , Rn )-topology.  Lemma 6. Let the function y : [0, T ] × Tn → Rn be defined by the formula y(t, θ) = Ytt,e (θ).

(33)

Forward-backward Stochastic System

57

Then, for every t ∈ [0, T ], there exists a set Ω ⊂ Ω of full P-measure, so that for all u ∈ [t, T ], for all ω ∈ Ω the following relation holds: Yut,e = y(u, · ) ◦ Zut,e .

(34)

Proof. Note that (25) implies that if ξ is Ft -measurable then Ytt,ξ = y(t, · ) ◦ ξ.

(35)

Further, for each fixed u ∈ [t, T ], is a solution of the following problem on [u, T ]: √
s . t,e Zs = Zut,e + u Yrt,e dr + 2ν (Ws − Wu ), √
T  
T   Yst,e = h ZTt,e ( · ) + s F r, Zrt,e ( · ) dr − 2ν s Xrt,edWr . (Zst,e , Yst,e , Xst,e )

t,e u,Zu

By the uniqueness of solution, it holds that Yst,e = Ys u,Z t,e Yu u

a.s. on [u, T ]. Next, by

(35), we obtain that = y(u, · ) ◦ This implies that there exists a set Ωu (which depends on u) of full P-measure such that (34) holds everywhere on Ωu . Clearly, one can find a set ΩQ , P(ΩQ ) = 1, such that (34) holds on ΩQ for all rational u ∈ [t, T ]. But the trajectories of Zst,e and Yst,e are a.s. continuous with respect to L2 (Tn , Rn )-topology by Lemma 4. Furthermore y(t, · ) is continuous in t with respect to (at least) the L2 (Tn , Rn )-topology. Therefore, (34) holds a.s. with respect to the L2 (Tn , Rn )-topology. Since both sides of (34) are continuous in θ ∈ Tn it also holds a.s. for all θ ∈ Tn .  Zut,e .

Finally the function y(s, · ) defined by (33) indeed verifies the Burgers equation. This is the content of the next lemma. Lemma 7. The function y defined by formula (33) is C 1 -smooth in t ∈ [0, T ], and is a solution of problem (1). Proof. Let δ > 0. We obtain: t+δ,e t+δ,e t,e t,e y(t + δ, · ) − y(t, · ) = Yt+δ − Ytt,e = Yt+δ − Yt+δ + Yt+δ − Ytt,e .

As before, let Gα be the group of H α -diffeomorphisms Tn → Tn , and let Yˆs be the right-invariant vector field on Gα generated by y(s, · ) (see [C-S]). Relation (34) allows us to represent the SDE (26) as an SDE on the manifold Gα . Indeed, by results of [G1] and [C-S], the SDE . t,e √ dZs = exp{Yˆs (Zst,e ) ds + 2ν dWs }, (36) Ztt,e = e where exp is the exponential map of the weak Riemannian metric on Gα (see [C-S]), has a unique Gα -valued solution. As it was proved in [C-S], the latter solution coincides with the unique solution of the H α (Tn , Rn )-valued SDE . t,e √ dZs = y(s, · ) ◦ Zst,e ds + 2ν dWs , Ztt,e = e.

58

A.B. Cruzeiro and E. Shamarova

Therefore, the Zst,e -part of the solution to (26), (27) is the unique solution to (36). t,e t,e By Lemma 6, a.s. Yt+δ = Yˆt+δ (Zt+δ ). Thus we obtain that a.s.  t,e  t,e y(t + δ, · ) − y(t, · ) = Yˆt+δ (e) − Yˆt+δ (Zt+δ ) + (Yt+δ − Ytt,e ). We use the BSDE (27) for the second difference and apply Itˆo’s formula to the first difference when considering Yˆt+δ as a C 2 -smooth function Gα → L2 (Tn , R2 ). We obtain:  t+δ t,e Yˆt+δ (Zt+δ ) − Yˆt+δ (e) = Yˆrt,e (Zrt,e )[Yˆt+δ (Zrt,e )] dr +

√



t

t+δ

2ν t

n  '

( ¯ ei Yˆt+δ (Zrt,e ) dWr + 2ν ∇



t+δ

t

i=1

n  '

( ¯ 2e Yˆt+δ (Zrt,e ) dr ∇ i

i=1

¯ is the covariant derivative on Gα , ei are regarded as constant vector fields where ∇ α on G , and the expression Yˆrt,e (Zrt,e )[Yˆt+δ (Zrt,e )] has the same meaning as in [C-S]. We obtain:  t+δ t,e Yˆt+δ (Zt+δ ) − Yˆt+δ (e) = dr ∇y(r, · ) y(t + δ, · ) ◦ Zrt,e 

t

t+δ

dr ν ∆ y(t + δ, · ) ◦

+ t

Zrt,e

√  + 2ν

t+δ t

Further we have: t,e Ytt,e − Yt+δ =



t+δ

dr F (r, · ) ◦ Zrt,e − t

n  ' ( ¯ ei Yˆt+δ (Z t,e ) dWr . ∇ r i=1

√



t+δ

Xrt,e dWr .

2ν t

Taking expectations implies:

  1 1 3 t+δ y(t + δ, · ) − y(t, · ) = − E dr [ (y(r, · ), ∇) y(t + δ, · ) δ δ t 4 + ν ∆ y(t + δ, · ) + F (r, · )] ◦ Zrt,e .

(37)

Note that Zrt,e , F (r, · ), and (y(r, · ), ∇) y(t + δ, · ) ◦ Zrt,e are continuous in r a.s. with respect to the L2 (Tn , R2 )-topology. By Lemma 5, ∇ y(t, · ) and ∆ y(t, · ) are continuous in t with respect to at least the L2 (Tn , R2 )-topology. Formula (37) and the fact that Ztt,e = e imply that in the L2 (Tn , R2 )-topology ∂t y(t, · ) = −[∇y(t, · ) y(t, · ) + ν ∆ y(t, · ) + F (t, · )].

(38)

Since the right-hand side of (38) is an H α−2 -map, so is the left-hand side. This implies that ∂t y(t, · ) is continuous in θ ∈ Tn . Therefore, (38) holds for any θ ∈ Tn . Relation (38) is obtained so far for the right derivative of y(t, θ) with respect to t. Note that the right-hand side of (38) is continuous in t which implies that the right derivative ∂t y(t, θ) is continuous in t on [0, T ). Hence, it is uniformly continuous on every compact subinterval of [0, T ). This implies the existence of the left derivative of y(t, θ) in t, and therefore, the existence of the continuous  derivative ∂t y(t, θ) everywhere on [0, T ].

Forward-backward Stochastic System

59

Remark. Note that at the same time we have proved that the process Zst,e takes values in the group Gα of H α -diffeomorphisms Tn → Tn . Acknowledgement The first author acknowledges the support of the Portuguese Foundation for Science and Technology through the project PTDC/MAT/69635/2006. The second author acknowledges the support of the Portuguese Foundation for Science and Technology through the Centro de Matem´ atica da Universidade do Porto.

References V.I. Arnold, Sur la g´eom´ etrie diff´erentielle des groupes de Lie de dimension infinie et ses applications a l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier 16 (1966), 316–361. [B] Ya.I. Belopolskaya, Yu.L. Dalecky, Stochastic equations and differential geometry, Series: Mathematics and its Applications, Kluwer Academic Publishers, Netherlands, (1989), 260 p. [C-S-T-V] P. Cheridito, H. Mete Soner, N. Touzi and N. Victoir, Second order backward stochastic differential equations and fully nonlinear parabolic PDEs, Comm. Pure Appl. Math. 60 (2007), 1081–1110. [C-S] A.B. Cruzeiro, E. Shamarova, Navier–Stokes equations and forward-backward SDEs on the group of diffeomorphisms of a torus, Stochastic Processes and their Applications, 119, (2009), 4034–4060. [D] F. Delarue, On the existence and uniqueness of solutions to the FBSDEs in a non-generate case, Stoch. Proc. and their Appl. 99, (2002), 209–286. [G] Yu.E. Gliklikh, Solutions of Burgers, Reynolds, and Navier–Stokes equations via stochastic perturbations of inviscid flows, Journal of Nonlinear Mathematical Physics, Vol. 17, No. Supplementary Issue 1 (2010) 15–29. [G1] Yu.E. Gliklikh, Global Analysis in Mathematical Physics: Geometric and Stochastic Methods, Springer (1997), 213 p. [E-M] D.G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 (1970), 102–163. [N-Y-Z] T. Nakagomi, K. Yasue, J.-C. Zambrini, Stochastic variational derivations of the Navier-Stokes equation, Lett. Math. Phys., 160 (1981), 337–365. [Y] K. Yasue, A variational principle for the Navier-Stokes equation, J. Funct. Anal., 51 (2), (1983), 133–141 [A]

Ana Bela Cruzeiro Dep. de Matem´ atica, IST-UTL and Grupo de F´ısica Matem´ atica da Universidade de Lisboa e-mail: [email protected] Evelina Shamarova Centro de Matem´ atica da Universidade do Porto e-mail: [email protected]

Progress in Probability, Vol. 65, 61–71 c 2011 Springer Basel AG


On the Estimate for Commutators in DiPerna–Lions Theory Shizan Fang and Huaiqian Lee Abstract. In this note, we will exhibit the estimates on the commutator of semi-groups, motivated by commutator estimates in Di Perna-Lions theory. Mathematics Subject Classification (2000). 60H30, 34A12, 60H10. Keywords. Sobolev regularity, OU semi-group, heat semi-group, commutator, renormalized solutions.

1. Introduction In 1989, R.J. Di Perna and P.L. Lions proved in [4] that for a vector field Z on Rd having only Sobolev space W1p regularity such that div(Z) ∈ L∞ and
|Z(x)| Rd 1+|x| dx < +∞, the differential equation dXt (x) = Z(Xt (x)) dt

(1)

has a unique solution Xt (x) for almost all initial x ∈ Rd ; moreover for each t ≥ 0, the Lebesgue measure dx on Rd admits a density Kt (x) under x → Xt (x); that is the push forward measure (Xt )∗ (dx) by Xt is absolutely continuous with respect to the Lebesgue measure: (Xt )∗ (dx) = Kt (x) dx. For the procedure of approximation in order to solve (1), instead of estimating the Jacobian of vector fields, they transformed the ordinary differential equation (1) to the transport equation dut + Z · ∇ut = 0. dt

(2)

We say that u ∈ L∞ ([0, T ], Lp(Rd )) solves (2) in distribution sense if   3 4  −α F ut − α div(F Z) ut dxdt = α(0)F u0 dx [0,T ]×Rd

where α ∈ Cc∞ ([0, T )) and F ∈ Cc∞ (Rd ).

Rd

(3)

62

S. Fang and H. Lee

A solution in distribution sense is generally difficult to be handled. A useful concept in this respect is the notion of renormalized solutions: for any β ∈ Cb1 (R), β(ut ) solves again (3). A basic result in the above-cited paper is Theorem 1.1. Let unt = ut ∗ χn . Then unt satisfies dunt + Z · ∇unt = cn (ut , Z), dt here cn (f, Z) = (DZ f ) ∗ χn − DZ (f ∗ χn ) and ||cn (f, Z)||L1 ≤ C ||f ||Lp (||∇Z||Lp + || div(Z)||Lp ).

(4)

(5)

Theorem 1.1 plays a key role in proving the existence and uniqueness of solutions to (2). By establishing the well-posedness for (2), they solved the differential equation (1). This concept of renormalized solutions has been substantially put forward by L. Ambrosio [1]: he took advantage the use of continuity equations ∂µt + Dx · (Zµt ) = 0, t > 0, µ|t=0 = µ0 ∂t and dealt successfully with vector fields having only the bounded variation. The main purpose of this note is to establish such kind of estimate for the heat semi-group in Riemannian manifolds. The main result is Theorem 3.3 in Section 3. In order to make the contrast with flat cases, we will include a short discussion on the Wiener space in Section 2.

2. The case of the Wiener space In the spirit of [1], Ambrosio and Figalli in [2] treated vector fields Z on the abstract Wiener space (W, H, µ). Let Pt be the Ornstein-Ulenbeck semi-group:  5 Pt f (x) = f (e−tx + 1 − e−2t y) dµ(y). W

They proved that if rt (f, Z) = et Z, ∇Pt (f ) − Pt (divµ (f Z)), then 3 Λ(p)t 4 ||rt (f, Z)||L1 ≤ 2 ||f ||Lp √ ||Z||Lq + || divµ (Z)||Lq + ||∇Z||Lq . 1 − e−2t Notice first that the divergence is defined by   f divµ (Z) dµ = − ∇f, ZH dµ. W

For h ∈ H,

W

Dh Pt f = e−t Pt (Dh f ).

Then rt (f, h) = −Pt (f div(h)). In this simple case: ||rt (f, h)||L1 ≤ ||f ||Lp || div(h)||Lq .

(6)

Commutators in DiPerna–Lions Theory

63

But for Z = h, the term ||Z||Lq = |h|H is not zero, the first term on the right hand in (6) is an extra term comparing to the above estimate. In the paper [9], we proved that if ct (f, Z) = DZ Pt f − Pt (DZ f ), then ||ct (f, Z)||L1 ≤ 2||f ||Lp (||∇Z||Lq + || divµ (Z)||Lq ).

(7)

In order to emphasize the difference with the case of Riemannian manifolds, let’s explain a bit the proof of (7). Let h ∈ H. Then  5 −t (Dh Pt f )(x) = e (∇f )(e−t x + 1 − e−2t y), hH dµ(y) W  5 e−t =√ f (e−t x + 1 − e−2t y) δh (y) dµ(y) 1 − e−2t W where δh (y) = h, y is a Gaussian Random variable of variance |h|2H . If we replace h by a vector field Z : W → H, then  5 e−t (DZ Pt f )(x) = √ f (e−tx + 1 − e−2t y) Zx , y dµ(y). (8) −2t 1−e W There is no convenient expression for Pt (DZ f ). Recall that for Z = fi hi , divµ (Z) = (−fi δ(hi ) + Dhi fi ). We have  5 Pt (Dhi fi )(x) = Dhi fi (e−t x + 1 − e−2t y) dµ(y) W  5 1 √ = fi (e−t x + 1 − e−2t y) hi , y dµ(y). 1 − e−2t W and  5 5 Pt (fi δ(hi )) = fi (e−t x + 1 − e−2t y)hi , e−t x + 1 − e−2t y dµ(y). W

Remark

5 5 y e−t √ − (e−t x + 1 − e−2t y) = √ (− 1 − e−2t x + e−t y). 1 − e−2t 1 − e−2t ˜ y) = Zx , y = δ(Zx )(y), then Let Z(x,  e−t ˜ t (x, y)) dµ(y), Pt (divµ (Z))(x) = √ Z(O (9) 1 − e−2t W where 5 5 Ot (x, y) = (e−t x + 1 − e−2t y, − 1 − e−2t x + e−t y). Replacing Z by f Z in (9), we get  5 e−t ˜ t (x, y)) dµ(y). (10) Pt (divµ (f Z)) = √ f (e−t x + 1 − e−2t y)Z(O 1 − e−2t W Now note divµ (f Z) = f divµ (Z) + DZ f . Then ct (f, Z) = DZ Pt f − Pt (DZ f ) = DZ Pt f − Pt (div µ (f Z)) + Pt (f divµ (Z)).

64

S. Fang and H. Lee Let Bt = DZ Pt f − Pt (divµ (f Z)), then  3 4 5 e−t ˜ y) − Z(O ˜ t (x, y)) dµ(y). Bt = √ f (e−t x + 1 − e−2t y) Z(x, 1 − e−2t W

Note that

dOt dt

Bt = − √

=

−t

If f˜(x, y) = f (x), then ) 1 * te−ts √ f˜(Ot (x, y)) I(Ots (x, y)) ds dµ(y) 1 − e−2ts 0

√ e R O. 1−e−2t −π/2 t

e−t 1 − e−2t



W

˜

where I = Z (R−π/2 ) and I(x, y) = Z  (x)y, y−Z(x), x. Therefore the estimate (7) follows (for more details, see [9]): ||ct (f, Z)||L1 ≤ 2||f ||Lp (||∇Z||Lq + || divµ (Z)||Lq ), where ct (f, Z) = DZ Pt f − Pt (DZ f ).

3. The case of Riemannian manifolds Let M be a smooth compact Riemannian manifold and consider the heat semigroup TtM := et∆ on M . Recall first a probabilistic construction for TtM . Let O(M ) be the bundle of orthonormal frames of M ; r ∈ O(M ) is an isometry from Rd onto Tπ(r) M . Endowing M with the Levi-Civita connection, denote by A1 , . . . , Ad the d canonical horizontal vector fields on O(M ): let {ε1 , . . . , εd } be the canonical basis of Rd , consider the geodesic ξi (t) on M , starting from x0 and tangent to r0 εi ∈ Tx0 M ; translating the frame r0 along ξi (t) gives a curve ri (t) ∈ O(M ), then dri (t) Ai (r0 ) = dt at t = 0. Let rt (w, r0 ) solve the following Stratanovich SDE on O(M ): drt (w) =

d 

Ai (rt (w)) ◦ dwti ,

r0 (w) = r0

(11)

i=1

where t → w(t) is the standard Rd -valued Brownian motion, defined on a probability space (Ω, F, P ). Let xt (w) = π(rt (w, r0 )). The law of w → x· (w) is independent of r0 and TtM f (x0 ) = E(f (xt )),

for any r0 ∈ π −1 (x0 ).

Now let’s state two basic formulae in path space analysis. The first one is due to J.M. Bismut [3]: Theorem 3.1. Let Z be a C 1 vector field on M . Then for any 0 ≤ t ≤ 1 and r0 ∈ π −1 (x0 ),  t 4  1 3 M DZ Tt f )(x0 ) = E f (xt ) Qs · r0−1 Zx0 , dws  , (12) t 0

Commutators in DiPerna–Lions Theory

65

where {Qt ; t ∈ [0, 1]} solves the following resolvent equation dQt 1 = − ricrt (w,r0 ) Qt , Q0 = Id. (13) dt 2 For more general study of such formula, we refer to S. Fang and P. Malliavin [10], D. Elworthy and X. Li [7], or the book [12]. Another formula due to B. Driver [5], concerns the forward derivative on the heat semi-group. Theorem 3.2. For any r0 ∈ π −1 (x0 ),  t   7* 1 )6 1 ¯ s , E(div(Z)(xt )) = E rt−1 Zxt (w) , I − s ricrs (w) dw t 2 0

(14)

¯ s denotes the Itˆo backward integral: where dw  t  ¯ s = lim As dw At∧τ+ (wt∧τ+ − wt∧τ ), |P|→0

0

τ ∈P

where τ+ denotes the point at the right side of τ in the partition P. The relation
t ¯ s and the usual Itˆo stochastic integral is given by between 0 As dw  t  t  t ¯ s= As dw As dws + dAs · dws , (15) 0

0

0

It seems that the formula (14) has the inconvenience that the derivative of the Ricci tensor will be involved according to (15). There is another formula given in [6]:  t 7* 1 )6 E(div(Z)(xt )) = − E rt−1 Zxt (w) , Qt Q−1 dw . s s t 0
t Set Mt = Qt 0 Q−1 o’s formula s dws . Then by Itˆ ) t * 1 dMt = − ricrt (w,r0 ) Qt Q−1 ds dt + dwt , s 2 0
t or Mt = wt − 12 0 ricrs (w,r0 ) Ms dws . So  t 7* 1 1 )6 −1 −1 E(div(Z)(xt )) = − E(rt Zxt , wt ) + E rt Zxt , ricrs Ms ds . (16) t 2t 0 we rewrite (14): E(div(Z)(xt )) =

 t * 1 1 ) E(rt−1 Zxt , wt ) − E rt−1 Zxt , s ricrs dws  t 2t 0  t ) * 1 − E rt−1 Zxt , sJrs ds 2t 0

where J(r) =

d   i=1

 LAi ric (r)εi .

(17)

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Note that when t → 0, the singular term in above two formulae is 1t E(rt−1 Zxt , wt ), but in opposite sign. To see the compatibility of these two formulae, note that 1 1 E(rt−1 Zxt , wt ) = E(rt−1 Zxt − r0−1 Zx0 , wt ). t t This means that the singular term will disappear, for the price that the derivative of Z will be involved. In order to get the term E(DZ f (xt )), we use the relation div(f Z) = DZ f + f div(Z); therefore when we replace Z by f Z in the formula (16), we get E(DZ f (xt )) = E(div(f Z)(xt )) − E(f (xt ) div(Z)(xt ))  t 7* 1 1 )6 = − E(f (xt ) rt−1 Zxt , wt ) + E rt−1 f (xt )Zxt , ricrs Ms ds t 2t 0 − E(f (xt ) div(Z)(xt )). (18) Recall the Bismut formula  t 4  1 3 M DZ Tt f )(x0 ) = E f (xt ) Qs · r0−1 Zx0 , dws  , t 0 In order to make a cancelation between the singular term in (18) and the term in above Bismut formula, we write down that 1 − E(f (xt ) rt−1 Zxt , wt ) t * 1 2 ) = E(f (xt ) rt−1 Zxt , wt ) − E rt−1 f (xt )Zxt − r0−1 f (x0 )Zx0 , wt  ; t t this last term gives rise the derivative of f . So the formula (16) is not convenient for our purpose. Theorem 3.3. Let q ≥ 2 and ε > 0. Then for t ∈ [0, 1], ||ct (f, Z)||L1 ≤ C1 ||f ||Lp+ε ||∇Z||Lq √ + C2 (ric) t ||f ||Lp+ε ||Z||Lq + ||f ||Lp || div(Z)||Lq . Proof. By Bismut and Driver’s formulae, ct (f, Z) admits the expression * 1 ) ct (f, Z)(x0 ) = E f (xt ) r0−1 Zx0 − rt−1 Zxt , w(t) t  t  s *   1 ) − E f (xt )  ricru Qu du (r0−1 Zx0 ), dws  2t 0 0  t * 1 ) −1 + E f (xt )rt Zxt , s ricrs dws  2t 0  t *   1 ) + E f (xt )rt−1 Zxt , sJrs ds + E (f div(Z))(xt ) . 2t 0

(19)

(20)

Let a1 (t, r0 ) be the first term on the right side. Using Itˆo’s formula will involve the second derivative of the vector field Z. To avoid this, we shall use the reversibility of the process rt (w, r0 ) with the initial law dr0 on O(M ) (see

Commutators in DiPerna–Lions Theory

67

D. Stroock [13]), which is locally dx ⊗ dg, where dx is renormalized Riemannian measure on M and dg the renormalized Haar measure on SO(d). Let {ε1 , · · · , εd } be the canonical basis of Rd ; then we have rt−1 Zxt , εi  = Zxt , rt εi . We know that π  (r)Ai (r) = rεi . Set ϕi (r) = Zπ(r) , π (r)Ai (r). Set f˜(r) = f (π(r)). Then a1 (t, r0 ) =

d   4 1 3˜ E f (rt (w, r0 )) wti · ϕi (rt (w, r0 )) − ϕi (r0 ) . t i=1

and d ) 3 ϕ(r (w, r )) − ϕ(r ) q 4*q /q 3 1   q 4   t 0 0  q √ E  wti ϕi (rt (w, r0 )) − ϕi (r0 )  ≤ Cq,q E    t t i=1

Now it remains to estimate  3 ϕ(r (w, r )) − ϕ(r ) q 4  t 0 0  √ E   dr0 . t O(M) By Lyons-Zheng decomposition formula [11], for each i, ϕi (rt ) − ϕi (r0 ) =

1 ¯ t ), Mt − M 2

(21)

¯ s is a where s → Ms is a martingale relative to Fs = σ(ru ; u ≤ s) and s → M ¯ martingale to Fs = σ(rt−u ; 0 ≤ u ≤ t). Their quadratic variation are given by  dMs · dMs =

d s

|LAj ϕi |2 (ru ) du,

0 j=1

 ¯ s · dM ¯s = dM

d s

|LAj ϕi |2 (rt−u ) du.

0 j=1

Set for a smooth function F : O(M ) → R, |∇H F (r)|2 =

d 

|LAj F (r)|2 .

j=1

By B¨ urkh¨older inequality, q 3) t *q/2 4 Mt q 1 || √ ||Lq ≤ Cq √ EPˆ |∇H ϕi |2 (ru ) du , t t 0

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where Pˆ is the probability for which rt is reversible. For q ≥ 2, the right-hand side is dominated by   31  t 4 * 1 t) H q Cq EPˆ |∇ ϕi | (ru ) du = Cq |∇H ϕi |q dr0 du t 0 t 0 O(M )  = Cq |∇H ϕi |q dr0 . O(M )

Therefore

8 8 8 ϕ(rt ) − ϕ(r0 ) 8q 8 8 √ 8 8 q t

L (Pˆ )

≤ Cq,d

d   i=1

|∇H ϕi |q dr0 .

(22)

O(M)

Now we shall compute the right-hand side of (22). Let Uj (s) be the flow associated to Aj on O(M ): dUj (s) = Aj (Uj (s)). ds Recall that ϕi (r) = Zπ(r) , π  (r)Ai (r). Then ϕi (Uj (s)) = Zπ(Uj (s)) , π (Uj (s))Ai (Uj (s)). Set mj (s) = π(Uj (s)). Then dmj (s) Uj (s) = π  (Uj (s)) · = π  (Uj (s)) · Aj (Uj (s)) = Uj (s)εj . ds ds So ϕi (Uj (s)) = Zmj (s) , Uj (s)εi  = r0 Uj (s)−1 Zmj (s) , r0 εi . Therefore, by taking the derivative with respect to s and at s = 0, we get the expression (LAj ϕi )(r0 ) = (∇r0 εj Z)(x0 ), r0 εi . (23) It follows that d 

|LAj ϕi |2 (r0 ) = |∇Z|2 (x0 ).

(24)

i,j=1

The other terms in (20) are easily obtained.



4. The case of Lie groups Now a natural question is what happens for compact Lie groups? Is there a simpler approach based on the group structure? Let G be a compact Lie group (for example closed subgroups of SO(N )), G its Lie algebra equipped with an AdG invariant metric  , . For a ∈ G, the left

Commutators in DiPerna–Lions Theory

69

invariant a ˜(g) = ga and the right invariant vector field a ˆ(g) = ag. Let {ε1 , · · · , εd } be an orthonormal basis of G. Then the Laplace operator is given by d d 1 2 1 2 Lε˜i = L . 2 i=1 2 i=1 εˆi

∆= Let pt (g) be the heat kernel:



f (g)pt (g0−1 g) dg.

Tt f (g0 ) = et∆ f (g0 ) = G

Let a ∈ G. Then

 d f (g)pt (e−εa g0−1 g) dg dε |ε=0 G  =− f (g0 g) (Laˆ pt )(g) dg.

(La˜ Tt f )(g0 ) =

G

On the other hand,



La˜ f (g)pt (g0−1 g) dg G  =− f (g0 g)(La˜ pt )(g) dg.

Tt (La˜ f )(g0 ) =

G

If G is not commutative, there is no reason to expect that La˜ Tt f = Tt (La˜ f ). Let (w(t))t≥0 be the G-valued standard Brownian motion. Consider the SDE on G dgw (t) = gw (t) ◦ dw(t), gw (0) = e. , Let Pe (G) = γ : [0, T ] → G continuous , γ(0) = e and ν the Wiener measure on Pe (G), the law of w → gw (·). Denote by  T ! " 2 ˙ H = h : [0, T ] → G; |h|2H = |h(s)| ds < +∞ . 0

For h ∈ H, consider



t

˙ Adgw (s)−1 h(s) ds.

Zh (w)(t) = 0

Then by elementary calculation (see [8]), for a cylindrical functional F : Pe (G) → R, d d F (eεh gw ) = F (gw+εZh (w) ). dε |ε=0 dε |ε=0 It follows that   ∂h F (γ) dν(γ) = F (γ)Kh (γ) dν(γ), (25) Pe (G)

Pe (G)

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S. Fang and H. Lee

where ∂h F (γ) =

d F (eεh γ) and dε |ε=0  T ˙ Kh (gw ) = Adgw (s)−1 h(s), dw(s). 0

But for the right derivative, we have ([8]),   ∂hr F (γ) dν(γ) = F (γ)Khr (γ) dν(γ), Pe (G)

(26)

Pe (G)

where

 Khr (gw )

T

˙ h(s), dw(s).

= 0

Now take F (γ) = f (γ(T )) and h(s) = Ts a with a ∈ G in (25), we get   ∇f (γ(T )), h(T )γ(T ) dν(γ) = f (γ(T ))Kh (γ) ν(γ), Pe (G)

or

Pe (G)



 ∇f (g), agpT (g) dg =

G

f (g)E(Kh (gw ) | gw (T ) = g) pT (g) dg. G

It follows that

 T * Laˆ pT (g) 1 ) = − E a, Adgw (s) dw(s) | gw (T ) = g . pT (g) T 0

(27)

Using (26), we get

* La˜ pT (g) 1 ) = − E a, w(T ) | gw (T ) = g . pT (g) T

(28)

It is difficult to conclude with these formulae (27) and (28). This means that the estimate in Theorem 3.3 is geometric: even in the case of Lie groups, we have to take into account the Riemannian structure. Acknowledgment The authors are grateful to referees for pertinent suggestions to improve the exposition of this note.

References [1] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math. 158 (2004), 227–260. [2] L. Ambrosio and A. Figalli, On flows associated to Sobolev vector fields in Wiener space: an approach a ` la DiPerna-Lions, Preprint 2008. [3] J.M. Bismut, Large deviations and Malliavin calculus, Birkh¨ auser, Boston/Basel/ Stuttgart, 1984. [4] R.J. Di Perna and P.L. Lions Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), 511–547.

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[5] B. Driver, Integration by parts for heat kernel measures revisited, J. Math. Pures Appl., 76 (1997), 703–737. [6] B. Driver and A. Thalmaier, Heat equation derivative formulas for vector bundles, J. Funct. Anal. 183 (2001), 42–108. [7] D. Elworthy and X. Li, Formulae for the derivative of heat semigroups, J. Funct. Anal. 125 (1994), 252–286. [8] S. Fang, Introduction to Malliavin Calculus, Math. Series for Graduate studients, vol. 3, Springer, Tsinghua University Press, 2005. [9] S. Fang and D. Luo, Transport equations and quasi-invariant flows on the Wiener space, to appear in Bulletin Sciences Math. [10] S. Fang and P. Malliavin, Stochastic analysis on the path space of a Riemannian manifold, J. Funct. Anal. 118 (1993), 249–274. [11] T. Lyons and W. Zheng, A croissing estimate for the canonical process on a Dirichlet space and tightness result, Colloque Paul L´evy, Ast´erisque, 157 (1988), 248–272. [12] P. Malliavin, Stochastic Analysis, Grund. Math. Wissen, vol. 313, Springer, 1997. [13] D. Stroock, An Introduction to the Analysis of Paths on a Riemannian manifold, Math. Survey and Monographs, vol. 74, AMS, 1999. Shizan Fang and Huaiqian Lee I.M.B. B.P. 47870 Universit´e de Bourgogne Dijon, France e-mail: [email protected]

Progress in Probability, Vol. 65, 73–81 c 2011 Springer Basel AG


Approximation Theorem for Stochastic Differential Equations Driven by G-Brownian Motion Fuqing Gao and Hui Jiang Abstract. We present an approximation theorem for stochastic differential equations driven by G-Brownian motion, i.e., solutions of stochastic differential equations driven by G-Brownian motion can be approximated by solutions of ordinary differential equations. Mathematics Subject Classification (2000). 60F10, 60H05, 60H10, 60J65. Keywords. G-Brownian motion, G-expectation, stochastic differential equations, approximation theorem.

1. Introduction Recently, G-Brownian motion was defined in [6] and [7]. The expectation E[·] associated with the G-Brownian motion is a sublinear expectation which is called G-expectation. In this framework, the related stochastic calculus has been established in [6] and [7]. The H¨older continuity and the homeomorphic property of the solution for stochastic differential equations driven by G-Brownian motion are studied in [2]. The large deviations for G-Brownian motion and stochastic differential equations driven by G-Brownian motion are established in [3]. The approximation theorem for stochastic differential equations driven by Brownian motion has been studied by many authors. For references, one can see Stroock and Varadhan ([11]), Ikeda and Watanabe ([4]). Ren ([10]) investigated this problem under Cr,p -capacity. The aim of this paper is to study the approximation theorem for stochastic differential equations driven by G-Brownian motion. 1.1. G-expectation and G-Brownian motion Let us briefly recall some basic conceptions and results about G-expectation and G-Brownian motion (see [1], [6], [7] for details). Research supported by the National Natural Science Foundation of China (10871153) .

74

F. Gao and H. Jiang Let Ω denote the space of all Rd -valued continuous paths ω : [0, +∞)  t −→ ωt ∈ Rd ,

with ω0 = 0, equipped with the distance

∞  1 2 −n 1 2 ρ(ω , ω ) := 2 max |ωt − ωt | ∧ 1 . n=1

t∈[0,n]

For each T > 0, set , Lip (FT ) := ϕ (ωt1 , ωt2 , . . . , ωtn ) : n ≥ 1, t1 , . . . , tn ∈ [0, T ], ϕ ∈ lip(Rd×n ) , where lip(Rd×n ) is the set of bounded Lipschitz continuous functions on Rd×n . 9∞ Define Lip (F ) := n=1 Lip (Fn ) ⊂ Cb (Ω). Sd denotes the space of d × d symmetric matrices. Γ is a given nonempty, bounded and closed subset of Rd×d which is the space of all d × d matrices. Set Σ := {γγ τ , γ ∈ Γ} ⊂ Sd and assume that Σ is a bounded, convex and closed subset. For A = (Aij )di,j=1 ∈ Sd given, set G(A) =

1 sup tr[γγ τ A]. 2 γ∈Γ

(1.1)

For each ϕ ∈ lip(Rd ), define E(ϕ) = u(1, 0) where u(t, x) is the viscosity solution of the following G-heat equation: ∂u − G(D2 u) = 0, on (t, x) ∈ [0, ∞) × Rd , ∂t

u(0, x) = ϕ(x),

(1.2)

and D2 u is the Hessian matrix of u, i.e., D 2 u = (∂x2i xj u)di,j=1 . Then E : lip(Rd ) → R is a sublinear expectation. This sublinear expectation is also called G-normal distribution on Rd and denoted by N (0, Σ)(cf. [9]). Let H be a vector lattice of real functions defined on Ω such that Lip (F) ⊂ H and if X1 , . . . , Xn ∈ H then ϕ(X1 , . . . , Xn ) ∈ H for each ϕ ∈ lip(Rn ). Let E[·] : H → R be a sublinear expectation on H. A d-dimensional random vector X with each component in H is said to be G-normal distributed under the sublinear expectation E[·] if for each ϕ ∈ lip(Rd ), √ u(t, x) := E(ϕ(x + tX)), t ≥ 0, x ∈ Rd is the viscosity solution of the G-heat equation (1.2). E[·] is called to be a Gexpectation if the d-dimensional canonical process {Bt(ω) = ωt , t ≥ 0} is a GBrownian motion under the sublinear expectation, that is, B0 = 0 and (i) For any s, t ≥ 0, Bt ∼ Bt+s − Bs ∼ N (0, tΣ); (ii) For any m ≥ 1, 0 = t0 < t1 < · · · < tm < ∞, the increment Btm − Btm−1 is independent from Bt1 , . . . , Btm−1 , i.e., for each ϕ ∈ lip(Rd×m), E(ϕ(Bt1 , . . . , Btm−1 , Btm − Btm−1 )) = E(ψ(Bt1 , . . . , Btm−1 )) where ψ(x1 , . . . , xm−1 ) = E(ϕ(x1 , . . . , xm−1 , Btm − Btm−1 )).

(1.3)

Approximation Theorem for SDEs Driven by G-Brownian motion

75

Throughout this paper, we assume that there exist constants 0 < σ ≤ σ < ∞ such that , Γ ⊂ γ ∈ Rd×d ; σId×d ≤ γγ τ ≤ σId×d . (1.4) The topological completion of Lip (Ft ) (resp. Lip (F )) under the Banach norm · p,G := (E(| · |p ))1/p is denoted by LpG (Ft ) (resp. LpG (F)), where p ≥ 1. E(·) can be extended uniquely to a sublinear expectation on LpG (F). We denote also by E the extension. p,0 Given T > 0. For p ∈ [1, ∞), let MG (0, T ) denote the space of R-valued n−1 piecewise constant processes ηt = i=0 ηti 1[ti ,ti+1 ) (t) where ηti ∈ LpG (Fti ), 0 = p,0 t0 < t1 < · · · < tn = T . For η ∈ MG (0, T ), the G-stochastic integral is defined by  t n−1  j j j j I (η) := ηs dBs := ηti (Bt∧t − Bt∧t ). i+1 i 0

Let

p MG (0, T )

i=0

be the closure of

p,0 MG (0, T )



η pM p (0,T ) := G

under the norm: T

E (|ηt |p ) dt. 0

2,0 Then the mapping I j : MG (0, T ) → L2G (FT ) is continuous, and so it can be 2 2 continuously extended to MG (0, T ). For any η = (η1 , . . . , η d ) ∈ (MG (0, T ))d , define  t d  t  ηs dBs = ηsi dBsi . 0

0

i=1

The quadratic variation process of G-Brownian motion is defined by

 t ij i j i j Bt := (Bt )1≤i,j≤d = Bt Bt − 2 Bs dBs . 0 ≤ t ≤ T, 0

1≤i,j≤d

The G-stochastic integral satisfies the following BDG inequality:  u p

 t   ¯  E sup  ηv dBv  ≤ Cp σ p/2 |t − s|p/2−1 E (|ηu |p ) du, s≤u≤t

s

(1.5)

s

where p ≥ 2, 0 < Cp < ∞ is a constant independent of Γ and η. 1.2. Main result Let us consider the following G-SDE dx(t) = σ(x(t))dBt + b(x(t))dt + h(x(t))dBt , x(0) = x0 ,

(1.6)

where σ = (σij ) : Rd → Rd ⊗ Rr , b = (bi ) : Rd → Rd , h = (hi,(j,l) ) : Rd → Rd ⊗ Rr

2

satisfy the following conditions: σij , bi , hi,(j,l) , ∂x∂ k σij , ∂x∂ k hi,(j,l) are bounded and ) *   d ∂ Lipschitz continuous. Denote by g = gi,(j,l) := ( σ )σ . ij kl k=1 ∂xk 1≤i≤d,1≤j,l≤r

76

F. Gao and H. Jiang

(1.6) can be written in Fisk-Stratonovich form:

1 dx(t) = σ(x(t)) ◦ dBt + b(x(t))dt + h(x(t)) − g(x(t)) dBt , x(0) = x0 , 2 where





t

t

σ(x(s)) ◦ dBs = 0

σ(x(s))dBs + 0

1 2



t

g(x(s))dBs . 0

Let xn be the solution of the following ordinary differential equation: 



 ˙ n (t) + b(xn (t)) + h(xn (t)) − 1 g(xn (t)) V˙ n (t) dt,  dxn (t) = σ(xn (t))W 2   xn (0) = x0 , (1.7) n

n

[2 t]+1 where for tn = [22nt] , t+ , n = 2n ) *   ij ˙ n (t) := 2n B(t+ ˙ ˙ W ) − B(t ) , V (t) := V (t) n n n n

d×d

  := 2n B(t+ n ) − B(tn ) .

Now we state our main result. Theorem 1.1. For any p ≥ 1, there exists some constant C(p) > 0 such that for all n ≥ 1,   C(p) 2p E sup |xn (t) − x(t)| ≤ np . 2 t∈[0,1] Throughout this paper, C with or without indices will denote different constants whose values are not important. Remark 1.1. Comparing with the classical case, we need to deal with the quadratic variation process B.

2. The proof of the approximation theorem In this section, we show Theorem 1.1. We will omit some estimates similar to the classical case. Lemma 2.1. For any p ≥ 1, there exists some constant C(p) > 0 such that for all n ≥ 1,   E

sup |x(t) − x(tn )|

2p

≤ C(p)2−np

(2.1)

≤ C(p)2−np .

(2.2)

t∈[0,1]

and

 E

 sup |xn (t) − xn (tn )| t∈[0,1]

2p

Approximation Theorem for SDEs Driven by G-Brownian motion Proof. It is obvious that  t  x(t) − x(tn ) = σ(x(s))dB(s) + tn



t

and



h(x(s))dBs tn



t

xn (t) − xn (tn ) =

t

b(x(s))ds + tn

77

t

˙ n (s)ds + σ(xn (s))W tn

b(xn (s))ds tn



1 h(xn (s)) − g(xn (s)) V˙ n (s)ds. 2 tn Then, by the assumptions and the BDG inequality for G-stochastic integral (cf. [2]), we can obtain (2.1) and (2.2) immediately.  

t

+

The following result is very useful for our proof. Lemma 2.2. For each 1 ≤ m ≤ d and 1 ≤ i, j ≤ r,

 tn  i +  j 1 ˙ i,j i ˙ gm,(i,j) (xn (sn )) Wn (sn ) − Wn (s) Wn (s) − Vn (s) ds 2 0  tn   1 = gm,(i,j) (xn (sn )) Bsi − Bsi n dBsj 2 0    1 tn + gm,(i,j) (xn (sn )) Bsj − Bsjn dBsi , 2 0 [2n s] + , sn 2n

n

= [2 2s]+1 , and for s ∈ [ k−1 , k ) and i = 1, . . . , r, n 2n 2n





k−1 k k−1 n i i k−1 Wni (s) = B i + 2 B − B ( ) s − . 2n 2n 2n 2n ) * Proof. Since V˙ ni,j (s) ≡ 2n B i , B j  kn − B i , B j  k−1 for s ∈ [ k−1 , k ), and 2n 2n 2 2n





k k k−1 i i i i k−1 n Wn − Wn (s) = B −B ( n ) 1−2 s− n , 2n 2n 2 2 where sn =

we have 

k 2n k−1 2n

=

gm,(i,j) (xn (sn ))



 j i ˙ (s) − 1 V˙ i,j (s) ds Wni (s+ ) − W (s) W n n n 2 n

1 k−1 gm,(i,j) (xn ( n )) 2) 2

× (B ikn − B ik−1 )(B jk − B jk−1 ) − B i , B j  kn + B i , B j  k−1 2 2n 2 2n 2n 2n  kn * ) 1 2 k−1 = gm,(i,j) (xn ( n )) Bsi − B ik−1 dBsj 2n 2 k−1 2 2n  k * 1 2n k−1 ) + gm,(i,j) (xn ( n )) Bsj − B jk−1 dBsi . 2 k−1 2 2n 2n

*

78

F. Gao and H. Jiang

Consequently,

 tn  i +  j 1 ˙ i,j i ˙ gm,(i,j) (xn (sn )) Wn (sn ) − Wn (s) Wn (s) − Vn (s) ds 2 0 n

2 tn  kn 2  i +  j 1 ˙ i,j i ˙ = gm,(i,j) (xn (sn )) Wn (sn ) − Wn (s) Wn (s) − Vn (s) ds k−1 2 2n k=1  tn   1 gm,(i,j) (xn (sn )) Bsi − Bsi n dBsj = 2 0    1 tn + gm,(i,j) (xn (sn )) Bsj − Bsjn dBsi , 2 0 

which completes the proof of this lemma.

Proof of Theorem 1.1. Since xn (t) − x(t) = xn (tn ) − x(tn ) + xn (t) − xn (tn ) − 2p x(t) + x(tn ), by Lemma 2.1, we only have to estimate E |xn (tn ) − x(tn )| . By the k−1 k Newton-Leibniz formula, for any s ∈ [ 2n , 2n ),

 d  s k−1 ∂ σij (xn (s)) = σij xn + σij (xn (u))dxm n (u) n k−1 2 ∂x m n 2 m=1 and



 d  s k−1 ∂ hi,(j,l) (xn (s)) = hi,(j,l) xn + hi,(j,l) (xn (u))dxm n (u). k−1 ∂xm 2n 2n m=1

Therefore, we have that





k−1 k k−1 j j xn B −B 2n 2n 2n

n

xin (tn ) = xi0 +

tn r 2 

σij

j=1 k=1

+

r  d  tn 

×

r  l=1

+

Wnj (s+ n)

r 

n tn r 2 

j,l=1 k=1

+





1 m,l ˙ hk,(m,l) (xn (s)) − gk,(m,l) (xn (s)) Vn (s) ds 2



) * k−1 hi,(j,l) xn B j , B l  kn − B j , B l  k−1 n n 2 2 2

r  d   j,l=1 k=1

∂ σij (xn (s)) ∂xk

˙ nl (s) + bk (xn (s)) σkl (xn (s))W

m,l=1

+

−



Wnj (s)

0

j=1 k=1





0

tn



 j,l Vnj,l (s+ n ) − Vn (s)



∂ hi,(j,l) (xn (s)) ∂xk

Approximation Theorem for SDEs Driven by G-Brownian motion  ×

r 

˙ j (s) + bk (xn (s)) σk,l (xn (s))W n

l=1



r  1 u,v ˙ + hk,(u,v) (xn (s)) − gk,(u,v) (xn (s)) Vn (s) ds 2 u,v=1  tn  r 1  tn + bi (xn (s))ds − gi,(j,l) (xn (s))V˙ nj,l (s)ds, 2 0 0 j,l=1

and can write that I(t) := xin (tn ) − xi (tn ) =

8 

Im (t).

m=1

where I1 (t) = I2 (t) =

r  

tn

(σi,j (xn (sn )) − σi,j (x(s))) dB j (s),

j=1 0 r  tn 

tn

I3 (t) =

 hi,(j,l) (xn (sn )) − hi,(j,l) (x(s)) dB j , B l s ,

0

j,l=1





bi (xn (s)) − bi (x(s))ds,

0

I4 (t) =

r  



tn

j,l=1

0

j,l=1

0

 gi,(j,l) (xn (s)) − gi,(j,l) (xn (sn ))

) *  l j ˙ (s) − V˙ j,l (s) ds, × Wnj (s+ ) − W (s) W n n n n  r t ) * n   l  j ˙ ˙ j,l I5 (t) = gi,(j,l) (xn (sn )) Wnj (s+ n ) − Wn (s) Wn (s) − Vn (s) ds,

I6 (t) =

r  d   j=1 k=1

I7 (t) =

and I8 (t) =

tn



 j Wnj (s+ n ) − Wn (s)



0

∂ σij (xn (s)) bk (xn (s))ds, ∂xk

∂ σij (xn (s)) ∂xk j,m,l=1 k=1 0   1 × hk,(m,l) (xn (s)) − gk,(m,l) (xn (s)) V˙ nm,l (s)ds, 2 r d   

r  d   j,l=1 k=1

0

tn

tn



j Wnj (s+ n ) − Wn (s)

 j,l +  Vn (sn ) − Vnj,l (s)







∂ hi,(j,l) (xn (s)) ∂xk

79

80

F. Gao and H. Jiang  ×

r  l=1

+

˙ j (s) + bk (xn (s)) σk,l (xn (s))W n

r 

u,v=1



1 u,v ˙ hk,(u,v) (xn (s)) − gk,(u,v) (xn (s)) Vn (s) ds. 2

Then, by the BDG inequality for G-stochastic integral and Lemma 2.1, it is easy to get that for any t ∈ [0, 1],

 t ) *

2p 2p E sup |I1 (s)| ds , ≤ C 2−np + E |I(s)| E

sup |I2 (s)|

2p



0≤s≤t

+E

sup |I3 (s)|

0≤s≤t

2p

0

≤C 2

−np



and

sup |I4 (t)|

2p

*

ds ,

0

E

)

E |I(s)|

+

0≤s≤t

t

2p

≤ C2−np .

0≤t≤1

By the BDG inequality for G-stochastic integral and Lemma 2.2, we also have

2p E sup |I5 (t)| ≤ C2−np . 0≤t≤1

) 2p * j  From E Wnj (s+ = C2−np (1 − 2n(s − sn ))2p , it is easy to see that n ) − Wn (s)



E sup |I6 (t)|2p + E sup |I7 (t)|2p ≤ C2−np . 0≤t≤1

0≤t≤1

j,l −n By |Vnj,l (s+ n ) − Vn (s)| ≤ C2

 

 ˙ j 2p and E W ≤ C2np , we have n

E

sup |I8 (t)|2p

≤ C2−np .

0≤t≤1

Together with above estimations, we obtain that





 t 2p −np 2p E sup |I(t)| ≤C 2 + E sup |I(u)| du 0≤s≤t

)

0

0≤u≤s

Consequently, by Gronwall’s inequality, E sup0≤t≤1 |I(t)| pletes the proof of this theorem.

2p

*

≤ C2−np which com

References [1] L. Denis, M.S. Hu, S. Peng, Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion pathes. Potential Analysis, 34 (2011), 139–161. [2] F.Q. Gao, Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion, Stoch. Proc. Appl., 119(2009), 3356–3382.

Approximation Theorem for SDEs Driven by G-Brownian motion

81

[3] F.Q. Gao, H. Jiang, Large Deviations for Stochastic Differential Equations driven by G-Brownian motion, Stoch. Proc. Appl., 120 (2010), 2212–2240. [4] N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland/Kodanska, Amsterdam, 1989. [5] M. Ledoux, M. Talagrand, Probability in Banach Space Isoperimetry and Processes. Springer-Verlag, 1991. [6] S. Peng, G-Expectation, G-Brownian motion and related stochastic calculus of Itˆ o’s type, in: Proceedings of the 2005 Abel Symposium 2, Edit. Benth et al., 541–567, Springer-Verlag, 2006. [7] S. Peng, Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stochastic Processes and their Applications, 118 (2008) 2223–2253. [8] S. Peng, G-Brownian motion and dynamic risk measure under volatility uncertainty. Preprint (pdf-file avaliable in arXiv: math.PR/0711.2834v1 19 Nov. 2007). [9] S. Peng, A new central limit theorem under sublinear expectations. arXiv: 0803.2656v1[math.PR], 18 Mar. 2008. [10] J.G. Ren, Analyse quasi-sˆ ure des ´equations diff´erentielles stochastiques. Bull. Sc. Math. 114(1990), 187–213. [11] D. Stroock, S.R.S. Varadhan, On the support on diffusion processes with applications to the stroock maximum principle. Proceedings of the 6th Berkeley symposium on mathematical statistics and probability. 3(1972), 333–359. [12] T. Taniguchi, Successive approximations to solutions of stochastic differential equations. J.Differential Equations. 96(1992), 152–169. [13] J. Xu, B. Zhang, Martingale characterization of G-Brownian motion. Stochastic Processes and their Applications, 119 (2009) 232–248. [14] X.C. Zhang, Euler-Maruyama approximations for SDEs with non-Lipschitz coefficients and applications. J. Math. Anal. Appl. 316(2006), 447–458. Fuqing Gao School of Mathematics and Statistics Wuhan University Wuhan 430072, P.R. China e-mail: [email protected] Hui Jiang School of Science Nanjing University of Aeronautics and Astronautics Nanjing 210016, P.R. China e-mail: [email protected]

Progress in Probability, Vol. 65, 83–97 c 2011 Springer Basel AG


Stochastic Flows for Nonlinear SPDEs Driven by Linear Multiplicative Space-time White Noises Benjamin Goldys and Xicheng Zhang Abstract. For a nonlinear stochastic partial differential equation driven by linear multiplicative space-time white noises, we prove that there exists a bicontinuous version of the solution with respect to the initial value and the time variable. Mathematics Subject Classification (2000). 60H15. Keywords. Stochastic flows, stochastic partial differential equations, spacetime white noises.

1. Introduction and main result Consider the following stochastic partial differential equation (SPDE) on [0, 1] with Dirichlet boundary conditions:  ∂ 2u ∂2W  ∂u = + f (x, u(t, x)) + [b(x)u(t, x) + c(x)] , (1) ∂t ∂x2 ∂t∂x  u(t, 0) = u(t, 1) = 0, u(0, x) = u0 (x), where W (t, x) is a space-time white noise on R+ × [0, 1] defined on some stochastic basis (Ω, F , P ; (Ft )t0 ), i.e., a two parameter Brownian sheet, and f (x, r) : [0, 1] × R → R and b(x), c(x) : [0, 1] → R are real measurable functions, and satisfy |f (x, r) − f (x, r )|  Cf |r − r |, ∀r, r ∈ R, x ∈ [0, 1]

(2)

f (x, 0), c(x) ∈ L2 (0, 1), b(x) ∈ L∞ (0, 1).

(3)

and

This work is supported by ARC Discovery grant DP0663153 of Australia and NSF of China (no. 10971076).

84

B. Goldys and X. Zhang 2

W Since ∂∂t∂x does not make sense in general, equation (1) is understood as the following mild form (cf. [9]):  1  t 1 u(t, x) = Gt (x, y)u0 (y)dy + Gt−s (x, y)f (y, u(s, y))dyds 0

 t

+

0

0

1

Gt−s (x, y)[b(y)u(s, y) + c(y)]W (ds, dy), 0

(4)

0

where Gt (x, y) is the fundamental solution of the associated homogeneous heat equation, i.e, equation (1) with f = b = c = 0 and u0 (x) = δy (x), which has the expression # $ # $& ∞ %  1 (y − x − 2n)2 (y + x − 2n)2 Gt (x, y) := √ exp − − exp − ; 4t 4t 4πt n=−∞ the third integral in (4) is the usual Itˆo stochastic integral with respect to the martingale measure W (ds, dy) (cf. [9]). It is well known that for any u0 ∈ L2 (0, 1), there exists a unique continuous (Ft )-adapted process t → u(t; u0 ) ∈ L2 (0, 1) such that equation (4) holds in L2 (0, 1) (cf. [9, Theorem 3.2]). However, it is not known whether the family of solutions {u(t; u0 ) : t ∈ R+ , u0 ∈ L2 (0, 1)} forms a stochastic flow. More precisely, does there exist a version u ˜(t; u0 ) of u(t; u0 ) such that R+ × L2 (0, 1)  (t, u0 ) → u ˜(t; u0 ) ∈ L2 (0, 1) is continuous almost surely? For finite-dimensional stochastic dynamic systems, the existence of stochastic flows has been studied extensively by employing Kolmogorov’s continuity criterion (cf. [6, 10, etc.]). However, the flow property for infinite-dimensional stochastic dynamic systems may not hold true (cf. [2, p. 5–6] or [8]). For linear stochastic evolution equations with multiplicative noises, this problem has been investigated by Flandoli in [2], see also [9, p. 332, Theorem 4.1]. For nonlinear SPDEs, there are few results and we only know the literatures [3] and [4]. In these two references, the authors considered the SPDEs driven by linear multiplicative noises with finite many Brownian motions, which can not be applied to equation (1). Our main result in this paper is stated as follows: Theorem 1.1. Under (2) and (3), for u0 ∈ L2 (0, 1), let u(t; u0 ) be the unique solution of equation (4). Then there exists a version u ˜(t; u0 ) of u(t; u0 ) such that for any p
2 and P -almost all ω ∈ Ω, (0, ∞) × L2 (0, 1)  (t, u0 ) → u ˜(t, ω; u0 ) ∈ Lp (0, 1) is continuous. In particular, the following flow property holds: for P -almost all ω ∈ Ω, u ˜(t + s, ω; u0 ) = u ˜(t, θs ω; u ˜(s, ω, u0 )), ∀t, s
0, u0 ∈ L2 (0, 1), where θs is the usual shift operator.

Stochastic Flows for Nonlinear SPDEs

85

We remark that for nonlinear multiplicative noises, the above problem is still open. Our proof of this theorem is mainly based on some careful estimations of the singularity for a family of random evolution linear operators studied in [8, 2] along the diagonal. In Section 2, we shall give an abstract result of the existence of stochastic flows for stochastic evolution equations. In Section 3, Theorem 1.1 will be proved.

2. An abstract result for stochastic evolution equations Let H be a real separable Hilbert space. Let L : D(L) ⊂ H → H be the infinitesimal generator of an analytic semigroup (Tt )t0 of positive type. Define a family of Hilbert spaces Hα := D(Lα/2 ), α ∈ R endowed with the graph norm x Hα := x H + Lα/2 x H . Since we are assuming that (Tt )t0 is of positive type, we have the following equivalent norm: x Hα  Lα/2 x H . Below, we shall always use this norm. Moreover, we can extend the operator Tt to Hα for any α ∈ R (cf. [7]). The following properties are well known (cf. [5, p. 24–27] or [7, p. 74]). Proposition 2.1. (i) Tt : H → Hα for each t > 0 and α > 0. (ii) For each t > 0, α ∈ R and every x ∈ H2α , Tt Lα x = Lα Ttx. (iii) For some δ > 0 and each t, α > 0, the operator Lα Tt is bounded in H and Lα Tt x H  Cα t−α e−δt x H , ∀x ∈ H.

(5)

(iv) Let α ∈ (0, 1] and x ∈ H2α , then Tt x − x H  Cα tα x H2α , t > 0.

(6)

Let {W k (t), t
0, k ∈ N} be an infinite sequence of independent standard (Ft )-Brownian motions on (Ω, F, P ). Let l2 be the usual sequence Hilbert space of real numbers. Then {W k (t), t
0, k ∈ N} can be regarded as a standard cylindrical Brownian motion on l2 . For α ∈ R, let L2 (l2 ; Hα ) be the space of all Hilbert-Schmidt operators from l 2 to Hα . Below, we fix β ∈ [0, 1) and let B = (Bk )k∈N : H → L2 (l2 ; H−β ) be a bounded linear operator. For r
0, consider the following linear SPDE . dX(t) = −LX(t)dt + Bk X(t)dW k (t), t
r, X(r) = x0 .

(7)

(8)

86

B. Goldys and X. Zhang

Here and below, we use the convention that the repeated indices are summed and the letter C with or without subscripts denotes a positive constant, whose value may change from one place to another. For any x0 ∈ H−β , it is well known that there exists a unique solution X(t, r; x0 ) for equation (8) such that for any γ ∈ (0, 1 − β) (cf. [2, p. 25, Theorem 5.4]) t → X(t, r; x0 ) ∈ C((r, ∞); Hγ ), P − a.s. and  t

X(t, r; x0 ) = Tt−r x0 +

Tt−s Bk X(s, r; x0 )dW k (s). r

It is easy to see that H−β  x0 → X(t, r; x0 ) ∈ L2 (Ω, P; Hγ ) is a bounded linear operator for any γ ∈ (0, 1 − β). By the uniqueness of solution we also have X(t, r, ω; x0 ) = X(t − r, 0, θr ω; x0 ), P − a.s.,

(9)

where θr is the usual shift operator. The following simple estimate will be used frequently: For α, β ∈ [0, 1) and 0  r < t,  t  1 ds ds 1−α−β = (t − r)  Cα,β · (t − r)1−α−β . (10) β α β α r (t − s) (s − r) 0 (1 − s) s We begin with the following Gronwall lemma of Volterra type. Lemma 2.2. Let T > 0 and f (t) : [0, T ] → R+ be a continuous function. Assume that for some α, β ∈ [0, 1) and γ > α + β − 1,  t f (s)ds f (t)  f0 + Ctγ , ∀t ∈ [0, T ]. β α 0 (t − s) s Then for some CT > 0, f (t)  CT · f0 , ∀t ∈ [0, T ].

(11)

Proof. Choose p > 1 so that p 1+γ 1 1 < ∧ ∧ . p−1 α+β α β By H¨older’s inequality we have  t

p f (s)ds p p γp f (t)  Cp f0 + Cp t β α 0 (t − s) s  t

p−1  t ds  Cp f0p + Cp tγp f (s)p ds qβ sqα (t − s) 0 0  t (10) p (1+γ−q(α+β))(p−1)+γ  Cp f0 + Cp t f (s)p ds 0  t p  Cp f0 + CT,p f (s)p ds. q :=

0

The estimate (11) now follows by the usual Gronwall inequality.



Stochastic Flows for Nonlinear SPDEs

87

We now prove several moment estimations about the mapping (t, r) → X(t, r; x0 ). Lemma 2.3. Let α ∈ [0, 1). For any p
1 and T > 0, there exists a positive constant CT,p such that for all x0 ∈ H ) * sup E (t − r)αp X(t, r; x0 ) 2p  CT,p · x0 2p . (12) H H−α 0
r
Proof. By (9) we may assume r = 0 and simply write X(t) := X(t, 0; x0 ). By BDG’s inequality and (5), we have  8 t 82p ) * 8 8 2p αp αp 8 k 8 E t X(t) H  Cp tαp Tt x0 2p + t E T B X(s)dW (s) t−s k H 8 8  αp  Cp x0 2p H−α + Cp · t E

H

0

0

p

t

Tt−s BX(s) 2L2 (l2 ;H) ds

.

The last expectation can be controlled by using H¨ older’s inequality as follows:  t

p E Tt−s BX(s) 2L2 (l2 ;H) ds 0

p 1 −β/2 2 L BX(s) ds L2 (l2 ;H) β 0 (t − s) 

p t (7) 1 2  CE X(s) ds H β 0 (t − s)  t

p 1 α 2 = CE · (s X(s) )ds H β α 0 (t − s) s  t

p−1  t 1 1 C ds · E(sα X(s) 2H)p ds β α β α 0 (t − s) s 0 (t − s) s  t (10) 1  Cp t(1−α−β)(p−1) · · E(sα X(s) 2H)p ds. β α 0 (t − s) s (5)



t

 CE

Hence, )



*

E tαp X(t) 2p  Cp x0 2p + Cp t(1−β)(p−1)+α H H−α

0

t

) * E sαp X(s) 2p ds H (t − s)β sα

. 

The desired estimate follows by Lemma 2.2.

Lemma 2.4. Let α ∈ [0, 1) and γ ∈ [0, 1 − β). For any p
1 and T > 0, there exists a positive constant CT,p such that for all x0 ∈ H and all 0  r  r1 < r2  t  T , 8  r2 82p 8 8 k 8 E8 T B X(s, r; x )dW (s) t−s k 0 8 8 r1

 CT,p



Hγ

r2

r1

ds (t − s)γ+β (s − r)α

p

· x0 2p . H−α

88

B. Goldys and X. Zhang

Proof. By BDG’s inequality and H¨ older’s inequality we have 8 r2 82p 8 8 k 8 E8 Tt−s Bk X(s, r; x0 )dW (s)8 8 r1



 Cp E

Hγ

r2

r1 r2

p

Tt−s BX(s, r; x0 ) 2L2 (l2 ;Hγ ) ds

p 1 −β/2 2 L BX(s, r; x ) ds 2 0 L2 (l ;H) γ+β r1 (t − s) 

p r2 (7) 1 2  Cp E X(s, r; x ) ds 0 H γ+β r1 (t − s)  r2

p−1  r2 ds E((s − r)α X(s, r; x0 ) 2H )p  Cp · ds γ+β α (s − r) (t − s)γ+β (s − r)α r1 (t − s) r1  r2

p (12) ds  CT,p · x0 2p H−α . γ+β (s − r)α (t − s) r1 (5)



 Cp E



The proof is thus complete.

Lemma 2.5. Let α ∈ [0, 1) and  ∈ (0, 1 − α). For any p
1 and T > 0, there exists a positive constant CT,p such that for all x0 ∈ H and 0  r < r < t  T , ) * E (t − r )(α+)p X(t, r ; x0 ) − X(t, r; x0 ) 2p H  CT,p · |r − r|(∧(1−β))p · x0 2p , H−α

(13)

where β is from (7). Proof. Let 0  r < r < t  T and set g(t, r , r) := (t − r )(α+)/2 (X(t, r ; x0 ) − X(t, r; x0 )). Then g(t, r , r) = (t − r )(α+)/2 (Tt−r − Tt−r )x0  r  (α+)/2 + (t − r ) Tt−s Bk X(s, r; x0 )dW k (s) r  t + (t − r )(α+)/2 Tt−s Bk (X(s, r ; x0 ) − X(s, r; x0 ))dW k (s) r 



= : I1 (t, r , r) + I2 (t, r , r) + I3 (t, r , r). For I1 , we have (6)

I1 (t, r , r) 2H  C(r − r) · (t − r )α+ · L/2 Tt−r x0 2H (5)

 C(r − r) · Lα/2 x0 2H

= C(r − r) · x0 2H−α .

Stochastic Flows for Nonlinear SPDEs For I2 , by Lemma 2.4 we have E I2 (t, r , r) 2p H

89

p ds  C(t − r )(α+)p · · x0 2p H−α (t − s)β (s − r)α r   p r ds  p  C(t − r ) · · x0 2p H−α β−α (s − r)α (t − s) r   p r ds  CT,p · · x0 2p H−α (r − s)β−α (s − r)α r 

r

(10)

 CT,p · |r − r|(1−β)p · x0 2p H−α .

As in the proof of Lemma 2.4, for I3 we have  2p   (α+)p ·E E I3 (t, r , r) H  C(t − r )  C(t − r )(α+)p  × (10)

C

t r



t

r



t

r

g(s, r , r) 2H ds β  α+ r  (t − s) (s − r )

p−1 ds (t − s)β (s − r )α+ t

(14)

p

1 E g(s, r , r) 2p H ds (t − s)β (s − r )α+ (t − r )(1−β)(p−1)+α+ E g(s, r , r) 2p H ds. (t − s)β (s − r )α+

Combining the above calculations, we obtain 3 4  p  (1−β)p E g(t, r , r) 2p  C (r − r) + (r − r) · x0 2p T,p H H−α  t (t − r )(1−β)(p−1)+α+ + CT,p E g(s, r , r) 2p H ds. β (s − r )α+ (t − s)  r 

The result follows by Lemma 2.2. Set f (t, r; x0 ) := (t − r)

α+γ 2



t

Tt−s Bk X(s, r; x0 )dW k (s). r

Lemma 2.6. Let α ∈ [0, 1), γ ∈ [0, 1 − β) and 0 <  < (α + γ) ∧ (1 − β − γ) ∧ (1 − α).

(15)

For any p
1 and T > 0, there exists a positive constant CT,p such that for all x0 ∈ H and 0  r  t  T , 0  r  t  T , 2p  p  p E f (t , r ; x0 ) − f (t, r; x0 ) 2p Hγ  CT,p (|t − t| + |r − r| ) · x0 H−α .

(16)

90

B. Goldys and X. Zhang

Proof. For 0  r < t  t  T , we have 3 4  t α+γ α+γ f (t , r; x0 ) − f (t, r; x0 ) = (t − r) 2 − (t − r) 2 Tt −s Bk X(s, r; x0 )dW k (s) + (t − r)



α+γ 2

r t

Tt −s Bk X(s, r; x0 )dW k (s) t

+ (t − r)



α+γ 2

t

(Tt −s − Tt−s )Bk X(s, r; x0 )dW k (s) r 

= : I1 (t , t) + I2 (t , t) + I3 (t , t). Note that |(t − r)

α+γ 2

− (t − r)

|  |t − t|

α+γ 2

α+γ 2

 |t − t| 2 · |t − r| 

α+γ− 2

.

For any p
1, by virtue of γ + β < 1 and (15), by Lemma 2.4 we have   p t ds 2p   p  (α+γ−)p E I1 (t ,t) Hγ  Cp |t − t| · |t − r| · · x0 2p H−α  γ+β (s − r)α r (t − s)  1

p (10) ds  Cp |t − t|p · |t − r|(1−−β)p · · x0 2p H−α γ+β sα (1 − s) 0  CT,p · |t − t|p · x0 2p . H−α Similarly, we have E I2 (t



, t) 2p Hγ

(17)



 Cp · (t − r)

(α+γ)p

·

 Cp · (t − r)

t

· t



ds

p

(t − s)γ+β (s − r)α p ds · x0 2p H−α (t − s)γ+β t

 γp

t

· x0 2p H−α

· x0 2p , H−α

 CT,p · |t − t|

(1−γ−β)p

and as in the proof of Lemma 2.4 E I3 (t , t) 2p Hγ



P (Tt −s − Tt−s )BX(s, r; x0 ) 2L2 (l2 ;Hγ ) ds

t

 Cp · (t − r)(α+γ)p · E r

  CT,p · (t − r)(α+γ)p · E r 

 CT,p · (t − t)

p

· (t − r)

t

p (t − t) L/2 Tt−s BX(s, r; x0 ) 2L2 (l2 ;Hγ ) ds

(α+γ)p

 · r

(10)

 CT,p · |t − t|p · x0 2p H−α .

t

ds (t − s)+γ+β (s − r)α

p · x0 2p H−α

Stochastic Flows for Nonlinear SPDEs

91

Combining the above calculations, we obtain that for all 0  r < t  t  T 2p  p E f (t , r; x0 ) − f (t, r; x0 ) 2p Hγ  CT,p · |t − t| · x0 H−α .

(18)

For 0  r < r < t  T we have 3 4 t α+γ α+γ f (t, r ; x0 ) − f (t, r; x0 ) = (t − r ) 2 − (t − r) 2 Tt−s Bk X(s, r; x0 )dW k (s) + (t − r )

α+γ 2

+ (t − r )

α+γ 2



r r

Tt−s Bk X(s, r; x0 )dW k (s) r



t

r 

Tt−s Bk (X(s, r ; x0 ) − X(s, r; x0 ))dW k (s)

=: J1 (r , r) + J2 (r , r) + J3 (r , r). As in the estimation of (17), we have  p E J1 (r , r) 2p Hγ  Cp · (r − r)

and as in the estimation of (14)  (1−β)p E J2 (r , r) 2p . Hγ  Cp · (r − r)

For J3 we have



t

1 X(s, r ; x0 ) − X(s, r; x0 ) 2H ds γ+β (t − s)  r  t

p−1 ds  (t − r )(α+γ)p γ+β (s − r )α+ r  (t − s)  t E((s − r )α+ X(s, r ; x0 ) − X(s, r; x0 ) 2H )p × ds (t − s)γ+β (s − r )α+ r  t (13) (r − r)p  CT (t − r )(1−−β)(p−1)+α+γ ds γ+β (s − r )α+ r (t − s)

 (α+γ)p E J3 (r , r) 2p E Hγ  (t − r )

p

(10)

 CT (t − r )(1−−β)p · (r − r)p .

Thus, for 0  r  r < t  T , we have 2p  p E f (t, r ; x0 ) − f (t, r; x0 ) 2p Hγ  CT,p · |r − r| · x0 H−α .

(19)

Moreover, similar to the above calculations, we also have for all 0  r < t  T 2p p E f (t, r; x0 ) 2p Hγ  CT,p · |t − r| · x0 H−α .

Estimation (16) now follows by (18), (19) and (20).

(20) 

Define a family of random linear operators by St,r (ω)x0 := X(t, r, ω; x0 ).

(21)

92

B. Goldys and X. Zhang Then we have:

Theorem 2.7. Assume that for some α ∈ (0, 1), L−α/2 is a Hilbert-Schmidt opera˜ t,r of St,r such that tor on H. Then for any γ ∈ [0, 1 − β), there exists a version S for P -almost all ω ˜ t,r (ω) ∈ L2 (H, Hγ ) is continuous, {(t, r) ∈ R2+ : r < t}  (t, r) → S and for any T > 0 and some random variable KT ∈ ∩p1 Lp (Ω, F, P ) ˜ t,r (ω) L (H,Hγ )  (t − r)−(α+γ)/2 · KT (ω), ∀0  r < t  T. S 2

(22)

Proof. We decompose St,r as the following two operators: (1)

(2)

St,r x0 = St,r x0 + St,r x0 , where

 (1) St,r x0

:= Tt−r x0 ,

(2) St,r x0

t

Tt−s Bk X(s, r; x0 )dW k (s).

:= r

Let {ek , k ∈ N} be an orthonormal basis of H. It is clear that  (1)   Cα Cα St,r ek 2Hγ = Tt−r ek 2Hγ  L−α/2 ek 2H  . α+γ (t − r) (t − r)α+γ k

k

k

(2)

Thus, we only need to prove that x0 → St,r (ω)x0 has the desired properties. For any p
1 and T > 0, by Minkowski’s inequality and Lemma 2.6 we have ∞ p  α+γ α+γ (2) (2) E (t − r ) 2 St ,r  ek − (t − r) 2 St,r ek 2Hγ k=1



=E  

∞ 

p 



f (t , r ; ek ) −

f (t, r; ek ) 2Hγ

k=1 ∞ ) 





E f (t , r ; ek ) −

f (t, r; ek ) 2p Hγ

k=1

 

 CT,p · (|t − t|

p



+ |r − r| ) p

∞ 

*1/p

p p

−α/2

L

ek 2H

k=1

 CT,p · (|t − t|p + |r − r|p ), ∀r  t, r  t ∈ [0, T ]. (2)

(2)

˜ By Kolmogorov’s continuity criterion, there exists a version S t,r of St,r such that ) * α+γ ˜ (2) ek {(t, r) ∈ R2+ : r  t}  (t, r) → (t − r) 2 S ∈ L2 (l 2 ; Hγ ) t,r k∈N

is locally H¨older continuous, and for any p
1 and T > 0  p ∞  α+γ (2) 2 ˜ 2 E sup (t − r) St,r ek Hγ < +∞. 0
r
t
T k=1

Stochastic Flows for Nonlinear SPDEs

93

Therefore, if we define for x0 ∈ H ˜ (2) (ω)x0 = S t,r

∞ 

(2)

˜ (ω)ek , x0 , ek H · S t,r

k=1

then

˜ (2) S t,r

(2) St,r

is a version of E sup

and for any p
1,

(t − r)

0
r
t
T

α+γ 2

˜ (2) L (H,Hγ ) S t,r 2

p < +∞. 

The proof is thus complete. We now turn to the following nonlinear stochastic evolution equation: dY (t) = −LY (t)dt + F (Y (t))dt + [Bk Y (t) + Dk (t)]dW k (t), Y (0) = y0 ,

(23)

where for some γ ∈ [0, 1 − β) (β from (7)), F (y1 ) − F (y2 ) H  CF · y1 − y2 Hγ and



(24)

T

D(t) 2L2 (l2 ;H−β ) dt < +∞, ∀T > 0.

0

(25)

Theorem 2.8. Assume that (7), (24) and (25) hold. Then, there exists a unique (Ft )-adapted solution Y (t, ω; y0 ) such that for P -almost all ω ∈ Ω (0, ∞) × H  (t, y0 ) → Y (t, ω; y0 ) ∈ Hγ is continuous. Proof. Let Z(t) ∈ C((0, ∞); Hγ ) solve the following equation:  t Z(t) = Tt−s [Bk Z(s) + Dk (s)]dW k (s).

(26)

0

Then the unique solution of equation (23) can be represented by using the variation of constant formula as  t ˜ t,0 (ω)x0 + ˜ t,s (ω)F (Y (s, ω; x0 ))ds + Z(t, ω), Y (t, ω; x0 ) = S S (27) 0

˜ t,s is given in Theorem 2.7. where S Indeed, note that by definition (21)  t ˜ s,0 x0 dW k (s) = St,0 x0 − Tt x0 Tt−s Bk S

(28)

0

and by stochastic Fubini’s theorem  s

 t ˜ s,r F (Y (r; x0 ))dr dW k (s) Tt−s Bk S 0

 t

0 t

˜ s,r F (Y (r; x0 ))dW k (s)dr Tt−s Bk S

= 

0

r t

˜ t,r F (Y (r; x0 )) − Tt−r F (Y (r; x0 ))]dr. [S

= 0

(29)

94

B. Goldys and X. Zhang

By (26)–(29), we find that

 t Y (t; x0 ) = Tt x0 + Tt−s F (Y (s; x0 ))ds 0  t + Tt−s [Bk Y (s; x0 ) + Dk (s)]dW k (s). 0

Thus, from (27) we have for any T > 0, ˜ t,0 (ω)(x0 − y0 ) Hγ Y (t, ω; x0 ) − Y (t, ω; y0 ) Hγ  S 8 t 8 8 8 ˜ 8 +8 St,s (ω)(F (Y (s, ω; x0 )) − F (Y (s, ω; y0 )))ds8 8 0

(22)(24)



t

− α+γ 2



t

+ 0

Hγ

KT (ω) x0 − y0 H CF · KT (ω) (t − s)

α+γ 2

Y (s, ω; x0 ) − Y (s, ω; y0 ) Hγ ds.

By [5, Lemma 7.1] we get Y (t, ω; x0 ) − Y (t, ω; y0 ) Hγ  CT t−

α+γ 2

KT (ω) x0 − y0 H . 

The proof is complete.

Remark 2.9. We shall use this abstract result to prove Theorem 1.1. We want to emphasize that Condition (7) is rather strong so that we can only apply it to the stochastic equation with spatial dimension 1 (see Lemma 3.1 below).

3. Proof of Theorem 1.1 2

∂ In order to use Theorem 2.8, we let H := L2 (0, 1) and L := − ∂x 2 with the domain

D(L) := {φ ∈ L2 (0, 1) : Lφ ∈ L2 (0, 1)} = H01 (0, 1) ∩ H 2 (0, 1), where H01 (0, 1) and H 2 (0, 1) are the usual Sobolev spaces. It is clear that (L, D(L)) is a positive self-adjoint and densely defined operator in L2 (0, 1), and generates an analytic semigroup (Tt )t0 , which is explicitly given by the fundamental solution Gt (x, y) as follows:  1 Tt φ(x) = Gt (x, y)φ(y)dy. (30) 0

The eigenvalues and normalized eigenfunctions of L are given by √ λn = π 2 n2 , en (x) = 2 sin(πnx), n ∈ N, i.e, Len (x) = λn en (x). For α ∈ R, the fractional power Lα is defined by Lα φ(x) =

∞  n=1

λα n φ, en H en ,

Stochastic Flows for Nonlinear SPDEs and Hα can be characterized by . H :=

φ ∈ L (0, 1) : φ Hα =

α

2

∞ 

95

: λα n

· |φ, en H | < +∞ . 2

n=1

From this, it is easy to see that L−α is a Hilbert-Schmidt operator on H for any α > 14 . Define



(31)

1

W k (t) =

ek (y)W (t, dy), 0

then {W k , k ∈ N} is a sequence of independent standard Brownian motions. Let g(s, x) be a measurable (Ft )-adapted process with  T 1 E |g(s, x)|2 dsdx < +∞, ∀T > 0. 0

0

By elementary process approximation, one may prove that  t 1  t g(s, y)W (ds, dy) = g(s), ek H dW k (s). 0

0

k

0

So, the stochastic integral occurring in (4) can be rewritten as  t 1 Gt−s (x, y)[b(y)u(s, y) + c(y)]dW (ds, dy) 0

0

=

 t  0

k (30)

=

 k

1

Gt−s (x, y)[b(y)u(s, y) + c(y)]ek (y)dydW k (ds)

0 t

[Tt−s (bu(s)ek + cek )](x)dW k (ds).

0

Thus, if we define F (u)(x) := f (x, u(x)) and Bk u = b · u · ek , Dk := c · ek , then equation (4) takes the form (23). Lemma 3.1. For any 1 > β > 12 , D ∈ L2 (l2 ; H−β ) and B is a bounded linear operator from H to L2 (l2 ; H−β )). Proof. Note that L−α also has the following expression:  ∞ 1 L−α = tα−1 Tt dt, Γ(α) 0 where Γ(α) is the usual Gamma function.

96

B. Goldys and X. Zhang By Minkowski’s inequality, we have

2    1  ∞ β 1 −β/2 2 −1 2 L (buek ) H = t [Tt (buek )](x)dt dx Γ(β/2) 0 0 k k   1/2 2  ∞ 1 β 1   t 2 −1 |[Tt (buek )](x)|2 dx dt Γ(β/2) 0 0 

1 = Γ(β/2)

(30)

k

∞

t



β 2 −1

0

1

0

1

1/2 2 |Gt (x, y)b(y)u(y)|2 dydx dt

0

1 =: (I1 + I2 )2 , Γ(β/2) where 

∞

I1 =

t 

β 2 −1

1 1

I2 =

β

t 2 −1

0



1



1/2

1

|Gt (x, y)b(y)u(y)| dydx 2

0



1

0



0

1

dt,

1/2 |Gt (x, y)b(y)u(y)|2 dydx dt.

0

From the expression of Gt (x, y), it is easy to see that % & 1 |x − y|2 Gt (x, y)  √ exp − . 4t 4πt Thanks to b ∈ L∞ (0, 1) and β > 12 , we have 1 I1  √ 4π and



∞

t

β 3 2−2



1

1



1/2

1

|b(y)u(y)| dy 2

0

dt  C u H

0

 ∞

1/2  1  1 |x|2 β 1 3 I2  √ t2 −2 e− 2t dx |b(y)u(y)|2 dy dt 4π 0 −∞ 0  1 β 5 C t 2 − 4 dt · u H  C u H . 0

Hence, for any u ∈ H = L2 (0, 1), Bu L2 (l2 ;H−β ) =



L−β/2 (buek ) 2H  C u 2H .

k



The proof is complete. Proof of Theorem 1.1. For any p
2, one chooses p. 243, (4.10)]) Hγ ⊂ Lp (0, 1).

1 2

− p1 < γ <

1 2

such that (cf. [7,

Stochastic Flows for Nonlinear SPDEs

97

Next we choose β with 12 < 1 − γ < β < 1. The first conclusion of Theorem 1.1 now follows from (31), Lemma 3.1 and Theorem 2.8. As for the flow property, it is a direct consequence of [1, p. 17, Theorem 1.3.2]. 

References [1] L. Arnold: Random dynamical systems. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. [2] F. Flandoli: Regularity theory and stochastic flows for parabolic SPDEs. Stochastics Monographs, 9. Gordon and Breach Science Publishers, Yverdon, 1995. x+79 pp. [3] F. Flandoli: Stochastic flows for nonlinear second-order parabolic SPDE. Ann. Probab., 24 (1996), no. 2, 547–558. [4] F. Flandoli and H. Lisei: Stationary Conjugation of Flows for Parabolic SPDEs with Multiplicative Noise and Some Applications. Stoch. Anal. and Appl., Vol. 22, No. 6, pp. 1385–1420, 2004. [5] D. Henry: Geometric Theory of Semilinear Parabolic Equations. Lect. Notes in Math., 840, Springer-Verlag, 1980. [6] H. Kunita: Stochastic flows and stochastic differential equations. Cambridge Univ. Press, 1990. [7] A. Pazy: Semi-groups of linear operators and applications. Berlin: Springer-Verlag, 1985. [8] A.V. Skorohod: Random Linear Operators. Reidel, Dordrecht, 1984. [9] J.B. Walsh: An introduction to stochastic partial differential equations. Lecture Notes in Math., 1180 Springer, 1986, pp. 266–437. [10] X. Zhang: Homeomorphic flows for multi-dimensional SDEs with non-Lipschitz coefficients. Stochastic Processes and their Applications, 115 (2005) 435–448, 116 (2006) 873–875. Benjamin Goldys School of Mathematics and Statistics The University of New South Wales Sydney, 2052, Australia e-mail: [email protected] Xicheng Zhang School of Mathematics and Statistics Wuhan University Wuhan, Hubei 430072, P.R. China e-mail: [email protected]

Progress in Probability, Vol. 65, 99–120 c 2011 Springer Basel AG


Optimal Stopping Problem Associated with Jump-diffusion Processes Yasushi Ishikawa Abstract. In this paper we study an optimal stopping problem associated with jump-diffusion processes. We use a viscosity solution approach for the solution to HJB equality, which the value function should obey. Using the penalty method we obtain the existence of the value function as a viscosity solution to the HJB equation, and the uniqueness. Mathematics Subject Classification (2000). 60J25, 60J75. Keywords. Optimal stopping; L´evy process; viscosity solution.

1. Introduction In this paper, we study the theory of optimal stopping problems arising from mathematical finance. Consider a wealth process X(t) consisting of a stock and a saving account components. To model this process we introduce the following processes. B(t) = one-dimensional the standard Brownian motion Z(t) = one-dimensional L´evy process with c`adl` ag path, σ = the non-negative diffusion constant, σ ≥ 0 We consider an optimal stopping problem associated with a L´evy process z(t) = rt + σB(t) + Z(t). Here rt (r ∈ R) denotes the drift term. The condition σ ≥ 0 means that it may lose the diffusion component (σ = 0). In the market a derivative asset called an American option is introduced. This is a contract between the two parts, in which the holder buys the right to sell or receive at time τ that he or she can choose. In the case of the put option, the reward function is given by g(x) = (K − x)+ . Let g(x) denote the reward function in general. An investor would like to fix the exercise time or stopping time τ to maximize the expected value of the sum in terms of g. The maximum expected value is called the value function. Let S denote the set of stopping times.

100

Y. Ishikawa

To this end we solve below a one-dimensional linear variational inequality (HJB equation, see (4) below) with one-sided boundary condition, associated with the above processes. We use an analytic method, called penalty method. We assume that the support of the L´evy measure is one-sided, which enables us to solve the boundary problem in the framework of pseudo-differential operators. See Cancelier [11] and [23] for previous treatments to such a problem. To solve the HJB equation, traditionally an heuristic method has been employed. That is, first we make a guess for the solution, and then we verify the guess is correct by the verification theorem (cf. [28]). Whereas the penalty method enables us to construct the solution to the HJB equation in the framework of the viscosity solution, even in the case that σ may be degenerate. The penalty method also can be extended to non-linear models such that σ depends on x. The formulation of the optimal stopping problem has been developed by Bensoussan and Lions [10]. We give the expression for the value function (Theorem 5.5). We obtain it as a “viscosity solution” of the HJB equation, and show (Theorem 5.3) the uniqueness in the class of continuous functions. However we could not give an expression for the optimal stopping time in closed form. See the second assertion of Theorem 5.5 and the comment (iii) in Sect. 2. As for the systematic study for the multi-dimensional HJB equation with boundary conditions, we refer to Pham [29]. See also Arisawa [1], Barles-Imbert [5]. In the case of the optimal stopping problem with respect to one-dimensional diffusion process, Alvarez [3] used a similar approach as ours. In the jump-diffusion case with arbitrary L´evy measure, Barles, Buckdahn and Pardoux [4] studies the uniqueness of the viscosity solution associated with the Backward S.D.E. However they did not considered the HJB equation. The organization of this paper is as follows. In Section 2, we state the framework of our theory. In Section 3 we introduce the penalized equation. We show the unique existence of the solution uε to this equation for each ε > 0. We further introduce the notion of the viscosity solution to the penalized equation, and proclaim that uε above is in fact a viscosity solution. In Section 4 we let ε → 0 to obtain the weak solution. We use an argument of approximation in the Banach space of continuous functions. In Section 5 we show that the weak solution in Section 4 is in fact a viscosity solution to the original variational inequality (HJB equation). Further we show the uniqueness. The solution v is expressed in terms of the stopping time, and we give at the last part of Section 5 an expression of this optimal stopping time. Due to this approach, the regularity of the value function can not be expected.

Optimal Stopping Problems

101

2. Problem setting We assume that the wealth price X = {X(t)} evolves according to the onedimensional stochastic differential equation of jump-diffusion type    dX(t) =

r+ |z|<1

(ez − 1 − z)µ(dz) X(t)dt + σX(t)dB(t)



+ X(t−) |z|<1

˜ (dtdz) (ez − 1)N

 + X(t−)

|z|≥1

(ez − 1)N (dtdz),

X(0) = x > 0,

(1)

on a complete probability space (Ω, F , P ), carrying a standard Brownian motion {B(t)} and a Poisson random measure N (dtdz), endowed with the natural filtration Ft generated by (B(s), s ≤ t) and (N (dsdz), s ≤ t). We assume {B(t)} and ˜ (dtdz) denotes the {N (dsdz)} are independent. Here σ ≥ 0 is a constant, and N ˜ (dtdz) = N (dtdz) − dtµ(dz), where µ(.) compensated Poisson radom measure N is a L´evy measure. The process X(t) has an exponential form with a modified log-process: %

& 1 2 X(t) = x exp rt − σ t + σB(t) + Z(t) , 2 where  t  t ˜ Z(t) = z N (dsdz) + zN (dsdz). 0

|z|<1

0

|z|≥1

We assume that the L´evy measure µ is non-degenerate, and satisfies  (ez − 1)µ(dz) < ∞. (A0) |z|≥1

Here “non-degenerate” means, for some β ∈ (0, 2), we have (A00)

lim inf →0

ϕ() > 0, β



where ϕ() =

|z|≤

|z|2 µ(dz).

Conditions (A0), (A00) are fairly general and we can choose, for example, a discrete measure ∞  µ(dz) = ck δ{ak } , k=0

where ck > 0 and ak → 0 as k → ∞ obeying (A0). Or, the L´evy measure of α-stable type (where α = 2 − β ∈ (0, 2)), and that of generalized inverse Gaussian type also apply. (The measure above is an extension of the one treated in [25, Sect. 3].)

102

Y. Ishikawa We put

 r˜ = r + R\{0}

(ez − 1 − z · 1|z|<1 )µ(dz)

for simplicity. The value of r˜, which corresponds to the intrinsic drift, will play an important role in the sequel. Then X(t) can be written  ˜ (dtdz), X(0) = x > 0. (1) dX(t) = r˜X(t)dt+σX(t)dB(t)+X(t−) (ez −1)N We assume supp µ ⊂ [0, +∞)

(A1)

for simplicity. (See the next section for the detailed reason.) This implies  (ez − 1 − z · 1|z|<1 )µ(dz) > 0. R\{0}

Further we assume (A2)

r˜ > 0.

By (A1) the process X(t) has no negative jumps. Hence, starting from x > 0, X(t) diffuses around {x} (when σ > 0), or moves rightward by jumps. The reward function g(x) is assumed to have the following property: g ≥ 0,

g ∈ C,

(2)

where C denotes the Banach space of all continuous functions on [0, ∞) vanishing at infinity, with norm h = supx∈[0,∞) |h(x)|. We fix a g in the sequel. A mathematical objective of this article is to find an optimal stopping time τ ∗ so as to maximize the expected reward function: J(τ ) = E[e−˜rτ g(X(τ ))]

(3)

over the class S of all stopping times τ associated with (Ft ), where e−˜rτ g(Xτ ) at τ = ∞ is interpreted as zero. Here and in what follows E denotes the expectation with respect to the probability law such that X(0) = x. This function, viewed as a function of the starting point x, is called a value function, denoted by v(x). That is, v(x) = supτ ∈S J(τ ). To solve this problem, we consider the variational inequality (HJB equation, for short): . max(Lu, g − u) = 0 in (0, ∞), (4) u(0) = g(0), with the interpretation u(0) = u(0+). Here L is an integro-differential operator given by 1 Lu = −˜ ru + σ 2 x2 u + r˜xu 2 +

{u(x + γ(x, z)) − u(x) − u (x) · γ(x, z)}µ(dz)

Optimal Stopping Problems

103

where γ(x, z) = x(ez − 1). We write Lu = −˜ r u + L0 u in the sequel. The condition (2) is fulfilled if the reward function is given by the bounded function g(x) = (K − x)+ · 1{x≥0} for the strike price K > 0 of an european put option. It may also hold for some type of call option g with an overhead barrier: choose g ∈ C such that g(x) = (x − K)+ .1{x>0} if x ≤ M , g(x) = 0 if x > M + 1 for some large M > K > 0. Remarks. (i) Suppose that the variational inequality (4) admits a solution v ∈ C 2 ([0, ∞)). Then the optimal stopping time τˆ is given by τˆ = inf{t > 0 : v(X(t)) ≤ g(X(t))}. Indeed, assuming τˆ < ∞ a.e., from (4) it follows that Lv = 0 if

v > g.

Hence Lv(X(t)) = 0 for t < τˆ. By Itˆ o’s formula, under some additional assumptions on v, we obtain  τˆ ' −˜rτˆ ( E e v(X(ˆ τ )) = v(x) + E e−˜rt Lv(X(t))dt #

0

+E 

τˆ

e−˜rt v (X(t))σX(t)dB(t)

0

τˆ

+



−˜ rt

e

$ ˜ (v(X(t) + γ(X(t), z)) − v(X(t)))N (dsdz)

0

= v(x). Thus

E[e−˜rτˆ g(X(ˆ τ ))] ≥ E[e−˜rτˆ v(X(ˆ τ ))] = v(x). On the other hand, since Lv ≤ 0,

Itˆ o’s formula gives

E[e−˜rτ v(X(τ ))] ≤ v(x), τ ∈ S. We assume v is bounded, and choose τ = τˆ. Therefore we might seem to obtain the optimality of τˆ, and we have Φ(x) = v(x), where Φ(x) = supτ J x (τ ). However, we remark that v ∈ C 2 may be violated, because v is connected at some point x to g who is supposed to be only continuous and because we admit σ to be degenerate. (In [28] Chap. 2 a similar problem is studied with σ = 1, cf. [28] p. 43. A similar work [22] is carried out under the assumption σ > 0.) This is the reason why we use the penalty method.

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(ii) There are several definitions of viscosity solutions concerning the HJB equation with respect to integro-differential operators, and some of them are equivalent. We adopt below the one introduced by Benth et al. [7], [8], [9], which goes in accordance with those given by Pham [29]. There are other attempts (e.g., [1], [2], [5]). (iii) Since L satisfies the positive maximum principle, L can be viewed as a pseudodifferential operator with the symbol a(x, ξ) given by a(x, ξ) = a1 (x, ξ) + a2 (x, ξ), where 1 a1 (x, ξ) = −˜ r − σ 2 x2 ξ 2 + i˜ rxξ 2  a2 (x, ξ) = {eiξγ(x,z) − 1 − iξ · γ(x, z)}µ(dz). The symbol of L0 is given by (a1 (x, ξ) + r˜) + a2 (x, ξ). By the assumption that σ ≥ 0, the symbol a1 may not be elliptic. On the other hand, we observe due to (A00) the symbol a2 satisfies |a2 (x, ξ)| ≥ c(x)|ξ|α for each x ∈ R. Here α = 2 − β ∈ (0, 2). Hence a(x, ξ) is a non-degenerate symbol of order α.

3. Penalized problem In this section, we show the existence of a unique solution u of the equation (4). To solve (4), we need to study the penality equation for ε > 0:  1   r˜u = L0 u + (u − g)− in (0, ∞), ε (5) g(0)   u(0) = , r˜ε + 1 originated by Bensoussan and Lions [10]. Here and in what follows we denote (x)+ = max(x, 0) and (x)− = max(−x, 0) for x ∈ R. At the point x = 0, it is known from [20] that the “trace” u(0+) exists finite under the assumption (A1), due to the transmission condition. Remark. In general we have ez − 1 ≥ −1 for z ∈ R, and hence ∆z(t) = z(t−)(ez − 1) ≥ −z(t−) if z(t−) ≥ 0 at each jump corresponding to z ∈ supp µ. This implies z(t) = z(t−) + ∆z(t) ≥ 0 if z(t−) ≥ 0 at each jump moment t. However, since we may have infinitely many jumps in a finite time interval, we can not guarantee in general the existence of the trace of u in (5) from the right at z = 0 if we admit the negative jumps (i.e., supp µ ∩ (−∞, 0) = ∅).

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Hence we may search the solution u in C (cf. Theorems 3.1, 4.1 below). We begin with a probabilistic penalty equation for u: # ∞ $ −(˜ r+ 1ε )t 1 u(x) = E e (u ∨ g)(X(t))dt , ε 0 for x ≥ 0, ε > 0, with the boundary condition u(0) =

(6)

g(0) . r˜ε+1

Theorem 3.1. We assume (2), (A0), (A00), (A1) and (A2). Then, for each ε > 0, there exists a unique nonnegative solution u = uε ∈ C of (6). Proof. Define

#

T h(x) = E

∞

0

$ 1 1 e−(˜r+ ε )t (h ∨ g)(X(t))dt ε

for h ∈ C+ ,

(7)

where C+ = {h ∈ C : h ≥ 0}. Clearly, C+ is a closed subset of C. By (7), we have # ∞ $ 1 1 0 ≤ T h(x) = E e−(˜r+ ε )t (h ∨ g)(X(t))dt ε 0  ∞ 1 1 ≤ h ∨ g e−(˜r+ ε )t dt ε 0 h ∨ g = ≤ h ∨ g . r˜ε + 1 Then,

#

∞

|T h(y) − T h(x)| ≤ E

e

−(˜ r+ 1ε )t 1

0

→0

ε

$ {|h ∨ g(X(t)) − h ∨ g(Y (t))|}dt

as y → x,

where {Y (t)} be the solution of (1) with the initial condition Y (0) = y > 0. This is true since |h ∨ g(X(t)) − h ∨ g(Y (t))| ≤ 2 h ∨ g where h ∨ g is bounded, and since we can use the dominated convergence theorem. Moreover, we have # ∞ $ −(˜ r+ 1ε )t 1 T h(x) = E e (h ∨ g)(X(t))dt → 0 as x → ∞. ε 0 Indeed, we denote X(t) = X x (t) for the dependence of X(t) on x. We see X x (t) → ∞ as x → ∞ by the expression of X x (t) in Section 2. Therefore (h∨g)(X x (t)) → 0 as x → ∞. Then  ∞ 1 1 e−(˜r+ ε )t (h ∨ g)(X x (t))dt → 0, x → ∞ a.s. ε 0 by the dominated convergence theorem. Therefore T h(x) → 0 as x → ∞ by the dominated convergence theorem again.

106

Y. Ishikawa Thus T maps C+ into C+ . Now, by (7), we have

|T (h1 − h2 )(x)| = |T h1 (x) − T h2 (x)| # ∞ $ 1 1 ≤E e−(˜r+ ε )t |h1 (X(t)) − h2 (X(t))|dt ε #0 ∞ $ 1 1 ≤E e−(˜r+ ε )t h1 − h2 dt ε 0 1 = h1 − h2 , h1 , h2 ∈ C+ . r˜ε + 1 This yields that T is a contraction mapping. Thus T has a fixed point u in C, which solves (6).  We fix ε > 0 temporarily. Theorem 3.2. We make the assumption of Theorem 3.1. Then the solution u in (6) is a viscosity solution of 1 r˜u = L0 u + (u − g)− ε

on

(0, ∞),

with boundary condition u(0) = r˜g(0) ε+1 . Further, for any τ ∈ S, we have # τ $ 1 u(x) = E e−˜rt (u − g)− (X(t))dt + e−˜rτ u(X(τ )) . ε 0 In particular,

# u(x) = E 0

∞

$ 1 e−˜rt (u − g)− (X(t))dt . ε

(8)

(9)

(10)

Here we use the notion of the viscosity solution for the penalty equation in the following sense. Definition 3.3. Let w ∈ C([0, ∞)) and w(0) = g(0). Then w is called a viscosity sub- or super-solution of (8) in the following manner; (a) w is a viscosity subsolution of (8), if for any φ ∈ C 2 ((0, ∞)) and any local maximum point z > 0 of w − φ, 1 r˜w(z) ≤ L0 φ(z) + (w − g)− (z), ε and (b) w is a viscosity supersolution of (8), if for any φ ∈ C 2 ((0, ∞)) and any local minimum point z¯ > 0 of w − φ, 1 r˜w(¯ z ) ≥ L0 φ(¯ z ) + (w − g)− (¯ z ). ε The function w is called a viscosity solution of (8) if it is viscosity both super- and sub-solutions of (8).

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107

We postpone the proof of this theorem to the last section. Here we remark that, since u ∨ g = u + (u − g)− , we can rewrite (8) as

1 1 r˜ + u = L0 u + (u ∨ g) ε ε

in (0, ∞).

(11)

Remark. In general we can not guarantee the regularity nor the uniqueness of the solution uε of (8), in case σ is possibly degenerate. If σ > 0, we will be able to use the method of Garroni et al. [18] and lead to the regularity (i.e., the C 2 property) of u.

4. Passing to the limit as ε → 0 In this section, granting Theorem 3.2 for the moment, we study the convergence of u = uε ∈ C+ as ε → 0, where uε is a solution to (8). Define the Green function # ∞ $ Gβ h(x) = E e−βt h(X(t))dt , β > 0, (12) 0

and let G = {Gβ (βh) : h ∈ C, β > r˜}.

(13)

Our objective is to prove the following. Theorem 4.1. We assume (2), (A0), (A1) and (A2). Let εn > 0 be any (A00), ∞ sequence such that εn → 0 and that n=1 εn < ∞. Then we have uε n

→

v

(14)

where v is some element in C. For the proof of this theorem, we prepare the following lemmas (Lemmas 4.2–4.4). Lemma 4.2. The subset G is dense in C. Proof. Step 1. Let h ∈ C be arbitrary. It is clear that Gβ (βh) ≤ h . By the dominated convergence theorem, we get # ∞ $ |Gβ (βh)(y) − Gβ (βh)(x)| ≤ E e−βt β|h(Y (t)) − h(X(t))|dt → 0 as y → x, 0

and

# Gβ (βh)(x) = E

∞

e

−βt

$ βh(X(t))dt

0

as in Section 3. Thus G ⊂ C.

→ 0

as x → ∞

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Y. Ishikawa

Step 2. On the other hand, for each x ≥ 0, it is easy to see that # ∞ $ −βt |Gβ (βh)(x) − h(x)| ≤ E e β|h(X(t)) − h(x)|dt #0 ∞ $ −s =E e |h(X(s/β)) − h(x)|ds → 0

as

β → ∞,

0

by the right continuity of X(t) as t → 0. We extend h ∈ C on the compactification [0, ∞) ∪ {∞} as h(∞) = 0. Hence Gβ (βh)(x) → h(x)

as β → ∞,

x ∈ [0, ∞) ∪ {∞}.

Let Λ be any element in the dual space C ∗ of C. By the Riesz representation theorem ([31] Theorem 6.19), there exists a measure µ = µΛ on [0, ∞) ∪ {∞} such that  Λ(h) = hdµ for all h ∈ C. Let Λ be such that Λ(f ) = 0 for all f = Gβ (βh) ∈ G. By the dominated convergence theorem, we have Λ(h) = lim Λ(βGβ h) = 0, β→∞

h ∈ C.

Now, suppose that G¯ = C, where G¯ is the closure of G. Then, there exists h0 ∈ C \ G¯ such that δ := inf f − h0 > 0. f ∈G¯

By the Hahn-Banach theorem, (cf. [31] Theorem 5.19), we can find some Λ∗ ∈ C ∗ such that ¯ Λ∗ (h0 ) = δ, Λ∗ (f ) = 0 for all f ∈ G. This is a contradiction, since h above ranges among C. Therefore, we deduce G¯ = C.  Lemma 4.3. Let u ˜ ∈ C+ be the solution of (8) with g˜ ∈ C+ replacing g. Then we have u − u ˜ ≤ g − g˜ . (15) Proof. Assume, on the contrary, u − u ˜ > g − g˜ . Let T˜ denote T in (7) with g˜ replacing g. Then, by Theorem 3.1, we observe u = T u and u ˜ = T˜ u ˜. Therefore we have  ∞ 1 1 1 |u − u ˜| = |T u − T˜ u ˜| ≤ e−(˜r+ ε )t dt (u ∨ g) − (˜ u ∨ g˜) ≤ u − u ˜ . ε r˜ε + 1 0 Here we used the relation |a ∨ b − c ∨ d| ≤ |b − d|

if

|a − c| ≤ |b − d|,

a, b, c, d ∈ R.

Hence we see that 1 u − u˜ . r˜ε + 1 Hence u − u ˜ = 0, and g − g˜ = 0. This is a contradiction. u − u ˜ ≤



Optimal Stopping Problems

109

uε (x) = sup E[e−˜rτ {g − (uε − g)− }(X(τ ))].

(16)

Lemma 4.4. We have τ ∈S

Proof. Let x > 0. By (9), we have # τ $ 1 uε (x) = E e−˜rt (uε − g)− (X(t))dt + e−˜rτ uε (X(τ )) ε 0 ≥ E[e−˜rτ (uε ∧ g)(X(τ ))]. On the other hand, we take τ = τε := inf{t : uε (X(t)) ≤ g(X(t))}, and consider τε ∧ T, T > 0. We have by (9)  τε ∧T 1 uε (x) = E e−˜rt (uε − g)− (X(t))dt + e−˜r(τε ∧T ) uε (X(τε ∧ T )) . ε 0 Since e−˜rτε uε (X(τε )) = e−˜rτε (uε ∧ g)(X(τε )), we let T → ∞, and have # τε $ 1 uε (x) = E e−˜rt (uε − g)− (X(t))dt + e−˜rτε (uε ∧ g)(X(τε )) ε 0 = E[e−˜rτε (uε ∧ g)(X(τε ))]. Therefore uε (x) ≤ sup E[e−˜rτ (uε ∧ g)(X(τ ))]. τ ∈S

Comparing this with the above inequality, we have the relation (16).



We now can prove Theorem 4.1 Proof of Theorem 4.1. Step 1. We claim that (uε − g)− ≤ ε βh + (˜ r − β)g ,

(17)

if g = Gβ (βh) ∈ G for some h ∈ C. Indeed, by the same line as the proof of Theorem 3.2, we observe that g is a unique viscosity solution of . βg = L0 g + βh in (0, ∞), g(0) = h(0), or equivalently,



1  ˆ + 1g  r ˜ + g = L0 g + h  ε ε  ε  ˆ  g(0) = {h(0) + r˜ε + 1

in (0, ∞), 1 g(0)}, ε

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Y. Ishikawa

ˆ = βh + (˜ where h r − β)g (see the proof of Theorem 5.3 below for ˆ + 1 g). Therefore, by (6) Hence we have g = Gr˜+ 1ε (h ε # ∞ 1 1 ˆ uε (x) − g(x) = E e−(˜r+ ε )t { (uε ∨ g)(X(t)) − (h(X(t)) + ε 0 # ∞ $ 1 ˆ ≥ −E e−(˜r+ ε )t h(X(t))dt

the uniqueness).

1 g(X(t)))}dt ε

$

0

ˆ ≥ −ε h ,

x > 0,

which implies (17). Step 2. Let g = Gβ (βh) ∈ G. Applying (17) to uεn+1 (x) and uεn (x), by Lemma 4.4, we have |uεn+1 (x) − uεn (x)| ≤ sup E[e−˜rτ |(uεn+1 − g)− − (uεn − g)− |(X(τ ))] τ ∈S

≤ (εn+1 + εn ) βh + (˜ r − β)g . Thus ∞ 

uεn+1 − uεn ≤

n=1

∞ 

(εn+1 + εn ) βh + (˜ r − β)g < ∞.

n=1

This implies that {uεn } is a Cauchy sequence in C, and we get (14). Step 3. Let g satisfy (2). By Lemma 4.2, there exists a sequence {gm } ⊂ G such that gm → g in C. Let um ε be the solution of (8) corresponding to gm . By Step 2, we see that um εn

→

vm ∈ C

as

n → ∞.

(18)

By Lemma 4.3, 

m um εn − uεn ≤ gm − gm .

Letting n → ∞, we have 

v m − vm ≤ gm − gm . Hence {v m } is a Cauchy sequence, and vm

→

v ∈ C.

(19)

Thus m m m uεn − v ≤ uεn − um εn + uεn − v + v − v m m ≤ g − gm + um εn − v + v − v .

Letting n → ∞ and then m → ∞, we obtain (14). The limit does not depend on the choice of (εn ) and {gm } as long as (εn ) satisfies the condition in the theorem. 

Optimal Stopping Problems

111

5. Viscosity solutions of variational inequalities In this section, we study the viscosity solution of the original variational inequality: . max(Lv, g − v) = 0 in (0, ∞), (20) v(0) = g(0). First we recall the notation Lv = −˜ r v + L0 v. We repeat the definition of the viscosity solution associated to the non-local operator. Definition 5.1 (cf. [7], [8]). Let v ∈ C([0, ∞)) satisfies v(0) = g(0). Then v is called a viscosity solution of (20), if the following assertions are satisfied: (a) For any φ ∈ C 2 and for any local minimum point z¯ > 0 of v − φ, −˜ rv(¯ z ) + L0 φ(¯ z ) ≤ 0, (b) v(x) ≥ g(x) for all x ≥ 0, (c) For any φ ∈ C 2 and for any local maximum point z > 0 of v − φ, (−˜ r v + L0 φ)(v − g)+ |x=z ≥ 0. Theorem 5.2. We assume (2), (A0), (A00), (A1) and (A2). Then the limit v in Theorem 4.1 is a viscosity solution of (20). Proof. Let φ ∈ C 2 and let z > 0 be a local maximum point of v − φ such that ¯δ (z), z = x v(z) − φ(z) > v(x) − φ(x), x ∈ B for some δ > 0. Here Bδ (z) denotes the neighborhood of z with radius δ > 0, and ¯δ (z) denotes its closure. B By the uniform convergence in Theorem 4.1, the function uεn − φ attains a ¯δ (z). local maximum at xn ∈ B We deduce xn → z as n → ∞. ¯δ (z) of (uεn − φ), Indeed, for the sequence of local maximum points (xn ) in B  ¯ choose a subsequence (xnk ) of (xn ) so that for some z ∈ Bδ (z) xnk → z  . By Theorem 4.1 (uεnk − φ)(xnk ) → (v − φ)(z  ) and max (uεnk (x) − φ(x)) → max (v(x) − φ(x)).

¯ δ (z) x∈B

¯ δ (z) x∈B

¯δ (z), and hence (v − φ)(z  ) ≥ (v − φ)(z). Hence (v − φ)(z  ) ≥ (v − φ)(x), x ∈ B  Hence we have z = z. Now, by Definition 3.3, we have 1 −˜ r uεn (x) + L0 φ(x) + (uεn − g)− (x)|x=xn ≥ 0. εn

112

Y. Ishikawa

Multiply both sides by (uεn − g)+ to obtain (−˜ r uεn (xn ) + L0 φ(xn ))(uεn − g)+ (xn ) ≥ 0. Letting n → ∞, we get (−˜ rv(z) + L0 φ(z))(v − g)+ (z) ≥ 0. Next, by (17), we have − (um r − β)gm , εn − gm ) ≤ εn βhm + (˜

where gm = Gβ (βhm ) for some hm ∈ C and um εn is as in the proof of Theorem 4.1. Letting n → ∞, by (18), we have since the norm · · · above does not depend on n, vm (x) ≥ gm (x),

x ≥ 0,

and then, by (19) v(x) ≥ g(x)

for all x ≥ 0.

Finally, let z¯ be the minimizer of v − φ, and x ¯n be the sequence of the local minimizers of uεn − φ such that x¯n → z¯. Then, by Definition 3.3 −˜ r uεn (x) + L0 φ(x) +

1 (uε − g)− (x)|x=¯xn ≤ 0, εn n

from which −˜ ruεn (¯ xn ) + L0 φ(¯ xn ) ≤ 0. Letting n → ∞, we deduce −˜ rv(¯ z ) + L0 φ(¯ z ) ≤ 0. Thus we get the assertion of the theorem.



Theorem 5.3. We make the assumption of Theorem 5.2. Let vi ∈ C, i = 1, 2, be two viscosity solutions of (20). Then we have v1 = v2 . Proof. Step 1. Let φ ∈ C 2 and let z > 0 be the local maximizer of v1 − φ. If v1 ≤ v2 , then (v1 − v2 )+ = 0. If v1 > v2 , then, as v1 > v2 ≥ g, we have (v1 − g)+ > 0. By Definition 5.1(c), we have (−˜ r v1 (z) + L0 φ(z))(v1 − g)+ (z) ≥ 0. Hence −˜ rv1 (z) + L0 φ(z) ≥ 0. Thus (−˜ r v1 (z) + L0 φ(z))(v1 − v2 )+ (z) ≥ 0.

Optimal Stopping Problems

113

Here we put the super jet J 2,+ v(x) and subjet J 2,− v(x) as follows. % 2,+ J v(x) = (p, q) ∈ R × S 0 ; v(y) ≤ v(x) + p · (y − x) & 1 2 2 + q(y − x) + o(|x − y| ) as y → x , 2 % J 2,− v(x) = (p, q) ∈ R × S 0 ; v(y) ≥ v(x) + p · (y − x) & 1 2 2 + q(y − x) + o(|x − y| ) as y → x , 2 2,+ 2,− ¯ ¯ and J v(x), J v(x) denote their closures. Let 1 F (x, u, p, q, B δ (x, u, p), Bδ (x, u, p)) = −˜ ru+ σ 2 x2 q+rxp+B δ (x, u, p)+Bδ (x, u, p), 2 where  δ B (x, u, p) = {u(x + γ(x, z)) − u(x) − p · γ(x, z)}µ(dz), |z|>δ

and

 Bδ (x, u, p) =

|z|≤δ

{u(x + γ(x, z)) − u(x) − p · γ(x, z)}µ(dz).

We have the following basic result concerning the characterisation of superjets (subjets). Lemma 5.4 (cf. [8] Lemmas 3.1, 3.2). Let v be a subsolution (supersolution) in (0, ∞). Then for all δ > 0, all x ∈ (0, ∞), and all (p, q) ∈ J¯2,+ v(x) (resp. J¯2,− v(x)), there exists φ ∈ C 2 such that F (x, v, p, q, B δ (x, v, p), Bδ (x, φ, φ )) ≥ 0 (≤ 0). Further φ must satisfy φ (x) = p and φ (x) = q. Therefore, we obtain

1 −˜ r v1 (x) + σ 2 x2 q + rxp + B δ (x, v1 , p) + Bδ (x, φ, φ ) (v1 − v2 )+ (x) ≥ 0, 2 ∀(p, q) ∈ J¯2,+ v1 (x), ∀x > 0. Step 2. Suppose there exists x0 > 0 such that v1 (x0 ) − v2 (x0 ) > 0. Then we find η > 0 such that M0 := sup[v1 (x) − v2 (x) − 2ηψ(x)] > 0, x≥0

where ψ(x) = (1 + x2 )1/2 + ψ0 for a constant ψ0 > 0 chosen later. Define k Ψk (x, y) = v1 (x) − v2 (y) − |x − y|2 − η{ψ(x) + ψ(y)} 2

114

Y. Ishikawa

for k ∈ N. Then there exists a maximizer (xk , yk ) ∈ [0, ∞)2 of Ψk (x, y) such that Ψk (xk , yk ) = sup Ψk (x, y) ≥ max(Ψk (¯ x, x ¯), Ψk (xk , xk )), (x,y)

where x ¯ > 0 is a maximum point of M0 . Since Ψk (xk , yk ) ≥ Ψk (¯ x, x ¯), we have v1 (xk ) − v2 (yk ) −

k |xk − yk |2 − η{ψ(xk ) + ψ(yk )} ≥ M0 , 2

and k |xk − yk |2 < v1 (xk ) − v2 (yk ) − η{ψ(xk ) + ψ(yk )} 2 < v1 (xk ) − v2 (yk ) < M for some M > 0. Hence we have |xk − yk | → 0 as k → ∞. Further, since Ψk (xk , yk ) ≥ Ψk (xk , xk ), we have k |xk − yk |2 ≤ v2 (xk ) − v2 (yk ) + η(ψ(xk ) − ψ(yk )). 2 Here we let k → ∞, then the above implies that k|xk − yk |2 → 0. By extracting subsequences xk , yk , we obtain that xk → x ˜ , yk  → x ˜ for some x ˜. We denote this (xk , yk ) by the same (xk , yk ). Therefore v1 (˜ x) − v2 (˜ x) − 2ηψ(˜ x) > 0,

x ˜ > 0.

(21)

Now, applying Ishii’s lemma (cf. [14] Theorem 3.2) to Ψk (x, y), we obtain q1 , q2 ∈ R such that (k(xk − yk ), q1 ) ∈ J¯2,+ v˜1 (xk ), (k(xk − yk ), q2 ) ∈ J¯2,− v˜2 (yk ),     q1 0 1 −1 ≤ 3k , 0 −q2 −1 1 where v˜1 (x) = v1 (x) − ηψ(x), v˜2 (y) = v2 (y) + ηψ(y). Hence (ˆ p1 , qˆ1 ) := (k(xk − yk ) + ηψ  (xk ), q1 + ηψ  (xk )) ∈ J¯2,+ v1 (xk ), (ˆ p2 , qˆ2 ) := (k(xk − yk ) − ηψ  (yk ), q2 − ηψ  (yk )) ∈ J¯2,− v2 (yk ). By Definition 5.1(a) and Lemma 5.4, we have 1 −˜ rv2 (yk ) + σ 2 yk2 qˆ2 + ryk pˆ2 + B δ (yk , v2 , pˆ2 ) + Bδ (yk , φ2 , φ2 ) ≤ 0. 2 Also, by Step 1

1 −˜ rv1 (xk ) + σ 2 x2k qˆ1 + rxk pˆ1 + B δ (xk ,v1 , pˆ1 ) + Bδ (xk ,φ1 ,φ1 ) (v1 − v2 )+ (xk ) ≥ 0. 2 Hence, by (21) 1 −˜ r v1 (xk ) + σ 2 x2k qˆ1 + rxk pˆ1 + B δ (xk , v1 , pˆ1 ) + Bδ (xk , φ1 , φ1 ) ≥ 0 2

Optimal Stopping Problems

115

for sufficiently large k. Putting these inequalities together, we get r˜{v1 (xk ) − v2 (yk )} 1 ≤ σ 2 (x2k qˆ1 − yk2 qˆ2 ) + r(xk pˆ1 − yk pˆ2 ) 2 !  + v1 (xk + γ(xk , z)) − v2 (yk + γ(yk , z)) |z|>δ

 + |z|≤δ

 " p1 · γ(xk , z) − pˆ2 · γ(yk , z) µ(dz) − v1 (xk ) − v2 (yk )) − (ˆ !    φ1 (xk + γ(xk , z)) − φ2 (yk + γ(yk , z)) − φ1 (xk ) − φ2 (yk )  " − φ1 (xk ) · γ(xk , z) − φ2 (yk ) · γ(yk , z) µ(dz)

≡ I1 + I2 + I3 + I4 (say). Here we remark that, from Lemma 5.4, we can choose φ1 and φ2 such that φ1 (xk ) = pˆ1 , φ1 (xk ) = qˆ1 , φ2 (yk ) = pˆ2 and φ2 (yk ) = qˆ2 . Letting k → ∞, we have I1 ≤

σ2 [3k|xk − yk |2 + η{x2k ψ  (xk ) + yk2 ψ  (yk )}] → σ 2 η˜ x2 ψ  (˜ x), 2

and I2 = r[k|xk − yk |2 + η{xk ψ  (xk ) + yk ψ  (yk )}]

→ 2rη˜ xψ  (˜ x).

On the other hand,  , I3 = (v1 (xk + γ(xk , z)) − v2 (yk + γ(yk , z))) − (v1 (xk ) − v2 (yk )) |z|>δ

− (k(xk − yk )2 (ez − 1) + η(γ(xk , z)ψ  (xk ) + γ(yk , z)ψ  (yk ))) µ(dz).

By (A0), (A00), µ is a bounded measure on {|z| > δ}, for which a constant times (ez −1) is integrable. As k → ∞, xk → x ˜ and yk → x ˜, and we have by the Lebesgue comvergence theorem   → −2ηψ (˜ x ) γ(˜ x, z)µ(dz). I3 |z|>δ

On the other hand,  , I4 = (φ1 (xk + γ(xk , z)) − φ2 (yk + γ(yk , z))) − (φ1 (xk ) − φ2 (yk )) |z|≤δ

− (φ1 (xk ) · γ(xk , z) − φ2 (yk ) · γ(yk , z)) µ(dz).

Here the integrand {· · · } is O((ez − 1)2 ) = O(|z|2 ) as |z| → 0 uniformly in k. Hence we can use the Lebesgue convergence theorem, and have  I4 → −2ηψ  (˜ x) γ(˜ x, z)µ(dz), k → ∞. |z|≤δ

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Thus, choosing ψ0 > 0 such that 12 σ 2 ≤ r˜ψ0 , we obtain % &  1 2 2    σ x r˜{v1 (˜ x) − v2 (˜ x)} ≤ 2η ˜ ψ (˜ x) + r˜ xψ (˜ x) − x ˜ψ (˜ x) γ(˜ x, z)µ(dz) 2

1 2 2 1 x ˜ ≤ 2η σ x ˜ + r˜ x√ 2 3/2 2 (1 + x˜ ) 1+x ˜2

1 2 σ + r˜x ≤ 2η ˜2 (1 + x ˜2 )−1/2 2 % & 1 2 2 1/2 2 ≤ 2η σ (1 + x ˜ ) + r˜(1 + x ˜ ) (1 + x ˜2 )−1/2 2 ≤ 2η{˜ rψ0 (1 + x˜2 )1/2 + r˜(1 + x ˜2 )}(1 + x ˜2 )−1/2 = 2˜ rηψ(˜ x). 

This is the contrary to (21). The proof is finished. Theorem 5.5. We make the assumption of Theorem 5.2. Then we have v(x) = sup E[e−˜rτ g(X(τ ))]. τ ∈S

Further, the optimal stopping time τ ∗ is given by τ ∗ = lim m m→∞

for x > 0, where m = inf{t ≥ 0 : v(X(t)) −

1 m

≤ g(X(t))}.

Proof. Let x > 0 and τ ∈ S. By (9), we get # τ $ −˜ rt 1 − −˜ rτ uεn (x) = E e (uε − g) (X(t))dt + e uεn (X(τ )) ε n 0 ' ( ≥ E e−˜rτ uεn (X(τ )) . Letting n → ∞, by Theorems 4.1 and 5.2, we have v(x) ≥ E[e−˜rτ v(X(τ ))] ≥ E[e−˜rτ g(X(τ ))]. Since v(X(t)) −

1 > g(X(t)) on m

{t < m },

we have # m $ # m $ 1 − −˜ rt − −˜ rt E e (uεn − g) (X(t))dt ≤ E e (uεn − (v − )) (X(t))dt m 0 #0 m $ 1 −˜ rt − ≤E e ( − uεn − v ) (X(t))dt m 0 =0 (22)

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117

for sufficiently large n. Hence, by (9) # m $ 1 uεn (x) = E e−˜rt (uεn − g)− (X(t))dt + e−˜rm uεn (X(m )) εn 0 ' −˜r ( m =E e uεn (X(m )) . Letting n → ∞, by (22), we get

# % &$ ( 1 −˜ rm v(x) = E e v(X(m )) ≤ E e g(X(m )) + m ' −˜rτ ( 1 ≤ sup E e g(X(τ )) + . m τ ∈S '

−˜ rm

Passing to the limit, we deduce v(x) ≤ sup E[e−˜rτ g(X(τ ))]. τ ∈S

For the latter statement, we set τ¯ = τR ∧ m , where τR = inf{t ≥ 0; X(t) > 1 R or X(t) < R }, R > 1. We then have τ¯ < ∞ a.s. By (22), it is clear that # τ¯ $ E e−˜rt (uεn − g)− (X(t))dt = 0 0

for sufficiently large n. Hence again by (9) E[e−˜rτ¯ uεn (X(¯ τ ))] = uεn . Letting n → ∞ and then R → ∞, we get E[e−˜rm v(X(m ))] = v(x). τ ∗ as m

Note that m

∞. Passing to the limit, we deduce

∗

∗

E[e−˜rτ g(X(τ ∗ ))] = E[e−˜rτ v(X(τ ∗ ))] = v(x). 

Thus, we obtain the assertion. This completes the proof.

6. Proof of Theorem 3.2 We have by (6) # u(x) = E 0

τ

e

−(˜ r + 1ε )t 1

ε

 (u ∨ g)(X(t))dt +

∞

−(˜ r+ 1ε )t 1

e τ

ε

$ (u ∨ g)(X(t))dt .

We have by the strong Markov property of X(t) # ∞ $ −(˜ r+ 1ε )τ −(˜ r+ 1ε )τ −(˜ r+ 1ε )τ 1 e u(X(τ )) = e E e (u ∨ g)(X(t + τ ))dt|Fτ ε 0 # ∞ $ −(˜ r+ 1ε )t 1 =E e (u ∨ g)(X(t))dt|Fτ a.s. ε τ

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Hence

#

u(x) = E

τ

e 0

−(˜ r+ 1ε )t 1

ε

(u ∨ g)(X(t))dt + e

−(˜ r+ 1ε )τ

$ u(Xτ ) , τ ∈ S.

(23)

This can be regarded as the dynamic programming principle for the solution u. This property and Itˆ o’s formula lead to that u is a viscosity solution to the equation

1 1 r˜ + u = L0 u(x) + (u ∨ g) in (0, ∞), ε ε (24) g(0) u(0) = , r˜ε + 1 as in the proof of Theorem 1 in [19]. This is equivalent to 1 r˜u = L0 u(x) + (u − g)− in (0, ∞), ε g(0) . u(0) = r˜ε + 1

(25)

Hence we have (9). By letting τ → ∞ we see that u in (6) can be written as (10).  Acknowledgement I thank the referee for suggesting several improvements of the manuscript.

References [1] M. Arisawa, A new definition of viscosity solutions for a class of second-order degenerate elliptic integro-differential equations, Ann. Inst. Henri Poincar´e, Anal. Non Lin´eaire 23 (2006) 695–711; Corrigendum: Corrigendum for comparison theorems in: “A new definition of viscosity solutions for a class of second-order degenerate elliptic integro-differential equations”, Ann. Inst. Henri Poincar´e, Anal. Non Lin´eaire 24 (2007), 167–169. [2] M. Arisawa, A remark on the definitions of viscosity solutions for the integrodifferential equations with L´evy operators, J. Math. Pures Appl. (9) 89 (2008), 567–574. [3] L.H.R. Alvarez, Solving optimal stopping problems of linear diffusions by applying convolution approximations, Math. Methods Oper. Res. 53 (2001), 89–99. [4] G. Barles, R. Buckdahn and E. Pardoux, Backward stochastic differential equations and integral-partial differential equations, Stochastics Stochastics Rep. 60 (1997), 57–83. [5] G. Barles and C. Imbert, Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited, Ann. Inst. Henri Poincar´e Analyse non lin´eare 25 (2008), 567–585. [6] N. Bellamy, Wealth optimization in an incomplete market driven by a jump-diffusion process, J. Math. Econom. 35 (2001) 259–287.

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[7] F. Benth, K. Karlsen and K. Reikvam, Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: a viscosity solution approach, Finance Stoch. 5 (2001) 275–303. [8] F. Benth, K. Karlsen and K. Reikvam, Optimal portfolio management rules in a nonGaussian market with durability and intertemporal substitution, Finance Stoch. 5 (2001) 447–467. [9] F. Benth, K. Karlsen and K. Reikvam, Portfolio optimization in a L´evy market with international substitution and transaction costs, Stoch. Stoch. Reports 74 (2002) 517–569. [10] A. Bensoussan and J.L. Lions, Applications of variational inequalities in stochastic control, North-Holland Pub. Co., New York, N.Y., 1982. [11] C. Cancelier, Problemes aux limites pseudo-diffrentiels donnant lieu au principe du maximum, Comm. Partial Differential Equations 11 (1986) 1677–1726. [12] M. Chesney and M. Jeanblanc, Pricing American currency options in an exponential L´evy model, Appl. Math. Finance 11 (2004) 207–225. [13] M. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. A.M.S. 277 (1983) 1–42. [14] M. Crandall, H. Ishii and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1–67. [15] N. Framstad, B. Oksendal and A. Sulem, Optimal consumption and portfolio in a jump diffusion market with proportional transaction costs, J. Math. Econom. 35 (2001) 233–257. [16] N. Framstad, B. Oksendal and A. Sulem, Sufficient stochastic maximum principle for the optimal control of jump diffusion and applications to finance, JOTA 121 (2004) 77–98. [17] W. Fleming and M. Soner, Controlled Markov processes and viscosity solutions (Second edition), Stochastic Modelling and Applied Probability 25, Springer, New York, 2006. [18] N.G. Garroni and J-L. Menaldi, Green functions for second order parabolic integrodifferential problems, Longman Scientific & Technical, Essex, 1992. [19] Y. Ishikawa, Optimal control problem associated with jump processes, Appl. Math. Optim. 50 (2004) 21–65. [20] Y. Ishikawa, A small example of non-local operators having no transmission property. Tsukuba J. Math. 25 (2001) 399–411. [21] N. Jacob, Pseudo-differential operators and Markov processes, Akademie Verlag, Berlin, 1996. [22] M. Jeanblanc, Financial markets in continuous time, Springer Finance. Berlin: Springer, 2003. [23] H. Kumano-go, Pseudodifferential operators. Translated from the Japanese by the author, R. Vaillancourt and M. Nagase, MIT Press, Cambridge, Mass.-London, 1981. [24] N.V. Krylov, Controlled diffusion processes, Applications of Mathematics, 14, Springer-Verlag, New York-Berlin, 1980. [25] E. Mordecki, Optimal stopping and perpetual options for Levy processes, Finance Stochast. 6 (2002) 473–493.

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[26] E. Mordecki and P. Salminen, Optimal stopping of Hunt and L´evy processes, Stochastics 79 (2007) 233–251. [27] H. Morimoto, Stochastic control and mathematical modeling: application to economics, Cambridge Univ. Press, Cambridge, 2010. [28] B. Oksendal and A. Sulem, Applied stochastic control of jump processes (Second edition), Universitext, Springer-Verlag, Berlin, 2007. [29] H. Pham, Optimal stopping of controlled jump diffusion processes: a viscosity solution approach, J. Math. System Estim. Control 8 (1998) 1–27. [30] Ph. Protter, Stochastic integration and differential equations. A new approach. Applications of Mathematics, 21. Springer-Verlag, Berlin, 1990. [31] W. Rudin, Real and complex analysis, third edition, McGraw-Hill, New York, 1986. Yasushi Ishikawa Department of Mathematics Faculty of Science Ehime University 790 85 77 Matsuyama, Japan e-mail: [email protected]

Progress in Probability, Vol. 65, 121–144 c 2011 Springer Basel AG


A Review of Recent Results on Approximation of Solutions of Stochastic Differential Equations Benjamin Jourdain and Arturo Kohatsu-Higa Abstract. In this article, we give a brief review of some recent results concerning the study of the Euler-Maruyama scheme and its high-order extensions. These numerical schemes are used to approximate solutions of stochastic differential equations, which enables to approximate various important quantities including solutions of partial differential equations. Some have been implemented in Premia [56]. In this article we mainly consider results about weak approximation, the most important for financial applications. Mathematics Subject Classification (2000). 60H35, 65C05,65C30. Keywords. Euler-Maruyama scheme, Kusuoka scheme, Milshtein scheme, weak approximations, stochastic equations.

1. Introduction The Euler-Maruyama scheme is a simple and natural approximation method for the solution of various types of stochastic differential equations. It helps not only to simulate the solutions of stochastic equations but it also serves theoretical purposes (see, e.g., the articles of E. Gobet [18, 19] on the local asymptotic mixed normality (LAMN) property in statistics). To introduce this notion consider the stochastic differential equation  t r  t  X(t) = x + b(X(s))ds + σj (X(s))dZ j (s), (1) 0

j=1

0

where b, σi : Rd → Rd , i = 1, . . . , r, are Lipschitz coefficients and Z = (Z 1 , . . . , Z r ) is an r-dimensional Wiener process. For a partition of the interval [0, T ] denoted as π : 0 = t0 < · · · < tn = T , we define the norm of the partition as π = max{ti+1 − ti ; i = 0, . . . , n − 1}

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and η(t) = sup{ti ; ti ≤ t} the last discretization time before t. Then the EulerMaruyama scheme is defined inductively by ∀t ∈ [ti , ti+1 ], X π (t) = X π (ti ) + b(X π (ti ))(t − ti ) +

r 

σj (X π (ti ))(Z j (t) − Z j (ti )).

j=1

The simplicity and the generality of the possible applications are the main attractions of this scheme. In practice, one only needs to simulate the Brownian increments (Z(ti+1 )− Z(ti ))0≤i≤n−1 in order to compute (X π (t1 ), X π (t2 ), . . . , X π (tn )). Let us first mention the strong convergence rate result. Theorem 1. Under the above assumptions # $ 2p π ∀p ≥ 1, E sup X(t) − X (t) ≤ C π p t≤T

where the constant C depends on p, T , x and the Lipschitz constants. The proof of this result is standard and essentially goes through the same methodology used to prove existence of solutions to (1). This result can also be generalized to various equations without changing the essential ideas. In this paper, we are mainly interested in analyzing different error terms which involve a test function. The corresponding results are called weak convergence results. Section two deals with weak convergence for the Euler scheme. In n case the partition is uniform (ti = iT inn ), denoting the Euler scheme by X π stead of X , we first state the convergence in law of the normalized error process √ n(X − X n ) to a process χ which writes as a stochastic integral with respect to a Brownian motion independent from Z. More precisely, for any √ continuous and bounded function F on the space of continuous paths, E (F ( n(X − X n ))) converges to E (F (χ)) as n → +∞. Analysis of the weak error E (f (X(T )))− E (f (X n(T ))) turns out to be more important for applications: for instance, the price of a European option with payoff function f and maturity T written on an underlying evolving according to (1) under the risk-neutral measure writes E(e−rT f (X(T ))) where r denotes the risk-free rate. In [60], Talay and Tubaro prove that this difference can be expanded in powers of n1 . This justifies the use of Romberg-Richardson extrapolations in order to speed  up the convergence: for instance E (f (X(T ))) − E 2f (X 2n (T )) − f (X n (T )) = O( n12 ) (see Pag`es [53] for a recent study devoted to the numerical implementation of these extrapolations). The proof given by Talay and Tubaro relies on the Feynman-Kac partial differential equation associated with (1). Here, we present another methodology introduced in [33] and based on the integration by parts formula of Malliavin calculus. In [10], a general framework relying on the study of the linear stochastic equation satisfied by the error process X − X n is presented. This new methodology enables to deal with a great variety of equations including some which seem beyond the scope of the former PDE approach. We illustrate this latter point on the example of stochastic differential equations with delay.

Approximation of Solutions of Stochastic Differential Equations

123

The third section is devoted to a method of exact (in law) simulation of (1) recently introduced by [4, 5] in the one-dimensional case r = d = 1. Then, for a smooth diffusion coefficient σ which does not vanish, one can make a change of variables which transforms (1) into a SDE with diffusion coefficient constant and equal to one. Under a new probability measure given by Girsanov theorem, the original Brownian motion Z solves the latter SDE. The exponential factor giving the change of probability measure is then simulated by a rejection/acceptation technique involving a Poisson point process. It is possible to obtain schemes with convergence order higher than the one of the Euler scheme by keeping more terms in the stochastic Taylor expansion of the solution of the SDE. Section four deals with such schemes. We first introduce Stratonovitch stochastic integrals in order to write nice Taylor expansions. Then, on the example of the Milshtein scheme, we illustrate the difficulty to simulate the iterated Brownian integrals which appear in the expansions and therefore to implement schemes with high-order of strong convergence. Recently, to overcome this difficulty, Kusuoka [40, 41] proposed to replace these iterated Brownian integrals by random variables with the same moments up to a given order. This leads to schemes with high order of weak convergence. These schemes and their application in finance are currently the subject of a consequent research activity: [42, 47, 48, 50, 17]. The last section adresses extensions of the results presented previously. The case where instead of Z we have a L´evy process, Z, is first considered. Discretization of reflecting stochastic differential equations is also discussed.

2. Weak errors: from Jacod-Kurtz-Protter to Milshtein-Talay If one is trying to approach the problem of weak convergence of the error process then the first natural approach is to study the weak convergence of the process √ n (X(t) − X n (t)) . This is done in a series of articles by Jacod, Kurtz and Protter (e.g., see Section 5 in [39]). To simplify the ideas suppose that we are dealing with the Wiener case in one dimension (r = 1), b ≡ 0 and that the partition is uniform: ti = iT . Since n the process X n solves  t X n (t) = x + σ(X n (η(s)))dZ(s), 0

we have that

 X(t) − X (t) = n

t

σ1n (s) (X(s) − X n (s)) dZ(s)  t σ2n (s) (Z(s) − Z(η(s))) dZ(s) + 0

0

(2)

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where

 σ1n (s)

1

=

σ  (αX(s) + (1 − α)X n (s))dα

0

 σ2n (s)

=

1

σ  (αX n (s) + (1 − α)X n (η(s)))dασ(X n (η(s))).

0

Given the strong convergence result and assuming smoothness of σ, one has that σ1n and σ2n converge in the Lp (C[0, T ], R)-norm to σ1 (s) = σ  (X(s)) σ2 (s) = σ  σ(X(s)). Solving (2), we obtain that



t

E n (s)−1 σ2n (s) (Z(s) − Z(η(s))) dZ(s) 0  t n − E (t) E n (s)−1 σ1n σ2n (s) (Z(s) − Z(η(s))) ds,

X(t) − X n (t) = E n (t)

0

where



 1 t n (σ1 (s))2 ds 2 0 0 is the Doleans-Dade exponential, solution of the linear equation  t E n (t) = 1 + σ1n (s)E n (s)dZ(s). E n (t) = exp

t

σ1n (s)dZ(s) −

(3)

0

Now consider the process  t √ n (Z(s) − Z(η(s))) dZ(s) 0   √ j(t)−1 n  2 = (Z(ti+1 ) − Z(ti ))2 + (Z(t) − Z(η(t))) − t , 2 i=0

where tj(t) = η(t). Then using Donsker’s theorem (see, e.g., Billingsley [6] p. 68) we have that  · √ n (Z(s) − Z(η(s))) dZ(s) =⇒ Z  0 ; 2  where T Z is a Wiener process independent of Z. Furthermore if we consider < =  ·  t √ √ Z, n (Z(s) − Z(η(s))) dZ(s) = n (Z(s) − Z(η(s))) ds 0

t

0

we have that this quadratic covariation converges to 0 in L2 . This points to the following convergence

 · √ (Z(s) − Z(η(s))) dZ(s) =⇒ (Z, Z  ) Z, n 0

Approximation of Solutions of Stochastic Differential Equations

125

; 2 where Z and Z  are two independent Wiener processes. Therefore one can T hint at the following result  t √ n n (X(t) − X (t)) =⇒ E(t) E(s)−1 σ2 (s)dZ  (s), 0

where



t

E(t) = exp ;

0 2 T

1 σ1 (s)dZ(s) − 2



t

(σ1 (s)) ds 2

0



and (Z, Z ) is a two-dimensional Wiener process. This result in a variety of forms and generalizations has been extensively proved by Jacod, Kurtz and Protter. In particular, from this result one obtains that for any continuous bounded √ functional F in C[0, T ] one has that E [F ( n (X − X n ))] converges to #

$  · E F E(·) E(s)−1 σ2 (s)dZ  (s) . 0

On one hand these results give more detail about the limit law of the error process. Nevertheless, this does not give full information about the rate of convergence of various other functionals such as E(X(t)p )−E(X n (t)p ), pX(t) (x)−pX n (t) (x) where p stands for the density function. For this reason other efforts have been directed into extending the type of convergence into stronger topologies than the one given by weak convergence of processes. In [29], the authors prove that for any continuous bounded functional F and any bounded real variable Y we have that #

$  · ' √ ( n −1  E YF n (X − X ) → E Y F E(·) E(s) σ2 (s)dZ (s) . 0

This type of convergence is called stable convergence in law. It is worth noting that if Y is restricted to a subfiltration this concept also allows the study of the convergence of the conditional expectation of the error process. This type of results are promising but still it does not allow the analysis of the convergence of quantities like the ones mentioned before. In order to analyze this problem, there is another “parallel” theory called weak approximation that deals particularly with the error E [f (X) − f (X n )] . The state of the art of this problem is more advanced than the one given previously by the theory of Jacod-Kurtz-Protter. In fact one is able to deal with non bounded, non continuous and even Schwartz distribution functions f (see Guyon [24]). On the other hand one is not able to give precise information on the distribution of the limit error. Just to explain in simple terms the ideas behind this approach, let’s explain in simple terms a complex result due to Bally and Talay [2, 3].

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To clarify the methodology, we consider a real diffusion process (that is Z is a one-dimensional Wiener process)  t X(t) = x + σ(X(s))dZ(s), t ∈ [0, T ], 0

and its Euler approximation



t

σ(X n (η(s)))dZ(s), t ∈ [0, T ],

n

X (t) = x + 0

where η(s) = kT /n for kT /n ≤ s < (k + 1)T /n. The error process Y = X − X n solves  t Y (t) = (σ(X(s)) − σ(X n (η(s))))dZ(s) 0

 t

1

= 0

σ  (aX(s) + (1 − a)X n (η(s)))da(X(s) − X n (η(s)))dZ(s),

0

this can be written



t

σ1n (s)Y (s)dZ(s) + G(t),

Y (t) =

0 ≤ t ≤ T,

0

with

 σ1n (s)

1

σ  (aX(s) + (1 − a)X n (η(s)))da

= 

0



0

t

σ1n (s)(X n (s) − X n (η(s)))dZ(s)

G(t) = t

σ1n (s)σ(X n (η(s)))(Z(s) − Z(η(s)))dZ(s).

= 0

In this simple case we have an explicit expression for Yt ,  t n Y (t) = E (t) E n (s)−1 (dG(s) − σ1n (s)d G, Zs ) 0

where E (t) is given by (3). Finally we obtain  t Y (t) = E n (t) E n (s)−1 σ1 (s)σ(X n (η(s)))(Z(s) − Z(η(s)))dZ(s) 0  t − E n (t) E(s)−1 σ1n (s)2 σ(X n (η(s)))(Z(s) − Z(η(s)))ds. n

0

Now let f be a smooth function with possibly polynomial growth at infinity. We are interested in obtaining the rate of convergence of Ef (X(T )) to Ef (X n (T )). We first write the difference # 1 $ n  n Ef (X(T )) − Ef (X (T )) = E f (aXT + (1 − a)X (t))daY (T ) . 0

Approximation of Solutions of Stochastic Differential Equations

127

Replacing Y (T ) by its expression, we obtain with the additional notation  1 n F = f  (aX(T ) + (1 − a)X n(T ))da, 0

Ef (X(T )) − Ef (X n (T ))   T = E F n E n (T ) E n (s)−1 σ1n (s)σ(X n (η(s)))(Z(s) − Z(η(s)))dZ(s) 0





T

− E F E (T ) n n

E n (s)−1 σ1n (s)2 σ(X n (η(s)))(Z(s) − Z(η(s)))ds .

(4)

0

Applying the duality formula for stochastic integrals (E[DF, uL2 [0,T ] ] = E[F δ(u)] see [51]) where D stands for the stochastic derivative and δ stands for the adjoint of the stochastic derivative, this gives Ef (X(T )) −Ef (X n(T ))  T =E Ds (F n E n (T ))E n (s)−1 σ1n (s)σ(X n (η(s)))(Z(s) − Z(η(s)))ds 0





− E F E (T ) n n

T

E n (s)−1 σ1n (s)2 σ(X n (η(s)))(Z(s) − Z(η(s))) .

0

Consequently, the differenceEf (X(T )) −Ef (X n (T )) has the simple expression  T n Ef (X(T )) −Ef (X (T )) =E U n (s)(Z(s) − Z(η(s)))ds , 0

with U n (s) = (Ds (F n E n (T )) − F n E n (T )σ1 (s))(E n (s)−1 σ1n (s)σ(X n (η(s)))). We finally obtain the rate of convergence by applying once more the duality for stochastic integrals   T s n Ef (XT ) −Ef (XT ) =E Du U n (s)duds . 0

η(s)

This last formula ensures that |Ef (X(T )) −Ef (X n (T ))| ≤ CT /n and leads to an expansion ofEf (X(T )) −Ef (X n(T )) with some additional work. Furthermore the above argument extends easily in the case that f is an irregular function through the use of the integration by parts formula of Malliavin Calculus. In other stochastic equations, one cannot explicitly solve the stochastic linear equation satisfied by Y , but in a recent article [10], one can find a general framework that allows treating a great variety of equations. As example we have developed the case of delay equations. The idea explained above appeared for the first time at some workshop proceedings (in [33]) and later was used by various

128

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authors (see [20, 22, 21]) to prove weak approximations errors in other contexts such as the Zakai equation or backward stochastic differential equations. In fact, the first time this argument appeared in [33], it was just considered as an alternative argument to prove the classical results of weak approximation of Milshtein [43] which are usually obtained through a PDE method. Later it has been shown that in fact this new approach can go beyond the classical proof method. To explain this with a concrete example, we will briefly describe the problem with delay equations which is solved in [10]. Notice that such equations have been introduced in finance by Rogers and Hobson [25] in order to propose a complete model with stochastic volatility. In few words the problem with the Euler approximation for delay equations is that if one tries to use the Milshtein method one gets into infinite-dimensional problems quite rapidly and therefore the degree of generalization is quite limited. In fact, consider (see the article of Buckwar and Shardlow [8]) the following one-dimensional delay equation  0

dX(t) = X(t + s)dm(s) + b(X(t)) dt + σ(X(t))dZ(t) −τ

where m is a deterministic finite measure on the interval [−τ, 0] and the initial conditions are X(s) = x(s) for s ∈ [−τ, 0]. Consider the integral operator A  0 Ax(t) = x(t + s)dm(s). −τ

Then using classical theory of stochastic differential equations in infinite dimensions (an extension of the variation of constants method, see Da Prato-Zabczyk) one obtains  t  t X(t) = S(t)x + S(t − s)b(X(s))ds + S(t − s)σ(X(s))dZ(s) 0

0

where S is the semigroup associated with the linear first term in the equation for X. The natural definition of the Euler scheme is obviously obtained by discretization of the integral in the drift term. That is, X n (ti+1 ) = X n (ti ) +

m 

X(ti + sj )m(sj , sj+1 ]

j=0

+ b(X n (ti ))(ti+1 − ti ) + σ(X n (ti ))(Z(ti+1 ) − Z(ti )) where sj is a partition of the interval [−τ, 0] such that ti + sj = tl for some l ≤ i. Similarly, one finds that X n is generated using instead of S the Yoshida approximations to this semigroup. That is the semigroup S n associated with n

A x(t) =

m  j=0

x(t + sj )(m(sj+1 ) − m(sj )).

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Then, as when n tends to infinity An x −→ Ax and S n x −→ Sx, one expects the strong convergence of the Euler scheme. In order to study the weak errors one has to go further and define the solution of the partial differential equation associated with this problem: ut (t, y) =

1 ux x (t, y)σ(x0 )2 + ux (t, y)Ax + ux0 (t, y)b(x0 ) 2 0 0

where x(0) = x0 for y = (x0 , x) ∈ R × L2 [−τ, 0]. The (non-trivial) argument is then similar to the classical Milshtein argument. Nevertheless, it is also clear from the above set-up that this approach has its limitations. For example, one cannot suppose that there is also a continuous delay in the diffusion coefficient or that the delay term is non-linear. In comparison, using the method explained previously, one obtains the following result: Let (Xt )t∈[0,T ] be the solution stochastic delay equation:   0

 0

  dX = σ Xt+s dν(s) dZt + b Xt+s dν(s) dt t −τ −τ   Xs = ξs , s ∈ [−τ, 0], where τ > 0, ξ ∈ C([−τ, 0], IR) and ν is a finite measure. We consider the Euler approximation of (Xt ) with step h = τ /n   0

 0

 n n  dX n = σ X dν(s) dZ + b X dν(s) dt t

t

η(t)+η(s)

−τ   X n = ξ , s ∈ [−τ, 0], s s

η(t)+η(s)

−τ

] with η(s) = [ns/τ n/τ , where [t] stands for the entire part of t. We assume that the functions f , σ and b are Cb3 . Then we obtain that

Ef (XT ) − Ef (XTn ) = hCf + I h (f ) + o(h)

(5)

where Cf = C(U 0 ) and I h (f ) = I h (U 0 ) are defined in [10]. In particular |I h (f )| ≤ Ch and  0

 0

0    Us = σ Xs+u dν(u) Ds f (XT ) + b Xs+u dν(u) f  (XT ) −r −r   



 + σ

0

−r

T

Xs+u dν(u) Ds

θt dt 0

+ b

0

−r

T

Xs+u dν(u)

and θ is the unique solution of       T  ∗  ∗ θt = α J f (XT ) + θs ds (t) + β E f  (XT ) + 0

θt dt s

.



T

θs ds|F.

(t)

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with



∗

0

α (X)(t) = E

σ



max(t−T,−r)

β ∗ (X)(t) = E



0

b

 

0 −r 0

−r

max(t−T,−r)



Xt−u+v dν(v) Xt−u dν(u)|Ft 

Xt−u+v dν(v) Xt−u dν(u)|Ft .

The above-quoted result (5) is an expansion of the error. This result is used in order to increase the rate of convergence of the method using the Romberg extrapolation (see [60] for details in the diffusion case). That is, if Ef (XT ) − n/2 Ef (XTn ) = hCf + o(h). Then if we define Y = 2f (XTn ) − f (XT ) one obviously obtains that Ef (XT ) − EY = o(h) therefore increasing the rate of convergence of the method. In order to know exactly what has been gained, it is also important to obtain the order of the term o(h) in (5). This can be done using the same method but through tedious work obtaining that in fact, Ef (XT )−Ef (XTn ) = hCf +O(h2 ) under enough smoothness conditions on σ and b. In the diffusion case, Pag`es [53] addresses variance issues in the context of multi-step Romberg extrapolation.

3. An exact simulation method for one-dimensional elliptic diffusions Recently in two articles by Beskos et al. [4, 5], an interesting exact method of simulation in dimension one has been introduced. Consider the one-dimensional diffusion  t  t X(t) = x + b(X(s))ds + σ(X(s))dZ(s) 0

0

where σ(x) ≥ c > 0 for any x ∈ R
and σ ∈ C 1 (R). Then perform the change z 1 of variables Yt = η(Xt ) where η(z) = x σ(u) du. By Itˆo’s formula, Y satisfies the following sde:  t Y (t) = α(Y (s))ds + Z(t) 0  b (η −1 (y))− σ2 (η−1 (y)). σ

where α(y) = Suppose that we want to compute E(f (XT )). Then using Girsanov’s theorem we have that     T 1 T 2 α(Bs )dBs − α(Bt ) dt (6) E(f (XT )) = E f (BT ) exp 2 0 0 where B is another Wiener process and here we assume that α is bounded. This idea is usually found when one proves existence of weak solutions for stochastic differential equations.

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u Next, one defines the function A(u) = 0 α(y)dy. With this definition we have, applying Itˆ o’s formula, that  T  1 T  A(BT ) = α(Bs )dBs + α (Bs )ds. 2 0 0 Therefore





1 Ef (XT ) = E f (BT ) exp A(BT ) − 2



T

  α(Bt )2 + α (Bt ) dt

 .

0

If one was to simulate the above quantity, one would need the whole path of the Wiener process B. In fact this is done in a series of papers by Detemple et al. [14, 15, 16] where the Doss-Sussman formula is used to improve the approximation and obtain a scheme which is of strong order one. Instead, Beskos et al. [5] propose to use a Poisson process to simulate the exponential in the above expression. In   fact, one assumes that φ(x) = 12 α(x)2 + α (x) is such that ∀x ∈ R, 0 ≤ φ(x) ≤ M and introduces a Poisson point process N with intensity ds×du on [0, T ]×[0, M ], independent of B. For any Borel subset S of [0, T ] × [0, M ], N (S) is a Poisson random variable with parameter the Lebesgue measure of S. Hence, the random variable N1 = N ({(s, u) ∈ [0, T ] × [0, M ] : 0 ≤ u ≤ φ(B(s))}) is such that    T P (N1 = 0|B) = exp − φ(Bs )ds . 0

The simulation scheme follows from the equality ' ( E(f (XT )) = E f (BT ) exp (A(BT )) E(1{N1 =0} |B) ' ( = E f (BT ) exp (A(BT )) 1{N1 =0} . How is the simulation done? First one simulates independent exponential random ν variables with parameter M say X1 , . . . , Xν until i=1 Xi > T . Then one simulates independent random variable Ui , . . . , Uν−1 uniformly distributed on the inter ν−1 val [0, M ]. The resulting point process N = i=1 δ(X1 +···+Xi ,Ui ) on [0, T ] × [0, M ] is Poisson with intensity ds × du. Now one simulates the independent increments B(X1 ), B(X1 + X2 ) − B(X1 ), . . . , B(T ) − B( ν−1 Xi ) of the Brownian motion i=1 ν−1 and computes N1 = i=1 1{Ui ≤B(X1 +···+Xi )} . Obviously there are various issues that have not been considered in this short introduction which rest as open problems or that had already been treated by the authors. Also as it was well known before, the one-dimensional case always permits various reductions that do not happen in higher dimensions. For instance, in higher dimensions, the so-called Doss transformation which permits to obtain a SDE with a constant diffusion coefficient is only possible when σ satisfies a restrictive commutativity condition. Notice that under that condition, the discretization scheme obtained by applying the Euler scheme to the SDE with constant diffusion coefficient and making the inverse change of variables is of strong order one (see [14]). Moreover, in higher dimensions the replacement of the stochastic integral

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in (6) by a standard integral thanks to Itˆ o’s formula is only possible when α is a gradient function. Nevertheless, the one-dimensional case always remains as a testing ground for new methodology as it was proven by our recent development in Section 2. And an exact Monte Carlo method for the pricing of Asian options in the Black-Scholes model inspired by the above ideas will be implemented by Jourdain and Sbai [30] in the version 10 of Premia [56].

4. Schemes with high-order of convergence 4.1. Stochastic Taylor expansions In order to make such expansions, it is more convenient to rewrite (1) in Stratonovich form. The interest is that the chain rule holds for Stratonovich integrals. We recall that for a regular adapted one-dimensional process (H(s))s≤t the Stratono
t vich integral 0 H(s) ◦ dZ j (s) is equal to the limit in probability of i 12 (H(ti+1 ∧ t) + H(ti ∧ t))(Z j (ti+1 ∧ t) − Z j (ti ∧ t)) as maxi |ti+1 − ti | tends to 0. Hence  t  t 7 16 H(s) ◦ dZ j (s) = H(s)dZ j (s) + H, Z j t 2 0 0 and (1) writes



t

X(t) = x +

σ0 (X(s))ds + 0

where σ0 = b − 12

r   j=1

t

σj (X(s)) ◦ dZ j (s)

(7)

0

)

r

j=1 ∂σj σj with ∂σj denoting the matrix

∂σij ∂xl

*

for σj = 1≤i,l≤d d (σ1j , . . . , σdj )∗ . Let us introduce the differential operators Vj = i=1 σij (x)∂xi for 0 ≤ j ≤ r. Since the chain rule holds for Stratonovich integrals, for f a smooth function on Rd ,  t r  t  f (X(t)) = f (x) + V0 f (X(s))ds + Vj f (X(s)) ◦ dZ j (s) 0

= f (x) +

r  t  j=0

j=1

0

Vj f (X(s)) ◦ dZ j (s),

0

where for notational convenience we set Z 0 (s) = s.
s r Now remarking that Vj f (X(s)) = Vj f (x) + l=0 0 Vl Vj f (X(u)) ◦ dZ l (u), one obtains  t r  f (X(t)) = f (x) + Vj f (x) ◦dZ j (s) +

j=0 r  j,l=0

0

 Vl Vj f (X(u)) ◦ dZ l (u) ◦ dZ j (s). 0≤u≤s≤t

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133

Iterating the reasoning, one obtains that for any positive integer m, f (X(t)) = f (x) +

+

m 

r 

Vj1 Vj2 . . . Vjk f (x)Z (j1 ,...,jk ) (t)

k=1 j1 ,...,jk =0  r  j1 ,...,jm+1 =0

0≤s1 ≤···≤sm+1 ≤t

Vj1 . . . Vjm+1 f (X(s1 ))

◦ dZ j1 (s1 ) ◦ · · · ◦ dZ jm+1 (sm+1 )


where Z (j1 ,...,jk ) (t) = 0≤s1 ≤···≤sk ≤t ◦dZ j1 (s1 ) ◦ · · · ◦ dZ jk (sk ). √ For 1 ≤ j ≤ r, Z j (s) is of order s while Z 0 (s) = s or, in other words, by scaling, Z (j1 ,...,jk ) (t) has the same distribution as t(k+#{1≤l≤k:jl =0})/2 Z (j1 ,...,jk ) (1). Hence to obtain terms with the same order of magnitude in the above expansion, 0 one has to count the 9 integrals withl respect to Z (s) twice. That is why for α = (j1 , . . . , jk ) ∈ A = l∈N ∗ {0, . . . , r} , we set |α| = k and α = k + #{1 ≤ l ≤ k : jl = 0}. Then we write  f (X(t)) = f (x) + Vj1 . . . Vjk f (x)Z α (t) + Rm,f (t) where α: α ≤m



Rm,f (t) =

Vj1 . . . Vjk f (x)Z α (t)

α:|α|≤m, α >m

+

r  j1 ,...,jm+1 =0

 0≤s1 ≤···≤sm+1 ≤t

Vj1 . . . Vjm+1 f (X(s1 ))

◦ dZ j1 (s1 ) ◦ · · · ◦ dZ jm+1 (sm+1 ).

(8)

Since the remainder Rm,f (t) involves termes scaling like t to a power greater or equal to (m + 1)/2, the following result (see Proposition 2.1 [42] for p = 1) is not surprising. Proposition 2. When the functions f , b and σj are smooth, the remainder Rm,f (t) m+1 is such that for p ≥ 1, E(|Rm,f (t)|2p )1/(2p) ≤ Ct 2 where the constant C depends on p, f , b, σj and their derivatives. 4.2. The Milshtein scheme The Milsthein scheme consists in choosing f (x) = x and m = 2 in the above expansion (8) and removing the remainder: ∀t ∈ [ti , ti+1 ], X π (t) = X π (ti ) + +

r 

σj (Xtπi )(Z j (t) − Z j (ti ))

j=0 r 

∂σj σl (Xtπi )(Z (l,j) (t) − Z (l,j) (ti+1 )).

(9)

j,l=1

The strong order of convergence of the Milshtein scheme is one (see, e.g., [32]):

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Theorem 3. Assume that the functions σj and b are C 2 with bounded derivatives. Then for p ≥ 1, sup E [ X(t) − X π (t) p ] ≤ C π p , t≤T

where the constant C does not depend on the partition π. To implement the Milshtein scheme, one faces the difficulty usually encountered when trying to construct practical discretization schemes from Taylor expansions: the need to simulate increments of the multiple stochastic integrals which appear. On
the one hand, by the fundamental theorem of calculus, for 1 ≤ j ≤ r, Z (j,j) (t) = 0≤u≤s≤t ◦dZuj ◦ dZsj is equal to 12 (Z j (t))2 . But on the other hand, no such nice expression in terms of Z j (t), Z l (t) holds for Z (l,j) (t) when j = l. The generalization of equality 2Z (j,j) (t) = (Z j (t))2 writes Z (l,j) (t)+Z (j,l) (t) = Z l Z j (t). Hence, for the Milshtein scheme to be simulable, one needs the following commutativity condition (C) ∀1 ≤ l < j ≤ r, ∂σj σl = ∂σl σj which always holds when r = 1 (single Brownian motion case). Under (C), it is enough to simulate the Brownian increments since the Milshtein scheme writes r  π π π X (ti+1 ) = X (ti ) + b(Xti )(ti+1 − ti ) + σj (Xtπi )(Z j (ti+1 ) − Z j (ti )) +

+

j=1



∂σj σl (Xtπi )(Z l Z j (ti+1 ) − Z l Z j (ti ))

1≤l<j≤r r 

1 2

∂σj σj [(Z j (ti+1 ))2 − (Z j (ti ))2 − (ti+1 − ti )].

j=1

In the elliptic case (when ∀x ∈ Rd , (σ1 (x), . . . , σr (x)) is a basis of Rd ), Cruzeiro, Malliavin and Thalmaier [11] have recently proposed a new version of the Milshtein ˜ π which does not involve iterated stochastic integrals of second order scheme X (even if commutativity (C) fails). It instead involves a process with values in orthogonal matrices which governs a dynamical rotation of the driving Brownian ˜ of the equation obtained from (1) by this rotation has motion. The solution of X ˜π − X ˜ t |2 ) ≤ C π 2 . the same distribution as X. They prove that E(supt≤T |X t 4.3. Schemes with high-order of weak convergence We just saw that the simulation of iterated Brownian integrals of order greater than one is a problem. To overcome this difficulty and obtain simulable schemes with high order of weak convergence, Kusuoka [40, 41] proposed to replace the iterated Brownian integrals which appear in the stochastic Taylor expansion (8) by random variables with the same moments up to order m. See also [47, 48] where Ninomiya discusses the numerical implementation and efficiency of the resulting scheme and [42] in which Lyons and Victoir propose another scheme based on similar ideas.

Approximation of Solutions of Stochastic Differential Equations

135

Definition 4. Let m ∈ N ∗ . A family (ζ α ) α ≤m of random variables with finite moments of any order is called m-moment like if ∀α1 , . . . , αk ∈ A such that α1 + · · · + αk ≤ m, E [ζ α1 . . . ζ αk ] = E [Z α1 (1) . . . Z αk (1)] .

(10)

The following 5-moment like family in dimension r = 1 is given in [48] where other examples are also presented: Example 5. Let η be a random variable such that P (η = 0) = 23 and P (η = √ ± 3) = 16 . Then one obtains a 5-moment like family in dimension r = 1 by setting 1 1 1 ζ 0 = 1, ζ 1 = η, ζ (1,1) = η 2 , ζ (1,0) = ζ (0,1) = η, ζ (1,1,1) = η 3 , 2 2 6 1 1 1 (1,1,0) (0,1,1) (0,0) (1,1,1,1) α ζ =ζ = , ζ = , ζ = and ζ = 0 otherwise. 4 2 8 Now replacing the mutiple Brownian integrals by a m-moment like family in (8) written for f (x) = I(x) where I(x) = x denotes the idendity function on Rd , one approximates the law of X(t) by the one of  Ytx = x + t α /2 Vj1 . . . Vjk I(x)ζ α . α ≤m

Let Qt f (x) =

E(f (Ytx ))

denote the corresponding approximation of E(f (X(t))).

Theorem 6. When the functions f , b and σj are smooth, n−1    E(f (X(T ))) − Qt1 Qt2 −t1 . . . QT −tn−1 f (x) ≤ C (ti+1 − ti )(m+1)/2 i=0

where the constant C depends on f , b, σj and their derivatives. Remark 7. (m+1)/2 (m+1)/2 • For a regular grid ti = iT , one has n−1 = Tn(m−1)/2 . Hence i=0 (ti+1 − ti ) n the order of weak convergence of the scheme is (m − 1)/2. • Setting ζ j = Z j (1) for 0 ≤ j ≤ r, ζ (j,j) = 12 for 1 ≤ j ≤ r, ζ (j,l) = 0 for 0 ≤ j = l ≤ r and ζ (j,k,l) = 0 for 1 ≤ j, k, l ≤ r, one obtains a m = 3-moment like family. According to the Markov property, this family is such that Qt1 Qt2 −t1 . . . QT −tn−1 f (x) = E(f (X π (T ))) where X π denotes the Euler-Maruyama scheme. The order of weak convergence of this scheme is 1 = (3 − 1)/2. • Of course, for the numerical approximation of E(f (X(T ))), m-moment like families which can be generated on finite probability spaces with as few elements as possible are preferable. For instance, the 5-moment like family given in example 5 can be generated on a probability space with 3 elements supporting the random variable η. For this choice, the exact computation of the approximation Qt1 Qt2 −t1 . . . QT −tn−1 f (x) is possible using a non-recombining

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trinomial tree with n time-steps and therefore 3n leaves. When m and r increase, so does the cardinality of the probability space necessary to support a m-moment like family. If exact computation of the approximation is no longer possible, one has to resort to a partial sampling technique such as the Monte Carlo method (see [48] which is devoted to that issue). An important open problem is how to generate m-moment like families of order higher than 5. • In [40, 41], Kusuoka works under a uniformly non-degeneracy assumption weaker than the uniform H¨ ormander condition (called the UFG condition) for (1) which ensures, thanks to the Malliavin calculus, that for t > 0 x → E(f (X(t))) is smooth in the directions given by the fields generated by the Lie brackets of Vj even if f is not. For the non uniform grid refined near the maturity T

γ

n−i ti = T 1 − with γ > m, n he obtains convergence of the approximation with order (m − 1)/2 for functions f only C 1 :   E(f (X(T ))) − Qt1 Qt2 −t1 . . . QT −tn−1 f (x) ≤ C ∇f ∞ . n(m−1)/2 Proof. Setting Pt f (x) = E(f (X(t))), one has the following decomposition of the error |E(f (X(T )) − Qt1 . . . QT −tn−1 f (x)| ≤ |E(PT −tn−1 f (X(tn−1 ))) − Qt1 . . . Qtn−1 −tn−2 PT −tn−1 f (x)|

(11)

+ |Qt1 Qt2 −t1 . . . Qtn−1 −tn−2 (PT −tn−1 f − QT −tn−1 f )(x)| We now prove the result by induction on n. For n = 1, one deals with Pt f (z) − Qt f (z) which by (8) is equal to 

 E f z+ Vj1 . . . Vjk I(z)Z α (t) + Rm,I (t) α ≤m



 −f z + t α /2 Vj1 . . . Vjk I(z)ζ α . α ≤m

Assuming for simplicity that d = 1 and making a standard Taylor expansion of the function f in the neighborhood of z, one deduces that Pt f (z) − Qt f (z) is equal to 

k m   f (k) (z) E Vj1 . . . Vjk I(z)Z α (t) + Rm,I (t) k! α ≤m

k=1

−



α ≤m

t α /2 Vj1 . . . Vjk I(z)ζ α

k + O(t

m+1 2

).

Approximation of Solutions of Stochastic Differential Equations

137

According to Proposition 2, Rm,I (t) scales like t(m+1)/2 . Now developping the powers and using (10) one obtains that the expectation of all terms scaling like tl/2 with l ≤ m vanish. Therefore |Pt f (z)−Qt f (z)| ≤ Ct(m+1)/2 and the induction hypothesis holds for n = 1. We now assume that the induction hypothesis holds at rank n − 1. Since when f is smooth, so is PT −tn−1 f , we deduce that the first term of the right-hand n−2 side of (11) is smaller than C i=0 (ti+1 − ti )(m+1)/2 . By the result proved for n = 1, PT −tn−1 f − QT −tn−1 f ∞ ≤ C(T − tn−1 )(m+1)/2 . Since for all t ≥ 0, Qt g ∞ ≤ g ∞ , one deduces that the second term of the r.h.s. of (11) is smaller than C(T − tn−1 )(m+1)/2 . This concludes the proof.  Let us briefly present the related approximation based on the notion of cubature proposed by Lyons and Victoir [42]. Definition 8. Let m ∈ N ∗ and t > 0. Continuous paths ωt,1 , . . . , ωt,N with bounded variation from [0, t] to Rr and positive weights λ1 , . . . , λN such that N l=1 λl = 1 define a cubature formula with degree m at time t if ∀α = (j1 , . . . , jk ) ∈ A such that α ≤ m,  N  (12) j1 jk α E(Z (t)) = λl dωt,l (s1 ) . . . dωt,l (sk ) l=1

where

j ωt,l (s)

0≤s1 ≤···≤sk ≤t

0 denotes the jth coordinate of ωt,l (s) when 1 ≤ j ≤ d and ωt,l (s) = s.

According to [42], there exists a cubature with degree m at time 1 such that N is smaller than the cardinality of {α ∈ A : α ≤ m}. Moreover, one deduces a cubature of degree m at time t by scaling. For l ∈ {1, . . . , N } let (yt,l (s, x))s≤t denote the solution of the ODE: r  j yt,l (0, x) = x and ∀s ∈ [0, t], dyt,l (s, x) = σj (yt,l (s, x))dωt,l (s). j=0

Lyons and Victoir propose to approximate E(f (X(t)) by Qt f (x) =

N 

λl f (yt,l (t, x)).

l=1

Theorem 6 still holds with this new definition of Qt f . The proof is based on a similar decomposition of the error but the analysis of E(f (X(t)) − Qt f (x) is easier. Indeed the Taylor expansion (8) holds for Z replaced by ωt,l . Multiplying by λl , summing over l and substracting (8) then using Proposition 2 and (12), one obtains that |E(f (X(t)) − Qt f (x)| ≤ Ct(m+1)/2 . Let us finally mention, interesting schemes with high weak order of convergence recently proposed by Ninomiya and Victoir [50] and Fujiwara [17]. Even if the idea of these schemes also comes from stochastic Taylor expansions, their implementation is different from the previous ones. It requires a sequence of independent uniform random variables (Ui )1≤i≤n independent from (Z 1 , . . . , Z r ).

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¯ π (ti ) to X ¯ π (ti+1 ), one repeats the following steps for For θ ∈ N ∗ , to go from X θ θ k ∈ {1, . . . , θ} d 1. integrate the ordinary differential equation dt x(t) = σ0 (x(t)) on an interval with length (ti+1 − ti )/2θ, 2. depending on whether Ui+1 ≤ 12 or not, integrate successively for j increasing d from 1 to r or for j decreasing from r to 1 the ODE dt x(t) = σj (x(t)) on an j k j k−1 k interval with random length Z (ti ) − Z (ti ) where ti = ti + k(ti+1 − ti )/θ. 3. do the first step again. ¯ π (T ) is an approximation of X(T ) with Ninomiya and Victoir [50] prove that X 1 weak order of convergence 2. The idea of Fujiwara [17] is to make Romberg-like extrapolations in order to improve the weak rate of convergence. Indeed, he proves ¯ n (T )) − f (X ¯ n (T ))) that E(f (X(T ))) is respectively approximated by 13 E(4f (X 2 1 1 n n n ¯ ¯ ¯ and 120 E(243f (X3 (T )) − 128f (X2 (T )) + 5f (X1 (T ))) with order of convergence 4 and 6. When, for some of the above ODEs, no analytical expression of the solution is available, one has to resort to discretization schemes. Those schemes have to be chosen carefully in order to preserve the weak order for the resulting scheme for (1). For instance, Fujiwara suggests a Runge-Kutta scheme with order 13 to preserve the weak order 6. More recently, Ninomiya and Ninomiya [49] have proposed a scheme with weak order 2 in which, for each time-step, only two ordinary differential equations have to be integrated on a random time-horizon. They have also analysed the effect in terms of weak error of the resort to Runge-Kutta schemes to integrate the ordinary differential equations. The schemes with weak order 2 proposed by Ninomiya and Victoir and by Ninomiya and Ninomiya are both implemented in Premia [56] for the pricing of Asian options under the Heston model of stochastic volatility.

5. Comments on some extensions We discuss first the case when Z is a L´evy process. That is, process with independent and stationary increments with characteristic function given by E [exp (i θ, Z(t))]

 1 = exp − θ, Γθ t + i b, θ t + (exp(i θ, x) − 1 − iθx1 {x ≤ 1}) ν(dx) 2 Rr where θ ∈ Rr ,)Γ ∈ Rr×r * is a symmetric non-negative matrix and ν is a measure
2 satisfying Rr 1 ∧ |x| ν(dx) < ∞. When b = ν = 0 and Γ is the identity matrix, then Z is a standard r-dimensional Wiener process. The constant b denotes the drift of the process and ν is the L´evy measure associated to the process Z. We note that in comparison with the Wiener process case
notk all moments of Z are finite. In fact the moment of order k of Z is finite if Rr |x| 1{x≥1} ν(dx) < ∞. The existence and uniqueness of the above equation (1) is ensured by standard theorems that can be found in, e.g., Protter [57] under Lipschitz assumptions on

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the coefficients b and σ. Nevertheless it is not clear under which conditions the moments of the solution are finite if Z is a L´evy process, except for the case of bounded coefficients. In particular, we do not know how the finite moment property transfers from Z into X when the coefficients are Lipschitz. These properties are important in order to determine the convergence properties of the Euler scheme. The situation in the case that σ is constant is already difficult enough. Nevertheless, this is an interesting problem. We quote here some results of the article Kohatsu-Yamazato [34] who study this problem in the particular case that σ is constant. For example, consider for simplicity the one-dimensional case r = d = 1 with Γ = 0, b = 0 and ν a measure concentrated on (0, ∞). The moment E(X(t)β ) is finite or not depending on whether the integral with respect to ν in the last column is finite or not. b(y) = y α

β

0≤α≤1

β>0

α>1

0 < β < α−1

α>1

β = α−1

α>1

β > α−1

Criterion for finiteness of E(X(t)β )
+∞ β y ν(dy) 1 always finite
+∞ log (y) ν(dy) 1
+∞ β−α+1 y ν(dy) 1

In the same lines of the above table, but in another set up, Grigoriu-Samorodnistsky [23] studied the tail behavior of X(t). In either case the conclusions are similar. The rule seems to be that if the drift coefficient is sublinear then the drift does not influence the finite moment property of Z and it transfers directly to X. If the drift is superlinear then the situation is different. That is, the finite moment property depends on the difference of power between the drift and the moment to be evaluated. Therefore, it can be conjectured that this is the situation in the Lipschitz cases. Currently, as far as our knowledge goes, it is not known if X has finite moments even if the exponential moments of Z are bounded unless one imposes a series of stringent conditions. In most papers found in the literature, besides this assumption, one also has to make the assumption that the moments of X are bounded which is an unaccomplished feature of this problem. For example one has that Theorem 9. Suppose that Z has exponential moments and that X has finite moments. Then # $ 2p p E sup X(t) − X π (t) ≤ C π t≤T

where the constant C depends on T , x and the Lipschitz constants.

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One remarkable different case from the discussion in this paper is the situation of reflecting stochastic differential equations. In general, if the domain is closed and convex and the reflection is normal, then the results can be usually obtained as generalizations of the non-reflecting case. The main difference lies in how the inequalities are obtained. In fact, instead of using strong type inequalities directly on the error process X(t) − X π (t), one has to use Itˆ o’s formula and the fact that contribution of the reflecting processes brings X n closer to X. If the domain is more general then the results are no longer valid. In fact, as proven by Pettersson [54] (later refined by Slominski [59]) the rates can decay slightly depending on the properties of the domain. The latest refined results on this can be found in a recent thesis by S. Menozzi [44]. Nevertheless there is no parallel theory in the style of Jacod-Kurtz-Protter. In finance, one-dimensional processes that remain non-negative are of particular interest. This non-negativity property comes from the choice of the coefficients in the SDE rather than from reflection. The typical example is the Cox-IngersollRoss process :  t  t5 X(t) = x + (a − kX(s))ds + σ X(s)dZ(s), with x, a, σ ≥ 0 and k ∈ R, 0

0

which is used to model short interest rates but also stochastic volatility in the Heston model. It is not possible to discretize this SDE by the standard Euler scheme: indeed, X π (t1 ) is negative with positive probability and then it is not possible to compute the square root in the diffusion coefficient in order to define X π (t2 ). To overcome this problem, Deelstra and Delbaen [12], propose to take the positive part before the square root and define recursively: 5 X π (ti+1 ) = X π (ti )(1 −k(ti+1 −ti ))+ a(ti+1 −ti )+σ (X π (ti ))+ (Z(ti+1 )− Z(ti )). In her thesis [13], Diop studies the symmetrized Euler scheme defined by   5   X π (ti+1 ) = X π (ti )(1 − k(ti+1 − ti )) + a(ti+1 − ti ) + σ X π (ti ) (Z(ti+1 ) − Z(ti )) . In [1], Alfonsi compares those schemes with some new ones that he proposes. He concludes that the following explicit scheme combines the best features when 2 a ≥ σ4 :  2

5 k σ(Z(t ) − Z(t )) i+1 i X π (ti+1 ) = 1 − (ti+1 − ti ) X π (ti ) + 2 2(1 − k2 (ti+1 − ti ))

σ2 + a− (ti+1 − ti ). 4 This scheme has been implemented in Premia [56] in order to discretize the SSRD model of credit risk [7]. Another interesting issue is the discussion about adaptive methods. That is, how to choose the partition as to improve the first term in the expansion of the strong error. For this, we refer the reader to [9] and subsequent articles [26] and [46].

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References [1] Alfonsi, A. On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl. 11(4):355–384, 2005. [2] Bally, V. and Talay, D. The law of the Euler scheme for stochastic differential equations (II): convergence rate of the density. Monte Carlo Methods and Applications, 2:93–128, 1996. [3] Bally, V. and Talay, D. The law of the Euler scheme for stochastic differential equations (I): convergence rate of the distribution function. Probability Theory and Related Fields, 104:43–60, 1995. [4] Beskos, A. and Roberts G.O. Exact Simulation of Diffusions, Ann. Appl. Probab., 15(4):2422–2444, 2005. [5] Beskos, A., Papaspiliopoulos, O. and Roberts G.O. Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12(6):1077–1098, 2006. [6] Billingsley, P. Convergence of Probability Measures. John Wiley and Sons, 1968. [7] Brigo, D. and Alfonsi, A. Credit default swap calibration and derivatives pricing with the SSRD stochastic intensity model. Finance Stoch. 9(1):29–42, 2005. [8] Buckwar, E. and Shardlow, T. Weak approximation of stochastic differential delay equations. IMA J. Numerical Analysis, 25:57–86, 2005. [9] Cambanis, S. and Hu, Y. Exact convergence rate of the Euler-Maruyama scheme, with application to sampling design. Stochastics Stochastics Rep. 59(3-4):211–240, 1996. [10] Clement, E., Kohatsu-Higa, A. and Lamberton, D. A duality approach for the weak approximation of stochastic differential equations. Ann. Appl. Probab. 16(3):1124– 1154, 2006. [11] Cruzeiro, A.B., Malliavin, P. and Thalmaier, A.: Geometrization of Monte Carlo numerical analysis of an elliptic operator: strong approximation. C. R. Math. Acad. Sci. Paris 338(6):481–486, 2004. [12] Deelstra, G. and Delbaen, F. Convergence of discretized stochastic (interest rate) processes with stochastic drift term. Appl. Stochastic Models Data Anal. 14(1):77– 84, 1998. [13] Diop, A. Sur la discr´etisation et le comportement a ` petit bruit d’EDS unidimensionnelles dont les coefficients sont ` a d´eriv´ees singuli`eres. Th`ese Universit´e de Nice Sophia-Antipolis, 2003. [14] Detemple, J., Garcia, R., M. Rindisbacher. Asymptotic properties of Monte Carlo estimators of diffusion processes, Journal of Econometrics: 134(1):349–367, 2005. [15] Detemple, J., Garcia, R., and M. Rindisbacher, “Representation Formulas for Malliavin Derivatives of Diffusion Processes”, Finance and Stochastics, 9(3):349–367, 2005. [16] Detemple, J., Garcia, R., and M. Rindisbacher, “Asymptotic Properties of Monte Carlo Estimators of Derivatives”, Management Science, 51:1657–1675, 2005. [17] Fujiwara, T. Sixth order method of Kusuoka approximation. Preprint, 2006. [18] Gobet, E. LAMN property for elliptic diffusion: a Malliavin Calculus approach. Bernoulli 7:899–912, 2001.

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[19] Gobet E. LAN property for ergodic diffusions with discrete observations. Ann. Inst. H. Poincar´e Probab. Statist. 38:711–737, 2002. [20] Gobet E. and Munos, R. Sensitivity analysis using Itˆ o-Malliavin calculus and martingales. Application to stochastic optimal control. SIAM Journal on Control and Optimization, 43(5):1676–1713, 2005. [21] Gobet, E., Pages, G., Pham, H. and Printemps, J. Discretization and simulation of the Zakai equation. SIAM J. Numer. Anal. 44(6):2505–2538, 2006. [22] Gobet E. and Labart C. Error expansion for the discretization of Backward Stochastic Differential Equations. Stochastic Process. Appl. 117(7):803–829, 2007. [23] Grigoriu M. and Samorodnitsky G. Tails of solutions of certain nonlinear stochastic differential equations driven by heavy tailed L´evy processes. Stochastic Proc. Appl. 105:69–97, 2003. [24] Guyon, J. Euler scheme and tempered distributions. Stochastic Process. Appl. 116(6)877–904, 2006. [25] Hobson, D.G. and Rogers L.C.G. Complete models with stochastic volatility. Math. Finance 8(1):27–48, 1998. [26] Hofmann, N., M¨ uller-Gronbach, T. and Ritter, K. Optimal approximation of stochastic differential equations by adaptive step-size control. Math. Comp. 69(231):1017– 1034, 2000. [27] Ikeda, N. and Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. Amsterdam Oxford New York: North-Holland/Kodansha 1989. [28] Jacod, J., Kurtz, T., M´el´eard, S. and Protter, P. The approximate Euler method for L´evy driven stochastic differential equations. Ann. Inst. H. Poincar´e, Probab. Statist. 41(3):523–558, 2005. [29] Jacod, J. and Protter, P. Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26(1):267–307, 1998. [30] Jourdain, B. and Sbai, M. Exact retrospective Monte Carlo computation of arithmetic average Asian options. Monte Carlo Methods Appl. 13(2):135–171, 2007. [31] Kebaier, A. Statistical Romberg extrapolation: a new variance reduction method and applications to option pricing. Ann. Appl. Probab. 15(4):2681–2705, 2005. [32] Kloeden, P.E. and Platen, E. Numerical Solution of Stochastic Differential Equations. Springer Berlin Heidelberg New York, 1992. [33] Kohatsu-Higa, A. and Pettersson, R. On the simulation of some functionals of diffusion processes. Proceedings of the “Fourth Workshop on Stochastic Numerics”. Research Institute for the Mathematical Sciences, Kyoto 2000. [34] Kohatsu-Higa, A. and Yamazato, M. On moments and tail behaviors of storage processes. Journal of Applied Probability, 20:1069–1086, 2003. [35] Kurtz, T. and Protter, P. Weak convergence of stochastic integrals and differential equations. II. Infinite-dimensional case. Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995), Lecture Notes in Math. 1627:197– 285, Springer, Berlin, 1996. [36] Kurtz, T. and Protter, P. Weak convergence of stochastic integrals and differential equations. Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995), Lecture Notes in Math. 1627:1–41, Springer, Berlin, 1996.

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[37] Kurtz, T. and Protter, P. Characterizing the weak convergence of stochastic integrals. Stochastic analysis (Durham, 1990), London Math. Soc. Lecture Note Ser. 167:255– 259, Cambridge Univ. Press, Cambridge, 1991. [38] Kurtz, T. and Protter, P. Wong-Zakai corrections, random evolutions, and simulation schemes for SDEs. Stochastic analysis, 331–346, Academic Press, Boston, MA, 1991. [39] Kurtz, T. and Protter, P. Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19(3): 1035–1070, 1991. [40] Kusuoka, S. Approximation of expectation of diffusion process and mathematical finance. Taniguchi Conference on Mathematics Nara ’98, Adv. Stud. Pure Math. 31:147–165, Math. Soc. Japan, Tokyo, 2001. [41] Kusuoka, S.: Approximation of expectation of diffusion processes based on Lie algebra and Malliavin calculus. Adv. Math. Econ., 6:69–83, Springer, Tokyo, 2004. [42] Lyons, T. and Victoir, N.: Cubature on Wiener space. Stochastic analysis with applications to mathematical finance. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460:169–198, 2004. [43] Milshtein, G.N.: A method of second order accuracy integration of stochastic differential equations. Theory Probab. Appl. 23:396–401, 1978. [44] Menozzi, S. Discr´etisations associ´ees ` a un processus dans un domaine et sch´emas num´eriques probabilistes pour les EDP paraboliques quasilin´eaires. Th`ese Universit´e Pierre et Marie Curie – Paris VI, 2004. [45] Mikulevicius, R. and Platen, E. Time discrete Taylor approximations for Itˆ o processes with jump component. Math. Nachr. 138(9):93–104, 1988. [46] M¨ uller-Gronbach, T. Optimal pointwise approximation of SDEs based on Brownian motion at discrete points. Ann. Appl. Probab. 14(4):1605–1642, 2004. [47] Ninomiya, S. A new simulation scheme of diffusion processes: application of the Kusuoka approximation to finance problems. Math. Comput. Simulation 62:479– 486, 2003. [48] Ninomiya, S. A partial sampling method applied to the Kusuoka approximation. Monte Carlo Methods Appl. 9(1):27–38, 2003. [49] Ninomiya, M. and Ninomiya, S.: A new higher-order weak approximation scheme for stochastic differential equations and the Runge-Kutta method. Finance and Stochastics 13:415–443, 2009. [50] Ninomiya, S. and Victoir, N. Weak approximation of stochastic differential equations and application to derivative pricing. Applied Mathematical Finance, 15(2):107–121, 2008. [51] Nualart, D. The Malliavin Calculus and Related Topics. Springer Berlin Heidelberg New York, 1995. [52] Nualart, D. Analysis on Wiener space and anticipating stochastic calculus. Lecture Notes in Math., 1690:123–227, 1998. [53] Pag`es, G. Multi-step Richardson-Romberg Extrapolation: Remarks on Variance Control and complexity. Monte Carlo Methods Appl. 13(1):37–70, 2007. [54] Pettersson, R. Approximations for stochastic differential equations with reflecting convex boundaries. Stochastic Proc. Appl. 59:295–308, 1995.

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[55] Platen, E. An approximation method for a class of Itˆ o processes with jump component. Litovsk. Mat. Sb. 22(2):124–136, 1982. [56] PREMIA: An Option Pricer, Mathfi Project (INRIA, CERMICS, UMLV). http://www-rocq.inria.fr/mathfi/Premia/ [57] Protter, P.E. Stochastic integration and differential equations. Applications of Mathematics (New York) 21. Stochastic Modelling and Applied Probability. SpringerVerlag, 2004. [58] Slomi´ nski, L. On approximation of solutions of multidimensional SDEs with reflecting boundary condition. Stochastic Proc. Appl. 50:197–219, 1994. [59] Slomi´ nski, L. Euler approximation of solutions of SDEs with reflecting boundary. Stochastic Proc. Appl. 94:317–337, 2001. [60] Talay, D. and Tubaro, L. Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Anal. Appl. 8(4):483–509, 1990. Benjamin Jourdain Universit´e Paris-Est, CERMICS Project team MathFi ENPC-INRIA-UMLV 6 et 8, avenue Blaise Pascal Cit´e Descartes – Champs-sur-Marne F-77455 Marne-la-Vall´ee Cedex 2, France e-mail: [email protected] Arturo Kohatsu-Higa Ritsumeikan University Japan Science and Technology Agency Department of Mathematical Sciences 1-1-1 Nojihigashi, Kusatsu Shiga, 525-8577, Japan e-mail: [email protected]

Progress in Probability, Vol. 65, 145–167 c 2011 Springer Basel AG


Strong Consistency of Bayesian Estimator Under Discrete Observations and Unknown Transition Density Arturo Kohatsu-Higa, Nicolas Vayatis and Kazuhiro Yasuda Abstract. We consider the asymptotic behavior of a Bayesian parameter estimation method under discrete stationary observations. We suppose that the transition density of the data is unknown, and therefore we approximate it using a kernel density estimation method applied to the Monte Carlo simulations of approximations of the theoretical random variables generating the observations. In this article, we estimate the error between the theoretical estimator, which assumes the knowledge of the transition density and its approximation which uses the simulation. We prove the strong consistency of the approximated estimator and find the order of the error. Most importantly, we give a parameter tuning result which relates the number of data, the number of time-steps used in the approximation process, the number of the Monte Carlo simulations and the bandwidth size of the kernel density estimation. Mathematics Subject Classification (2000). 62F12, 62F15, 65C60. Keywords. Bayesian inference; strong consistency; discrete observations.

1. Introduction We consider a parameter estimation method of Bayesian type under discrete observations. That is, our goal is to estimate the posterior expectation of some function f given the observed data Y0N = (Y0 , Y1 , . . . , YN );
f (θ)φθ (Y0N )π(θ)dθ EN [f ] := Eθ [f |Y0 , . . . , YN ] :=
, (1.1) φθ (Y0N )π(θ)dθ where φθ (Y0N ) is the joint-density of the model process for Y0N = (Y0 , Y1 , . . . , YN ) and π(θ) is a prior distribution. This method can be applied when the joint-density φθ (Y0N ) is known a priori. In this article, we suppose that this is not the case.

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Suppose that Yi is a stationary Markov chain and therefore φθ (y0N ) = >N−1 µθ (y0 ) i=0 pθ (yi , yi+1 ), where µθ is a probability density function which is the invariant measure of Yi and pθ (y, z) is the transition density from y to z. But this expression (1.1) is still theoretical, as we do not know the transition density pθ . So we propose to estimate this quantity based on the simulation of the underlying process. Usually the underlying process is also approximated using, for example, the Euler-Maruyama scheme in the case that Y is generated by a diffusion. Then we approximate the transition density of the Euler-Maruyama approximation through the kernel density estimation method. Under these settings, we consider an approximation of the posterior expectation (1.1):
N f (θ)φˆN n θ (Y0 )π(θ)dθ ˆ EN,m [f ] :=
, N φˆN θ (Y0 )π(θ)dθ >N N N where φˆN ˆθ (Yj−1 , Yj ) and pˆN θ (Y0 ) := µθ (Y0 ) θ (y, z) is an approximation of j=1 p the transition density obtained using the kernel density estimation method. The problem we want to address is that there exists an appropriate choice of all the parameters that appear in the approximation of the posterior expectation so that convergence is obtained. In particular, the issue that is new in this research in comparison with previous results is that we give an estimate of how good the approximation of the transition density has to be as the amount of observations increase to infinity (therefore the number of arguments in qθ tend to infinity). In reality one also needs to approximate the invariant measure but this problem can be solved with an extra term. The quality of approximation is studied in Talay [10] and the references therein. Approximating posterior distribution can also be interpreted as a type of filtering problem for a diffusion process. This approach is considered in a general framework by Del Moral, Jacod, Protter [6]. Cano et al. [5] considered this problem based on Bayesian inference for a stochastic differential equation with parameter θ under similar settings to ours. They use direct calculations through an error estimation between the transition density of the true stochastic differential equation and the transition density of the Euler-Maruyama approximation given by Bally and Talay [1]. They prove that for fixed N the approximate posterior distribution which uses the exact density of the Euler approximation converges to the exact posterior distribution as the number of steps increase. Yasuda [12] improved this result and gave a general framework where the rate of convergence is N −1/2 under a condition of approximation for the simulated density and the density of the approximation. This condition ((6)-(b) in this article) hides the tuning relationship between m and N . The result is stated in general terms so that it can be adapted to various diffusion cases. In this article, we consider the problem of approximation the transition densities using the kernel density estimation method and therefore we clarify the tuning process that is required for the convergence of approximation of posterior expectations. Therefore, we give an explicit expression that shows how to choose

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parameters (number of Monte Carlo simulation n, bandwidth size of the kernel density estimation h and number of approximation parameter of the process m) based on the number of observations N . In this paper, we assume that observation data Y0 , Y1 , . . . , YN comes from a stationary and α-mixing process and the time interval between Yi and Yi+1 is fixed for all i. We give the relationship between the number of data and the parameters of the transition density approximation and we estimate the error as follows;   Ξ   n ˆN,m [f ] ≤ √ a.s. ω, ω ˆ, EN [f ] − E N where Ξ is some positive random variable with Ξ < +∞ a.s. ω, ω ˆ (here ω denotes the randomness associated with the data and ω ˆ denotes the randomness associated with the simulations). And also we prove that EN [f ] converges to f (θ0 ) with order √1 , where θ0 is the true value. N This paper is structured as follows. In Section 2, we will give the setting of our problem and precisely state our main theorem. In Section 3, we will prove our main theorem stated in Section 2 by using Laplace method dividing the proof in four estimations expressed in Proposition 3.1. This decomposition plays a central role in the proof. In Section 4, we will show how to deal with each term in the decomposition. In Section 5, we will give the tuning result which states the relationship between the number of Monte Carlo simulation n and the bandwidth size h with related to N so that a certain rate of convergence is achieved. The proofs of various statements are involved and therefore we have tried to give the main line of thought in the proofs in this article. One can find proofs of theorems, propositions and lemmas of this paper in Kohatsu-Higa et al. [8] and Yasuda [12]. And also one can find an example, the Ornstein-Uhlenbeck process case, of this paper in Kohatsu-Higa et al. [9].

2. Settings and main theorem 2.1. Settings First we recall the definition of α-mixing process; Definition 2.1 (Billingsley [2], p. 315). For a sequence X1 , X2 , . . . of random variables, let αn be a number such that |P (A ∩ B) − P (A)P (B)| ≤ αn for A ∈ σ(X1 , . . . , Xk ), B ∈ σ(Xk+n , Xk+n+1 , . . . ) and k, n ∈ N. Suppose that αn → 0 as n → ∞, then we call the sequence X1 , X2 , . . . , α-mixing process. For convenience, we set α0 = 1. Here we consider the following setting: Let θ0 ∈ Θ := [θl , θu ], (θl < θ u ) be a parameter that we want to estimate, where Θ is a compact subset in R ˙ where Θ ˙ denotes the interior of the set Θ and Θ0 = Θ − {θ0 }. and θ0 ∈ Θ, ¯ ¯ ˆ F, ˆ Pˆ ) be three probability spaces, where the Let (Ω, F, Pθ0 ), (Ω, F, P¯ ) and (Ω,

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probability measure Pθ0 is parametrized by θ0 . ∆ > 0 is a fixed parameter that represents the time between observations. The probability space (Ω, F, Pθ0 ) is ¯ F¯ , P¯ ) is used for the process that defines the process used for the observations, (Ω, ˆ Fˆ , Pˆ ) is used for the simulations that are used in with law Pθ and the space (Ω, estimating the transition density. (i) (Observation process) Let {Yi∆ }i=0,1,...,N be a sequence of N +1-observations of a Markov chain having transition density pθ0 (y, z), y, z ∈ R and invariant measure µθ0 . This sequence is defined on the probability space (Ω, F , Pθ0 ). We write Yi := Yi∆ for i = 0, 1, . . . , N . (ii) (Model process) Denote by X y (θ) a random variable defined on the proba¯ F, ¯ P¯ ) such that its law is given by pθ (y, ·). bility space (Ω, ˆ ˆ (iii) Denote by (Ω, F, Pˆ ) the probability space which generates the simulation of the random variable X y (θ). y (iv) (Approximating process) Denote by X(m) (θ) the simulation of an approximay ¯ F¯ , P¯ ). H ˆ t := Fˆt ⊗ F¯ tion of the model process X (θ), which is defined on (Ω, satisfies the usual condition. m is the parameter that determines the quality of the approximation. p˜N ˜N θ (y, ·) = p θ (y, ·; m(N )) is the density for the random y variable X(m) (θ). (v) (Approximated transition density) Let K : R → R+ be a kernel which sat
isfies K(x)dx = 1 and K(x) > 0 for all x. Denote by pˆN θ (y, z), the kernel y density estimation of p˜N (y, z) based on n simulation (independent) of X(m) (θ) θ ˆ F, ˆ Pˆ ). The outcomes of n simulations are denoted which are defined on (Ω, y,(k)

by X(m) (θ), k = 1, . . . , n. For h(N ) > 0, pˆN ˆN ˆ ; m(N ), h(N ), n(N )) θ (y, z) := p θ (y, z; ω   y,(k) n(N )  ˆ) − z X(m(N )) (θ, ω 1 . := K n(N )h(N ) h(N ) k=1

(vi) For given m, we introduce the “average” transition density over all trajectories with respect to the kernel K; p¯N ¯N θ (y, z) := p θ (y, z; m(N ), h(N ))    y,(1) X(m(N )) (θ, ·) − z ' N ( 1 ˆ pˆ (y, z) = E ˆ  , := E K θ h(N ) h(N ) ˆ means the expectation with respect to Pˆ . where E As it can be deduced from the above set-up, we have preferred to state our problem in abstract terms without explicitly defining the dynamics that generate y X y (θ) or how the approximation X(m) (θ) is defined. This is done in order not to obscure the arguments that follow and to avoid making the paper excessively long.

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All the properties that will be required for pθ are p˜N θ that will be satisfied for a subclass of diffusion processes. Remark 2.2. (i) Without loss of generality, we can consider the product of the above three probability spaces so that all random variables are defined on the same probability space. We do this without any further mentioning. (ii) Note that from the definition and the properties of the kernel K, p¯N θ (y, ·) satisfies for all θ ∈ Θ and y ∈ R  p¯N ¯N θ (y, z)dz = 1 and p θ (y, z) > 0 for all z ∈ R. Our purpose is to estimate the posterior expectation of some function f ∈ C 1 (Θ) given the data:
f (θ)φθ (Y0N )π(θ)dθ IN (f ) EN [f ] := Eθ [f |Y0 , . . . , YN ] = :=
, IN (1) φθ (Y0N )π(θ)dθ >N where φθ (Y0N ) = φθ (Y0 , . . . , YN ) = µθ0 (Y0 ) j=1 pθ0 (Yj−1 , Yj ) is the joint density of (Y0 , Y1 , . . . , YN ). We propose to estimate this quantity based on the simulation of the process:
n N IˆN,m (f ) f (θ)φˆN n θ (Y0 )π(θ)dθ ˆ EN,m [f ] := n :=
, N IˆN,m (1) φˆN θ (Y0 )π(θ)dθ >N N N where φˆN ˆθ (Yj−1 , Yj ). θ (Y0 ) := µθ (Y0 ) j=1 p 2.2. Main theorem Assumption 2.3. We assume the following (1) (Observation process) {Yi }i=0,1,...,N is an α-mixing process with αn = O(n−5 ). (2) (The prior distribution) The prior distribution π is continuous in θ. And the support is Θ, that is, for all θ ∈ Θ, π(θ) > 0. (3) (Density regularity) The transition densities , p, p¯N ∈ C 2,0,0 (Θ -× R2 ; R+ ), and for all θ ∈ Θ, y, z ∈ R, we have that min pθ (y, z), p¯N θ (y, z) > 0. And 0,0 pθ admits an invariant measure µ ∈ Cb (Θ × R; R+ ), and for all θ ∈ Θ, µθ (y) > 0 for every y ∈ R. (4) (Identifiability) Assume that there exist c1 and c2 : R → (0, ∞) such that for all θ ∈ Θ,  inf |qθi (y, z) − qθi 0 (y, z)|dz ≥ ci (y)|θ − θ0 |, N

and Ci (θ0 ) := qθ2 = p¯N θ .




ci (y)2 µθ0 (y)dy ∈ (0, +∞) for i = 1, 2 and qθ1 = pθ and

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(5) (Regularity of the log-density) We assume that for qθ = pθ ,¯ pN θ

12  i ∂ sup sup ln qθ (y, z) pθ0 (y, z)µθ0 (y)dydz < +∞, f or i = 0, 1, 2, ∂θi N θ∈Θ  2   ∂   < +∞, sup sup  2 (ln qθ (y, z)) p¯N (y, z)µ (y)dydz θ0 θ0  N θ∈Θ ∂θ    i  N ∂   sup sup  ∂θi ln qθ (y, z) p¯θ0 (y, z)µθ0 (y)dydz < +∞, f or i = 0, 1, N θ∈Θ 0

∂ 1 where ∂θ 0 qθ = qθ . (6) (Parameter tuning) (a) We assume the following boundedness condition; 

 −1  1 N ∂ ∂   N N sup sup  √ ln pˆθ (Yi , Yi+1 ) − ln p¯θ (Yi , Yi+1 )  < +∞ a.s.  ∂θ ∂θ N θ∈Θ  N i=0

(b) Assume that for each y, z ∈ R, there exists a factor C1N (y, z) and c1 (y, z) such that   N pθ0 (y, z) − p¯N  θ0 (y, z) ≤ C1 (y, z)a1 (N ), where supN C1N (y, z) < +∞ and a1 (N ) → 0 as N → ∞, and √ C1N (y, z)a1 (N ) N < c1 (y, z), where c1 satisfies the following;    ∂   ln p¯N (y, z) c1 (y, z)µθ0 (y)dydz < +∞. sup sup θ   ∂θ N θ∈Θ (c) There exist some function g N : R2 → R and constant a2 (N ), which depends on N , such that for all y, z ∈ R   ∂  ∂  ≤ |g N (y, z)|a2 (N ), sup  ln p¯N (y, z) − ln p (y, z) θ θ  ∂θ θ∈Θ ∂θ where supN Eθ0 [|g N (Y0 , Y1 )|4 ] < +∞ and a2 (N ) → 0 as N → ∞. Remark 2.4. (i) Assumption 2.3 (4) is needed in order to be able to obtain that the density can be used in order to discern the value of θ from the observations. This type of assumption is natural in statistics and can be assured in the case of one-dimensional stochastic differential equations under differentiability of the coefficients. Assumption 2.3 (5) will be satisfied under enough regularity of the transition density function pθ and its approximation p¯N θ . This can be achieved with Malliavin Calculus techniques in the case of diffusion equations. 1 The

power 12 is needed to prove a central limit theorem (Proposition 4.4.).

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(ii) Assumption 2.3 (6)-(a) will be crucial in what follows and it is the property that will determine the rate of convergence and the tuning properties. Note that all other hypothesis deal with the transition density pθ or the average of its approximation p¯N θ . Therefore the needed properties essentially follow from similar properties of pθ and some limit arguments. Assumption 2.3 (6)(a) is the only condition that deals with the approximation itself pˆN θ , which is random. In particular, obtaining a lower bound for pˆN will be the important θ problem to solve. This will be further discussed in Section 5. (iii) In this problem, we need to study two approximation problems, one is a difference between the transition densities of the observation process and the approximated process, and the other is a difference of the transition density of the approximated process and the expectation for the approximation based on kernel density estimation. Assumption 2.3 (6)-(b) and (c) state the rate of convergence of the density of the approximation and its derivatives. This problem was studied in Bally, Talay [1] and Guyon [7], and the second approximation problem can be dealt using the kernel density estimation theory. For example, in the case that the data {Yi } comes from a stochastic differential equation, the Euler-Maruyama approximation is most commonly used. And in that case, we have the following results. Assume that the drift and diffusion coefficients belong to Cb∞ (R), and the diffusion coefficient satisfies the uniform ellipticity condition. Then for α, β ∈ N, there exist c1 ≥ 0 and c2 > 0 such that for all N ≥ 1, ∆ ∈ (0, 1] and y, z ∈ R, 1 m(N ) α β ∂yα ∂zβ p˜N ∂ α ∂ β πθ (∆, y, z) + rθ0 (∆, y, z), θ0 (y, z) − ∂y ∂z pθ0 (y, z) ≤ m(N ) y z 0 where let D be a differential operator and ptθ0 (y, z) be the transition density of Yt , set  ∆ ∞ πθ0 (∆, y, z) := psθ0 (y, w)D(p∆−s θ0 (·, z))(w)dwds, 0

and

−∞



1 1 c2 |y − z|2 exp − . m(N )2 ∆ α+β+5 ∆ 2 For more details, see Proposition 1 in Guyon [7]. For the second problem, in the case that the kernel K satisfies   xK(x)dx = 0 and x2 K(x)dx < +∞,    m(N )  rθ0 (∆, y, z) ≤ c1

if p˜N θ0 (y, z) is uniformly bounded in y, z ∈ R and N ∈ N and twice continuously differentiable with respect to z for all y ∈ R and N ∈ N, then we have, for all y ∈ R and N ∈ N,  ∂ 2 p˜N 1 θ0 N N 2 p¯θ0 (y, z) − p˜θ0 (y, z) = h(N ) (y, z) x2 K(x)dx + o(h(N )2 ). 2 ∂z 2 For more details, see Wand, Jones [11].

152

A. Kohatsu-Higa, N. Vayatis and K. Yasuda 1 Therefore roughly speaking, a1 (N ) = m(N + h(N )2 . And the choice for ) √ m(N ) = N will satisfy Assumption 2.3 (6)-(b).

Now we state the main result of the paper. Theorem 2.5. Under Assumption 2.3, there exists some positive random variable Ξ such that Ξ < +∞ a.s. and   Ξ   n [f ] ≤ √ a.s. EN [f ] − EˆN,m N And also, we have   Ξ1 Ξ2  n  |EN [f ] − f (θ0 )| ≤ √ a.s. and EˆN,m [f ] − f (θ0 ) ≤ √ a.s., N N where Ξ1 is some positive random variable with Ξ1 < +∞ a.s. and Ξ2 is some positive random variable with Ξ2 < +∞ a.s.

3. Idea of the proof of Theorem 2.5 First we introduce some notation. Let p, q : R2 → R+ be strictly positive functions of two variables. Then let  ) * ln p(y, z) q(y, z)µθ0 (y)dydz, H(p, q) := N −1 " 1 ! ZN (θ) := √ ln pθ (Yi , Yi+1 ) − H(pθ , pθ0 ) , N i=0

ε(θ) := H(pθ , pθ0 ) − H(pθ0 , pθ0 ), βN (θ) := ZN (θ) − ZN (θ0 ). And also we set; N −1  N N  1  Z¯N (θ) := √ ln pˆN ¯θ , p¯θ0 , θ (Yi , Yi+1 ) − H p N i=0    N N ε¯N (θ) := H p¯N ¯N ¯θ0 , p¯θ0 , θ ,p θ0 − H p ¯ ¯ ¯ βN (θ) := ZN (θ) − ZN (θ0 ).

Set Θ0 := Θ \ {θ0 }. The following proposition states the properties that are needed to achieve the proof of Theorem 2.5 Proposition 3.1. Under Assumption 2.3, we have the following results. (i) There exist some strictly negative constants c1 , c2 such that c1 ≤ inf

θ∈Θ0

ε(θ) ε(θ) ≤ sup ≤ c2 < 0. (θ − θ0 )2 (θ − θ0 )2 θ∈Θ0

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(ii) There exist some random variables d1 , d2 such that d1 ≤ inf inf

N θ∈Θ0

βN (θ) βN (θ) ≤ sup sup ≤ d2 a.s. θ − θ0 N θ∈Θ0 θ − θ0

(iii) There exist some strictly negative constants c3 , c4 such that c3 ≤ inf inf

N θ∈Θ0

ε¯N (θ) ε¯N (θ) ≤ sup sup ≤ c4 < 0. 2 (θ − θ0 )2 N θ∈Θ0 (θ − θ0 )

(iv) There exist some random variables d3 , d4 such that d3 ≤ inf inf

N θ∈Θ0

β¯N (θ) β¯N (θ) ≤ sup sup ≤ d4 a.s. θ − θ0 N θ∈Θ0 θ − θ0

We will give an idea of the proof of this proposition after this section. We give the proof of Theorem 2.5 using the results of Proposition 3.1. Idea of the Proof of Theorem 2.5. We decompose the approximation error as follows; 

 ˆn n IN,m (f ) − f (θ0 )IˆN,m (1) I (f ) − f (θ )I (1) N 0 N n ˆ EN [f ] − E − . N,m [f ] = n IN (1) IˆN,m (1) The goal is then to prove that there exists some random variable C1 and C2 such that      Iˆn (f ) − f (θ0 )Iˆn (1)   IN (f ) − f (θ0 )IN (1)  C1 C2  N,m  N,m  ≤ √ a.s., and   ≤ √ a.s.   n   IN (1) N N IˆN,m (1) n Indeed, we can write IN (f ) and IˆN,m (f ) as follows;

IN (f ) = e

√ N H(pθ0 ,pθ0 )+ N ZN (θ0 ) N

N

n (f ) = eN H(p¯θ0 ,p¯θ0 )+ IN,m

√ ¯N (θ0 ) NZ

 f (θ)eN ε(θ)+

√ N βN (θ)

Θ0



f (θ)eN ε¯(θ)+

√ N β¯N (θ)

µθ (Y0 )π(θ)dθ, µθ (Y0 )π(θ)dθ.

Θ0

Then by using the Laplace method and Proposition 3.1, we have our conclusion. In fact, as N goes to infinity the leading term in the quotients

IN (f ) IN (1)

and

n IˆN,m (f ) ˆ I n (1)

are

N,m

determined by ε(θ) and ε¯(θ) due to Proposition 3.1. Their behaviors are similar to Gaussian integrals where the variance tends to zero and therefore the integrals will tend to the value in their “mean” which in this case is θ0 as it follows from Proposition 3.1. Details can be found in Kohatsu-Higa et al. [8]. 

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4. Proof of Proposition 3.1 4.1. Ideas for the proof of Proposition 3.1 (i) and (iii) In order to prove the upper bounds for the statements in Proposition 3.1 (i) and (iii), one uses the Pinsker’s inequality; 

2  1 pθ (y, z) |pθ (y, z) − pθ0 (y, z)| dz ≤ ln 0 pθ (y, z)dz, 2 pθ (y, z) 0 and the identifiability condition. Therefore under Assumptions 2.3 (4), we obtain the upper bound in (i) and under Assumption 2.3 (4) and (5), we obtain the upper bound in (iii). For the lower bounds, we give a useful lemma for the first derivative of H(pθ , pθ0 ) in θ; Lemma 4.1. Let q be a transition density, which depends on a parameter θ. We assume that for all θ ∈ Θ,  ∂ (ln q(y, z; θ)) q(y, z; θ0 )µθ0 (y)dydz ∂θ

 ∂ = ln q(y, z; θ) q(y, z; θ0 )µθ0 (y)dydz. ∂θ Then ∂ ∂θ



  (ln q(y, z; θ)) q(y, z; θ0 )µθ0 (y)dydz 

= 0.

θ=θ0

Using the above lemma together with Taylor’s expansion we obtain the lower bound for (i) under Assumption 2.3 (3), (4) and (5), and under Assumption 2.3 (4) and (5) for (iii). 4.2. Ideas for the proof of Proposition 3.1 (ii) In this section, we consider Proposition 3.1 (ii). We prove this boundedness by using a central limit theorem in C(Θ; R∞ ). Note that C(Θ; R∞ ) is a complete and separable metric space with metric η defined as follows; For x = (x1 , x2 , . . . ), y = (y1 , y2 , . . . ) ∈ C(Θ; R∞ ), ∞  1 {|xi (θ) − yi (θ)| ∧ 1} . i θ∈Θ i=1 2

η(x, y) := sup For N ∈ N, set

 β (θ) ZN (θ) − ZN (θ0 )   N = if θ = θ0 , θ − θ0 θ − θ0 β˜N (θ) :=   ∂ ZN (θ0 ) if θ = θ0 . ∂θ And set γN := (β˜N , β˜N −1 , . . . , β˜1 , 0, . . . ) ∈ C(Θ; R∞ ). The reason why we need to use R∞ in the above setting can be seen from the following lemmas. In fact, in order to prove the boundedness of β˜N (θ) in N , we need to consider another

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random vector which has the same “joint”-distribution as γN when we apply the Skorohod representation theorem. The idea of the proof consists of proving that the sequence γN converges weakly in C(Θ; R∞ ). Therefore the limit γ = (γ 1 , γ 2 , . . . ) should satisfy that there exist some random variables d1 , d2 such that d1 ≤ inf γ 1 (θ) ≤ sup γ 1 (θ) ≤ d2 a.s. θ∈Θ

θ∈Θ

Now, without loss of generality we can use the Skorohod representation theorem, 1 so that the first component γN = β˜N of γN will satisfy a similar property. And finally, from the convergence, we obtain the boundedness of β˜N in N . In particular, we use the following lemmas. Lemma 4.2. Let X = (X1 , X2 , . . . ), Y = (Y1 , Y2 , . . . ) ∈ C(Θ; R∞ ) be random d variables such that X = Y . Then we have d

sup sup |Xn (θ)| = sup sup |Yn (θ)|. n θ∈Θ

n θ∈Θ

Lemma 4.3. Let (S, · ) be a complete, separable metric normed space. Let X be an S-valued random variable on a probability space (Ω, F , P ) and let Y be an d

S-valued random variable on a probability space (Ω , F  , P  ). Supposed that X = Y and there exists an R+ -valued random variable d on (Ω , F  , P  ) such that Y (ω  ) ≤ d(ω  )

for all ω  ∈ Ω .

Then there exists a positive random variable M on (Ω, F, P ) such that X(ω) ≤ M (ω) a.s. ω ∈ Ω. In order to prove the weak convergence of γN we extend Theorem 7.1 and Theorem 7.3 in Billingsley [3] which imply that it is enough to prove convergence of marginals and tightness of the sequence γN . In order to prove convergence of marginals, we use Assumption 2.3 (1) and (5), so that for every r ∈ N and θ1 , . . . , θr ∈ Θ, (γN (θ1 ), . . . , γN (θr )) converges weakly. The proof uses the Cr´amer-Wold device and the following extension of the Central limit theorem for α-mixing processes (the proof is an extension of Theorem 27.5 in Billingsley [2], p. 316.) Proposition 4.4. Suppose that X1 , X2 , . . . is stationary and α-mixing with αn = O(n−5 ), where we set α0 = 1 and f is a B(R2 )-measurable function which satisfies E[f (X0 , X1 )] = 0 and E[f (X0 , X1 )12 ] < +∞. If we set fi := f (Xi , Xi+1 ) and Sn := f1 + · · · + fn , then ∞  ' ( 1 V ar(Sn ) −→ σ 2 := E f12 + 2 E [f1 f1+k ] , n k=1

where the series converges absolutely. And ⇒ M , where M is a normal distributed random variable with mean 0 and variance σ 2 . If σ = 0, we define M = 0. Sn √ n

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To prove the tightness of γN , we use Assumption 2.3 (1), (3) and (5). One needs to prove that for all ε > 0,   N  "  1 ! ˜   ˜ lim lim sup Pθ0 sup βN −i+1 (θ) − βN −i+1 (θ ) ∧ 1 ≥ ε = 0. i δ→0 N →∞ |θ−θ  |≤δ i=1 2 In the proof, the basic ingredient is the Garsia, Rodemich, Rumsey lemma. 4.3. Decomposition for the estimation for Proposition 3.1 (iv) Set  . : N −1  pˆN 1  1 pˆN  θ0 θ  ln N (Yi , Yi+1 ) − ln N (Yi , Yi+1 ) , θ = θ0   √N θ − θ0 p¯θ p¯θ0 1 i=0 JN (θ) :=  N −1 N    1 ∂ pˆ    √ ln θN (Yi , Yi+1 ) , θ = θ0 ,  ∂θ p ¯ N i=0 θ θ=θ0  √   ln p¯N , ¯N  θ − ln p θ0   N (y, z) p¯N θ = θ0 θ0 − pθ0 (y, z)µθ0 (y)dydz, θ − θ0 2   JN (θ) := √  , N ∂    ln p¯N p¯θ0 − pθ0 (y, z)µθ0 (y)dydz, θ = θ0 ,  N θ (y, z) ∂θ θ=θ0 .  N −1 N N  1  ln p¯θ − ln p¯θ0   √ (Yi , Yi+1 )   θ − θ0   N i=0 :      ln p¯N − ln p¯N θ θ  0  − (y, z)pθ0 (y, z)µθ0 (y)dydz , θ = θ0  θ − θ0 3 JN (θ) :=  N −1%     1  ∂ N   √ ln p ¯ (Y , Y )  i i+1 θ    N ∂θ  θ=θ0 i=0   &     ∂  N   − ln p¯θ (y, z) pθ0 (y, z)µθ0 (y)dydz , θ = θ0 ,  ∂θ θ=θ0

Then we define: β¯N (θ) 1 2 3 := JN (θ) − JN (θ) + JN (θ). θ − θ0 1 We can easily prove boundedness of the first term JN (θ) from Assumption 2.3 2 (6)-(a), and also we can easily prove boundedness of JN (θ) from Assumption 2.3 3 3 (6)-(b). Finally we consider the third term JN (θ). To prove boundedness of JN (θ), we prove a weak convergence by using the similar argument as Section 4.2. Then we can prove Proposition 3.1 (iv).

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5. Parameter tuning and the Assumption 2.3 (6)-(a) This section is devoted to proving that Assumption 2.3 (6)-(a) is satisfied under sufficient smoothness hypothesis on the random variables and processes that appear in the problem as well as the correct parameter tuning. That is, we need to prove that the following condition (Assumption 2.3 (6)-(a) in Section 2.2) is satisfied 

 −1  1 N ∂ ∂   sup sup  √ ln pˆN ln p¯N θ (Yi , Yi+1 ) − θ (Yi , Yi+1 )  < +∞ a.s. (5.1)   ∂θ ∂θ N i=0 N θ∈Θ In order to understand the role of all the approximation parameters, we rewrite pˆN ¯N θ and p θ as follows y,(k) n X(m) (θ) − z

1  N pˆθ (y, z) := K nh h k=1  y,(1)

X(m) (θ, ·) − z 1 N p¯θ (y, z) := E K . h h Here m ≡ m(N ), n ≡ n(N ) and h ≡ h(N ) are parameters that depend on N . n is the number of Monte Carlo simulations used in order to estimate the density and m is the parameter of approximation (in the Euler scheme this is the number y,(1) of time steps used in the simulation of X(m) (θ)) and h is the bandwidth size associated to the kernel density estimation method. In this sense we will always think of hypotheses in terms of N although we will drop them from the notation and just use m, n and h. The goal of this section is to prove that under certain hypotheses, there is a choice of m, n and h that ensures that condition (5.1) is satisfied. As the main problem is to obtained upper and lower bounds for pˆN θ for random arguments, we will first restrict the values for the random variables Yi , i = 0, . . . , N − 1 to a compact set. This is obtained using an exponential type Chebyshev’s inequality and the Borel-Cantelli lemma (Theorem 4.3 in pp. 53 of Billingsley [2]) as follows Lemma 5.1. Assume the following hypothesis 2 (H0) mc1 := supi E[ec1 |Yi | ] < ∞ for some constant c1 > 0. Furthermore let aN ≥ θu − θl be a sequence of strictly positive numbers such that ∞   N exp −c1 a2N < ∞. N=1

Then we have that for a.s. ω ∈ Ω, there exists N big enough such that maxi=1,...,N |Yi | < aN . That is, for AN := {ω ∈ Ω; ∃i = 1, . . . , N s.t. |Yi | > aN } we have

) * P lim sup AN = 0. N →∞

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The decomposition that we will use in order to prove (5.1) is the same decomposition as in the proof of Theorem 3.2 in p. 73 of Bosq [4]. That is,    ∂θ pˆN  ∂θ p¯N θ θ  sup sup  N (x, y) − N (x, y) pˆθ p¯θ θ∈Θ |x|,|y|≤aN    supθ∈Θ sup|x|,|y|≤aN ∂θ pˆN ¯N θ (x, y) − ∂θ p θ (x, y) ≤ N inf θ∈Θ inf |x|,|y|≤aN p¯θ (x, y)      ∂θ pˆN  supθ∈Θ sup|x|,|y|≤aN pˆN (x, y) − p¯N (x, y) θ θ θ + sup sup  N (x, y) pˆθ inf θ∈Θ inf |x|,|y|≤aN p¯N θ∈Θ |x|,|y|≤aN θ (x, y) =:

A D +C , B B

(5.2)

where we remark that   y,(k) n  X(m) (θ) − z 1  ∂θ X y,(k) (θ) , ∂θ pˆN K  θ (x, y) = (m) nh2 h k=1



   y,(k) X (θ) − z 1  K   (m)  ∂θ X y,(k) (θ) . ∂θ p¯N θ (x, y) = E (m) h2 h √ A  Therefore in order to prove the finiteness of (5.1), we need to bound N B +C D B . This will be done in a series of lemmas using Borel-Cantelli arguments together with the modulus of continuity of the quantities p¯N ˆN θ and p θ . First, we start anaD lyzing the difficult part C B . 5.1. Upper bound for C D in (5.2) B We work in this section under the following hypotheses: (H1) Assume that there exist some positive constants ϕ1 , ϕ2 , where ϕ1 is independent of N and ϕ2 is independent of N and ∆, such that the following holds: inf

(x,θ)∈B N

p¯N θ (x, y)



ϕ2 a2N ≥ ϕ1 exp − , ∆

where we set , B N := (x, θ) ∈ R2 × Θ; x < aN , where · is the max-norm.

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(H2) Assume that the kernel K is the Gaussian kernel;

1 1 K(z) := √ exp − z 2 . 2 2π (H3) Assume that for some constant r3 > 0 and a sequence {b3,N ; N ∈ N} ⊂ [1, ∞), ∞ na2r3 E[|Z3,N (·)|r3 ] we have that N =1 N (h2 b3,N < ∞, where )r3 

(k) Z3,N

(ω) :=

a−2 N

sup (x,θ)∈B N

   x,(k)  X(m) (θ, ω) + 1



sup (x,θ)∈B N

    x,(k) ∂θ X(m) (θ, ω) .

(H4) Assume that for some constant r4 > 0 and a sequence {b4,N ; N ∈ N} ⊂ [1, ∞), r 4 3  4 (k) (·) ∞ nE Z4,N we have that N =1 < ∞, where (b4,N )r4 

(k) Z4,N

(ω) := a−1 N

sup (x,θ)∈B N

    x,(k) ∂x X(m) (θ; ω) +

sup (x,θ)∈B N

     x,(k) ∂θ X(m) (θ; ω) .

(H5) Assume that there exists some positive constant C5 > 0 such that for all x, y ∈ R, m ∈ N and θ ∈ Θ,       ∂x p¯N (x, y) , ∂y p¯N (x, y) , ∂θ p¯N (x, y) ≤ C5 < +∞. θ θ θ (H6) Assume that ηN and νN are sequences of positive numbers so that

∞  (ηN )2 nh2 3 νN exp − < ∞, 16 K ∞ N=1

where · ∞ denotes the sup-norm. Note that from the assumption (H1), we have a lower bound of B in (5.2). Lemma 5.2. Assume hypothesis (H2) and (H3), then we have that  . :    ∂θ pˆN  θ P lim sup sup sup  N (x, y) > b3,N = 0. pˆθ N →∞ θ∈Θ |x|,|y|≤aN The proof of the lemma follows by rewriting the fraction that it can be bounded using the upper bound of approximating process with respect to θ.

K K

∂θ pˆN θ pˆN θ

and show

and the derivative of the

5.2. Upper bound of D in (5.2) In this section, we use the modulus of continuity of pˆN ¯N θ and p θ in order to find an upper bound for B. Similar ideas appear in Theorem 2.2 on p. 49 of Bosq [4].

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Lemma 5.3. Set % & aN θ u − θl N BlN1 l2 := (x, θ) ∈ R2 × Θ; x − xN ≤ , |θ − θ | ≤ , l1 l2 νN νN 2 l1 = 1, . . . , νN , l2 = 1, . . . , νN ,

˚N ∩ B ˚N  = ∅ ((l1 , l2 ) = (l1 , l2 )) and appropriate set of points xN , θN , such that B l1 l2 l1 l2 l l 1 2

ν2

2 l1 = 1, . . . , νN and l2 = 1, . . . , νN such that ∪l1N=1 ∪νl2N=1 BlN1 l2 = B N . Then    N   pˆ (x) − p¯N (x) sup pˆN ¯N sup θ (x) − p θ (x) = max2 θ θ 1≤l1 ≤ν N 1≤l2 ≤νN

(x,θ)∈B N

≤ max2

1≤l1 ≤ν N 1≤l2 ≤νN

+ max2

1≤l1 ≤ν N 1≤l2 ≤νN

sup (x,θ)∈BlN l

1 2

sup (x,θ)∈BlN l

1 2

(x,θ)∈BlN l

1 2

   N   N   N  N  N N x + max p ˆ x − p ¯ xl1  pˆθ (x) − pˆN   N N N l1 l1 θ θ θ 2 l2

   N  N  pθ N xl1 − p¯N (x) ¯ . θ

1≤l1 ≤ν N 1≤l2 ≤νN

l2

l2

(5.3)

l2

The proof of the above lemma is straightforward. We consider the first term of (5.3). Lemma 5.4. Under (H2) and (H4), then we have that         2  N  2 K ∞ aN  N  = 0. P lim sup max2 sup pθ (x) − pˆN xl1  > b ˆ N 4,N θ l2  h2 νN N →∞  1≤l1 ≤νN (x,θ)∈B N 1≤l2 ≤νN

l1 l2

Now we consider the third term in (5.3). Lemma 5.5. Assume (H5), then, max2

1≤l1 ≤ν N 1≤l2 ≤νN

sup (x,θ)∈BlN l

1 2

  aN  N  N  pθ N xl1 − p¯N (x) ≤ 3C5 ¯ θ l2 νN

Finally, we consider the second term of (5.3). Lemma 5.6. Assume (H2) and that ηN satisfies (H6), then we have that             N  P lim sup max2 ˆ pθN xN − p¯N xN  > ηN  = 0. N l l θ 1 1 l2 l2  N →∞  1≤l1 ≤νN 1≤l2 ≤νN

Now we can conclude this section with the following upper bound for C D B. Theorem 5.7. Assume conditions (H2), (H4), (H5) and (H6), then there exists N big enough such that

D 1 ϕ2 a2N 2 K  ∞ a2N aN (3) C ≤ bN × exp × b4,N + ηN + 3C5 . B ϕ1 ∆ h2 νN νN

Strong Consistency of Bayesian Estimator 5.3. Upper bound for

A B

161

in (5.2)

The proof in this case is simpler on the one hand because many of the previous estimates can be used. On the other hand, when considering the analogous result of Lemma 5.6 for the derivatives, ∂θ pˆN ¯N θ and ∂θ p θ , has to be reworked as the condition N N ˆ pθ ∞ < +∞ and ¯ pθ ∞ < +∞ are not valid. From the condition (H1), we have   A 1 ϕ2 a2N  ≤ e ∆ × sup sup ∂θ pˆN ¯N θ (x, y) − ∂θ p θ (x, y) . B ϕ1 θ∈Θ |x|,|y|≤aN Here we consider the sup-term as before. Lemma 5.8. We use the same notations as the previous section. Then we have    sup sup ∂θ pˆN ¯N θ (x) − ∂θ p θ (x) θ∈Θ |x|,|y|≤aN

≤ max

1≤l1 ≤ν 2 N 1≤l2 ≤νN

  N   N xl1  ∂θ pˆθ (x) − ∂θ pˆN N θ

sup (x,θ)∈BlN l

l2

1 2

  N  N   N + max2 ∂θ pˆN x − ∂ p ¯ xl1  N N θ l1 θ θ 1≤l1 ≤ν N 1≤l2 ≤νN

+ max2

1≤l1 ≤ν N 1≤l2 ≤νN

l2

l2

sup (x,θ)∈BlN l

   N  N  (x) ∂θ p¯θ N xl1 − ∂θ p¯N . θ

(5.4)

l2

1 2

As in previous sections, we define that          (k) x,(k) x,(k) −1 ˙ Z4,N := aN h sup sup ∂x ∂θ X(m) (θ) + h sup sup ∂θ ∂θ X(m) (θ) θ∈Θ |x|≤aN

θ∈Θ |x|≤aN

   ) *   (k) x,(k) + Z4,N + 1 sup sup ∂θ X(m) (θ) . θ∈Θ |x|≤aN

(k) Note that {Z˙ 4,N }k=1,2,... is a sequence of independent and identically distributed random variables. Then we set the following hypothesis. (H4
). Assume that for some constant r˙ > 0 and a sequence {b˙ 4,N ; N ∈ N} ⊂ # $ 4

[1, ∞), we have that

   (k)  r˙ 4 nE  Z˙ 4,N 

∞ N =1

(b˙ 4,N )r˙ 4

< ∞.

Lemma 5.9. Under the above hypothesis and (H4
), we have that  .   N   P lim sup max2 sup ˆN ∂θ pˆN θ (x) − ∂θ p θ N xl1  N→∞

1≤l1 ≤ν N 1≤l2 ≤νN

(x,θ)∈BlN l

1 2

K  ∞ ∨ K  ∞ a2N ˙ ≥ b4,N h3 νN

l2

: = 0.

162

A. Kohatsu-Higa, N. Vayatis and K. Yasuda We consider the third term of (5.4).    N  N  max2 sup ∂θ p¯θ N xl1 − ∂θ p¯N θ (x) 1≤l1 ≤ν N 1≤l2 ≤νN

≤

aN νN

(x,θ)∈BlN l

1 2

max2

1≤l1 ≤ν N 1≤l2 ≤νN

l2

% sup

(x,θ)∈BlN l

1 2

   N  sup ∂x ∂θ p¯N θ εx + (1 − ε)xl1 , y

0≤ε≤1

  N  N  + sup ∂y ∂θ p¯N θ xl1 , εy + (1 − ε)yl1 0≤ε≤1

&   N   N + sup ∂θ ∂θ p¯εθ+(1−ε)θ N xl1  . 0≤ε≤1

l2

(H5
). Assume that there exists some positive constant C˙ 5 > 0 such that for all x, y ∈ R, m ∈ N and θ ∈ Θ,      2 N  ∂x ∂θ p¯N     ¯θ (x, y) ≤ C˙ 5 < +∞. ¯N θ (x, y) , ∂y ∂θ p θ (x, y) , ∂θ p Lemma 5.10. Assume (H5
). Then, we have    N aN ∨ (θu − θl )   N ˙ max2 sup x − ∂ p ¯ (x) ≤ 3 C . ∂θ p¯N  N θ 5 l1 θ θl 1≤l1 ≤ν νN 2 N (x,θ)∈B N l l 1≤l2 ≤νN

1 2

Finally, we consider the second term of (5.4). Set   x,(j) X (θ) − y 1 (m) (j),x  ˙  ∂θ X x,(j)(θ) W m,h (θ) := 2 K (m) h h     x,(1) 1    X(m) (θ; ·) − y  x,(1) − 2E K ∂θ X(m) (θ) . h h ˙ (j),x (θ)}j=1,2,... is a sequence of independent and identically disNote that {W m,h 3 4 ˙ (j),x (θ) = 0. tributed random variables with E W m,h (H6
). There exists C˙ 6 > 0 and α˙ 6 > 0 and a sequence of positive numbers b˙ 6,N such that  . :n ∞  n(ηN )2 h4 C˙ 6 3 νN exp − 1 + 1+α˙ 6 < ∞. 2 n 2 K  ∞ (b˙ 6,N )2 a2N N=1 (H6a
). Assume that there exists r˙6 > 0 and4 a sequence of positive numbers b˙ 6,N 3 r˙ E |Z˙ 6,N | 6 ∞ in (H6
) satisfy, such that N=1 n (b˙ )r˙6 < ∞ for 6,N  4" ! 3     (j) x,(j) x,(1) Z˙ 6,N := a−1 sup sup X (θ) + E X (θ) ∂  ∂  . θ (m) θ (m) N θ∈Θ |x|≤aN

Strong Consistency of Bayesian Estimator

163

#  $  ˙ q˙6 (H6b ). Assume that for some q˙6 > 1, supN ∈N E Z6,N  < +∞ and for α˙ 6 > 0,


C˙ 6 > 0 and b˙ 6,N given in (H6
) the following is satisfied   q˙6 C˙ 6 ηN h2 (ηN )2 exp − K  ≤ 1+α˙ 6 .  2 ( K ∞ daN ,m ) aN n 2( 2 ∞ b˙ 6,N aN )2 h

Applying Lemma 6.2 in Appendix, we obtain the following important lemma. Lemma 5.11. Assume (H6), (H6a
) and (H6b
). Then there exists N big enough such that   N  N   N max2 ∂θ pˆN x − ∂ p ¯ xl1  ≤ ηN . N N θ l θ θ 1 1≤l1 ≤ν N 1≤l2 ≤νN

l2

l2

Theorem 5.12. Assume conditions (H1), (H2), (H4
), (H5
), (H6
), (H6a
) and (H6b
). Then we have that for a.s. ω, there exists N0 ≡ N0 (ω) such that for all N ≥ N0 we have

K  ∞ ∨ K  ∞ a2N ˙ A 1 ϕ2 a2N aN ˙ ≤ e ∆ × b + η + 3 C . 4,N N 5 B ϕ1 h3 νN νN Finally putting all our results together, we have (see Theorem 5.7). Theorem 5.13. Assume conditions (H0), (H1), (H2), (H3), (H4), (H5), (H6), (H4
), (H5
), (H6
), (H6a
) and (H6b
). Then for a.s. ω, there exists N0 ≡ N0 (ˆ ω) such that for all N ≥ N0 we have   

√ √ 1 ϕ2 a2N K  ∞ ∨ K  ∞ a2N b˙ 4,N A D N +C ≤6 N e ∆ × + b4,N b3,N B B ϕ1 h2 νN h    C5 ∨ C˙ 5 + ηN + b3,N νN 5.4. Conclusion: Tuning for n and h We need to find now a sequence of values for n and h such that all the hypothesis in the previous theorem are satisfied and that the upper bound is uniformly bounded in N . We rewrite some needed conditions that are related to the parameters n and h. We assume stronger hypothesis that may help us understand better the existence of the right choice of parameters n and h. As we are only interested in the relationship between n and h with N , we will denote by C1 , C2 etc., various constants that may change from one equation to next. These constants depend on K, ∆ and Θ. They are independent of n, h and N but they depend continuously on other parameters. (i) There exists some positive constant CK,∆,Θ ≥ 0, which depends on K, ∆, Θ, and is independent of N such that  

2 ˙ 4,N √ ϕ2 a2 a b a N N N Ne ∆ × b4,N b3,N + + ηN + b3,N ≤ CK,∆,Θ . (5.5) νN h2 h νN

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A. Kohatsu-Higa, N. Vayatis and K. Yasuda

(ii) (Borel-Cantelli for Yi , (H0)) 2

mc1 := E[ec1 |Y1 | ] < +∞ for some constant c1 > 0 and {aN }N ∈N ⊂ [θu − ∞ l θ , ∞) is a sequence such that for the same c1 , N =1 exp(cN1 a2 ) < +∞. N

(k)

(iii) (Borel-Cantelli for Z3,N (ω), (H3)) For some r3 > 0, ∞  N =1

3 na2r r N < +∞ and sup E [|Z3,N (·)| 3 ] < +∞ for each fixed m ∈ N. 2 (h b3,N )r3 N ∈N

(k)

(iv) (Borel-Cantelli for Z4,N (ω), (H4)) For some r4 > 0, ∞  N =1

n r < +∞ and sup E [|Z4,N (·)| 4 ] < +∞ for each fixed m ∈ N. (b4,N )r4 N∈N

(v) (Borel-Cantelli for |ˆ pN ¯N θ (x) − p θ (x)|, (H6))

∞  (ηN )2 nh2 3 νN exp − < +∞. 16 K ∞ N =1

(k) (vi) (Borel-Cantelli for Z˙ 4,N (ω), (H4
)) For some r˙4 > 0, ∞ 

n

N =1

(b˙ 4,N )r˙4

# r˙4 $ ˙  < +∞ and sup E Z4,N (·) < +∞ for each fixed m ∈ N. N ∈N


(vii) (Borel-Cantelli for |∂θ pˆN ¯N ˙ 6 > 0, and a θ (x) − ∂θ p θ (x)|, (H6 )) For some α constant C˙ 6 ,  . :n ∞  n(ηN )2 h4 C˙ 6 3 νN exp − 1 + 1+α˙ 6 < +∞. n 2 K  2 (b˙ 6,N )2 a2 ∞

N =1

N

(k) (viii) (Borel-Cantelli for Z˙ 6,N (ω), (H6a
)) For some r˙6 > 0, # ∞ r˙6 $  n   < +∞ and sup E Z˙ 6,N (·) < +∞ for each fixed m ∈ N. ˙ 6,N )r˙6 ( b N ∈N N =1

(ix) ((H6b
)) For some q˙6 > 1,   q˙6 ηN h2 (ηN )2 C˙ 6 exp − K  ≤ 1+α˙ 6 ∞  2 n ( K ∞ b˙ 6,N ) aN 2( h2 b˙ 6,N aN )2 and

# q˙6 $ ˙  sup E Z6,N (·) < +∞.

N ∈N

where C˙ 6 and α˙ 6 are the same as (vii) above.

Strong Consistency of Bayesian Estimator

165

√ 5.4.1. Parameter tuning. Set aN := c2 ln N for some positive constant c2 . Set n = C1 N α1 for α1 , C1 > 0 and h = C2 N −α2 for α2 , C2 > 0. For (ii) to be satisfied, we need to have ∞  N =1

∞  N 1 = . exp(c1 c2 ln N ) N c1 c2 −1 N =1

Then we need c1 > c22 . Note that if we choose c2 large enough, then c1 can be chosen as small as needed. With the above specifications, we can check that all the conditions in Section 5.4 are satisfied if the following parameter condition is satisfied for N sufficiently large

1 γ3 α1 + γ˙ 6 ϕ2 c2 α1 4α2 + 2 + + + + q˙6 > α1 , (5.6) r˙6 ∆ 2 r3 r3 which has to be satisfied together with

2 2 4ϕ2 c2 2γ˙ 3 γ˙ 6 α1 1 − − > 8α2 + 1 + + +2 . r3 r˙6 ∆ r3 r˙6

(5.7)

Notice that the above two inequalities will be satisfied if we take c2 or c big enough. Here for (iii), we assume that there exist some r3 > 0, γ3 > 1 and some constant C3 = 0 such that 1

n(c2 ln N )r3 C3 C3 (N γ3 n) r3 c2 ln N = and therefore b . 3,N = 2 r γ (h b3,N ) 3 N 3 h2 And for (viii), we assume that there exists some r˙6 > 0, γ˙ 6 > 1 and some constant C˙ 6 = 0 such that ) * r˙1 n C˙ 6 6 = γ˙ 6 and therefore b˙ 6,N = C˙ 6 nN γ˙ 6 . N (b˙ 6,N )r˙6 For (iv) and (vi), we assume that for g = b4,N , b˙ 4,N , there exists some rg > 0, γg > 1 and some constant Cg = 0 such that n Cg = γg . (g)rg N These constants do not appear in the main restriction but the fact that the convergence is of the above form is used in the proof of the following theorem. Theorem 5.14. Assume that the constants are chosen so as to satisfy c1 > c22 , (5.6) and (5.7). Also assume that the moment conditions stated in (ii), (iii), (iv), (vi), (viii) and (ix) above are satisfied. Then (H0), (H3), (H4), (H4
), (H6), (H6
), (H6a
) and (H6b
) are satisfied. Furthermore, if we assume (H1), (H2), (H5) and (H5
), then Assumption 2.3 (6)-(a) is satisfied.

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And also if all other conditions on Assumption 2.3 are satisfied then there exist some positive finite random variables Ξ, Ξ1 and Ξ2 such that Ξ1 |EN [f ] − f (θ0 )| ≤ √ a.s. N

 n  Ξ2 EN,m [f ] − f (θ0 ) ≤ √ a.s., N

and

and therefore  EN [f ] − E n

  ≤ √Ξ a.s. N

N,m [f ]

Remark 5.15. x (i) In (5.7), r3 and r˙6 represent moment conditions on the derivatives of X(m) (θ), −1 x ϕ2 represents the variance of X(m) (θ), ∆ represents the length of the time interval between observations. Finally c2 > 2c−1 1 expresses a moment condix tion on Yi . In (5.6), recall that q˙6 determines a moment condition on X(m) (θ). (ii) Roughly speaking, if r3 , r˙6 and q˙6 are big enough (which √ implies a condition on n), and we choose α1 > 8α2 + 1 + 4ϕ∆2 c2 , m = N , h = C2 N −α2 and n = C1 N α1 , then Assumption 2.3 (6)-(a) and (6)-(b) are satisfied. Then conditions contain the main tuning requirements.

6. Appendix 6.1. Refinements of Markov’s inequalities In this section we state a refinement of Markov’s inequality that is applied in this n article. For λ > 0, let Sn := i=1 Xi where Xi is a sequence of independent and identically distributed random variables with E[Xi ] = 0. Lemma 6.1. Let X be a random variable with E[X] = 0. Then, for λ ∈ R, c > 0 and p := P (|X| < c), we have ' ( eλc − e−λc λ2 c 2 E eλX 1 (|X| < c) ≤ − E[X1(|X| ≥ c)] + pe 2 . 2c Lemma 6.2. Let q1−1 + q2−1 = 1 and assume that E[|Xi |q1 ] < C¯q1 , then for all 0 < ε < 1 and {fn}n∈N ⊂ R+ satisfying that εfn−1 ≤ K we have P (|Sn | > nε; |Xi | < fn , i = 1, . . . , n) %

q &n 2 ε ¯ q1−1 − 2fε22 1 − nε2 −1 2fn n ≤ 2e 1 + (q2 − 1) q2 K1 2 Cq1 e . fn ! K −K " Here K1 = max 1, e −e . 2K

(6.1)

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167

References [1] V. Bally, D. Talay, The law of the Euler scheme for stochastic differential equations II. Convergence rate of the density, Monte Carlo Methods Appl. 2, no. 2, 93–128, 1996. [2] P. Billingsley, Probability and measure, John Wiley & Sons, 1979. [3] P. Billingsley, Convergence of Probability Measures (Second Edition), John Wiley & Sons, 1999. [4] D. Bosq, Non-parametric Statistics for Stochastic Processes. Estimation and Prediction (second edition), Springer-Verlag, 1998. [5] J.A. Cano, M. Kessler, D. Salmeron, Approximation of the posterior density for diffusion processes, Statist. Probab. Lett. 76, no. 1, 39–44, 2006. [6] P. Del Moral, J. Jacod, P. Protter The Monte Carlo method for filtering with discretetime observations, Probab. Theory and Related Fields, 120, 346–368, 2001. [7] J. Guyon, Euler scheme and tempered distributions, Stochastic Process. Appl., 116, no. 6, 877–904, 2006. [8] A. Kohatsu-Higa, N. Vayatis, K. Yasuda, Tuning of a Bayesian Estimator under Discrete Time Observations and Unknown Transition Density, submitted. [9] A. Kohatsu-Higa, N. Vayatis, K. Yasuda, Strong consistency of Bayesian estimator for the Ornstein-Uhlenbeck process, accepted. [10] D. Talay, Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure and discretization by the implicit Euler scheme, Markov Processes and Related Fields, 8, 163–198, 2002. [11] M.P. Wand, M.C. Jones, Kernel Smoothing, Chapman & Hall, 1995. [12] K. Yasuda, Kernel Density Estimation. The Malliavin-Thalmaier formula and Bayesian parameter estimation, Ph.D. thesis, 2008. Arturo Kohatsu-Higa Ritsumeikan University, Japan Science and Technology Agency Department of Mathematical Sciences 1-1-1 Nojihigashi, Kusatsu Shiga, 525-8577, Japan e-mail: [email protected] Nicolas Vayatis Ecole Normale Sup´erieure de Cachan Centre de Math´ematiques et de Leurs Applications 61 avenue du pr´esident Wilson F-94 235 Cachan cedex, France e-mail: [email protected] Kazuhiro Yasuda Hosei University, Faculty of Science and Engineering 3-7-2 Kajino-cho Koganei-shi Tokyo 184-8584, Japan e-mail: k [email protected]

Progress in Probability, Vol. 65, 169–178 c 2011 Springer Basel AG


Exponentially Stable Stationary Solutions for Delay Stochastic Evolution Equations Jiaowan Luo Abstract. We establish some sufficient conditions ensuring existence, uniqueness and exponential stability of non-trivial stationary mild solutions for a class of delay stochastic partial differential equations. Some known results are generalized and improved. Mathematics Subject Classification (2000). 60H15; 34K21; 34K50. Keywords. Stochastic partial differential equations; stationary mild solutions; exponential stability; delay.

1. Introduction In this paper a stochastic evolution equation with memory is considered. The drift coefficient and the diffusion term of the equation are nonlinear functionals of the history of the solution. Sufficient conditions for the existence and uniqueness of a non-trivial stationary exponentially stable mild solution are given. This work is motivated by the recent papers [6, 8, 9] and [20, 21, 22]. In [6, 8, 9], by using a general random fixed point theorem, Caraballo et al. considered the exponential stability of non-constant stationary mild solutions of semilinear stochastic partial differential equations with delay and without delay, respectively. In [20, 21, 22], Liu studied the existence of a non-trivial stationary mild solution of retarded linear evolution equations with additive noise by means of Green operators. The approach of [20, 21, 22] works only for linear systems with additive noise and is not applicable to the cases of nonlinear systems and to linear systems with multiplicative noise. The exponential stability of stochastic partial differential equations is an important task which has been receiving considerable attention during the last decades. In particular, stochastic evolution equations containing some sort of delay or retarded argument have also been extensively studied due to their importance in applications (see, for example, [4, 11, 18] and [27]). However, even in the nondelay framework, most results in the literature are concerned with the exponential

170

J. Luo

stability of constant stationary solutions, mainly the trivial one (see [15, 25] in the finite-dimensional context, and [13, 16, 17, 23] among others in the infinitedimensional framework). It should be mentioned that there are a few papers already published for the stationary solutions of stochastic partial differential equations. Bakhtin and Mattingly in [1] proved the existence and uniqueness of the stationary solution of stochastically forced dissipative partial differential equations such as the stochastic Navier-Stokes equation and stochastic Ginsburg-Landau equation; Bessaih in [2] studied the existence of stationary martingale solutions of a two-dimensional dissipative Euler equation subjected to a random perturbation; Bl¨ omker and Hairer in [3] considered the existence of stationary mild solutions for a model of amorphous thin-film growth; Caraballo, Garrido-Atienza and Schmalfuß in [6] and Caraballo, Kloeden and Schmalfuß in [8, 9] considered the exponential stability of a nontrivial stationary solution of stochastic evolution equations with/without memory on some separable Hilbert space; Liu in [20, 21] considered the existence of stationary solution of retarded Ornstein-Uhlenbeck processes in Hilbert spaces; Liu in [22] studied the stationary solution of a class of Langevin equations driven by L´evy processes with time delay. In this note, some new techniques are applied to obtain the existence and uniqueness of a non-trivial stationary mild solution which is exponentially stable in the mean square sense. Our result improves and generalizes the results in [6, 8, 9, 10, 20, 21]. The content of the paper is as follows. In Section 2, we recall some definitions and do some preliminaries. In Section 3, we consider a linear stochastic evolution equation, and obtain the existence of a non-trivial stationary exponentially stable mild solution. At last, in Section 4, we consider a nonlinear stochastic evolution equation. Based on the result of Section 3, we prove the existence of a unique nontrivial stationary mild solution which is exponential stable in the mean square sense.

2. Definitions and preliminaries Let H and K be two real separable Hilbert spaces and denote by ·, ·H , ·, ·K their inner products and by · H , · K their vector norms, respectively. We denote by L(K, H) the set of all linear bounded operators from K into H, equipped with the usual operator norm · . In this paper, we always use the same symbol · to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises. Let Q be a linear, symmetric and nonnegative bounded operator on K. Suppose that W (t), t ∈ R, is a K-valued, Q-Wiener process on a probability space (Ω, F , {Ft }, P) with covariance operator Q, adapted to the filtration Ft and with increments W (t) − W (s) independent of Fs (see Da Prato and Zabczyk (1992)).

Stationary Solutions of Stochastic Equations

171

Let us fix an r ∈ [0, +∞] and Cr := C([−r, 0]; H) denote the family of all continuous functions ϕ from [−r, 0] into H. As usual, the space C([−r, 0]; H) is assumed to be equipped with the supremum norm ϕ Cr = sup−r≤θ≤0 ϕ(θ) H for any ϕ ∈ Cr . If we have a function X ∈ Cr , for each t ∈ R we denote by Xt ∈ Cr the function defined by Xt (θ) = X(t + θ), −r ≤ θ ≤ 0. In this paper, we consider the following infinite-dimensional stochastic differential equation dX(t) = AX(t)dt + F (Xt )dt + G(Xt )dW (t),

t ∈ R,

(2.1)

where A is the infinitesimal generator of a strongly continuous semigroup etA , t ∈ R, in H satisfying that etA ≤ M e−γt ,

(2.2)

where M > 0 and γ > 0 are constants. The operators F : Cr −→ H, G : Cr −→ L(K, H) are supposed to satisfy the following properties: for any positive continuous function ρ over (−r, +∞) there exist constants LF , LG ≥ 0 such that for all x, y ∈ C([−r, +∞); H) and all t ≥ s  t  t 2 2 ρ(u) F (xu ) − F (yu ) H du ≤ LF ρ(u) x(u) − y(u) 2H du (2.3) s

and

s−r





t

ρ(u) G(xu ) − G(yu ) du ≤ 2

s

t

ρ(u) x(u) − y(u) 2H du,

L2G

(2.4)

s−r

where G(x) − G(y) 2 := tr((G(x) − G(y))Q(G(x) − G(y))∗ ) (see Da Prato and Zabczyk [13] Chapter 4). We mention that Assumptions (2.3) and (2.4) are often assumed in the context of delay partial differential equations. See [5, 7, 8, 12] for some particular examples. Definition 2.1. A stochastic process X(t, s), s ≤ t, (we simply write as X(t) and omit the initial time s) defined on (Ω, F , {Ft}, P) is called a mild solution, of Equation (2.1) if (i) X(t), s ≤ t, is adapted to Ft , and for arbitrary t ≥ s, %  t & 2 P ω: X(u) H du < ∞ = 1; s

(ii) for any t ≥ s, we have 

e(t−u)A F (Xu )du + s

(iii) X(s) ∈ H.



t

X(t) = e(t−s)A X(s) +

t

e(t−u)A G(Xu )dW (u); s

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J. Luo

Definition 2.2. A solution X(t), t ≥ s, of Equation (2.1) is called strongly stationary, or simply stationary, if the finite-dimensional distributions are invariant under time translations, i.e., P {X(t + tk ) ∈ Γk , k = 1, 2, . . . , n} = P {X(tk ) ∈ Γk , k = 1, 2, . . . , n} for all t ≥ s, n ≥ 1, t1 , . . . , tn ≥ 0 and all Borel sets Γk ∈ B(H), k = 1, . . . , n, or equivalently, for any h1 , . . . , hn ∈ H,   n    n    E exp i X(tk + t), hk H = E exp i X(tk ), hk H . k=1

k=1

We say that (2.1) has a stationary solution X(t) if there is an initial X(s) ∈ H such that X(t), t ≥ s, is a stationary solution of Equation (2.1).

3. Linear stochastic evolution equations In this section we consider the linear version of equation (2.1): dx(t) = Ax(t)dt + α(t)dt + β(t)dW (t),

t ∈ R,

(3.1)

where α(t) and β(t) are continuous stochastic processes such that α(t) and β(t) are Ft -measurable for every t and sup E α(t) 2H < ∞, t∈R

sup E β(t) 2 < ∞.

(3.2)

t∈R

Theorem 3.1. Let (2.2) and (3.2) hold. Moreover, suppose that η(t) := (α(t), β(t)) is stationary. Then equation (3.1) has a unique non-trivial stationary mild solution which is exponentially stable in the mean square sense. Proof. From Proposition 1.3.3 in [19], we know that there exists a mild solution of equation (3.1). Define x∗ (t) =





t

t

e(t−u)A α(u)du + −∞

e(t−u)A β(u)dW (u).

(3.3)

−∞

The right-hand side of (3.3) exists for all t by the assumptions (2.2) and (3.2). Now, we show that x∗ (t), t ∈ R, is a mild solution of Equation (3.1). In fact, it is obvious that the process x∗ (t) verifies the following integral equation  x(t) = e

(t−s)A



t

x(s) +

e s

(t−u)A

t

e(t−u)A β(u)dW (u).

α(u)du + s

(3.4)

Stationary Solutions of Stochastic Equations

173

Next we prove that x∗ (t) is mean square bounded by estimating every term in (3.3). In fact, it follows from the Cauchy-Bunyakovskii inequality that 8 t 82  t

2 8 8 (t−u)A (t−u)A 8 8 E8 e α(u)du8 ≤ E e H · α(u) H du −∞

−∞

H



≤M E

t

2

e

 ≤ M2 ≤

e

−∞ t

e−γ(t−u)du

−∞



t

−∞

2 · α(u) H du

e−γ(t−u) E α(u) 2H du

M2 sup E α(t) 2H < ∞. γ 2 t∈R

Properties of stochastic integrals imply that 8 t 82  8 8 (t−u)A 8 E8 e β(u)dW (u)8 ≤ 8 −∞

− γ2 (t−u) − γ2 (t−u)

H

≤

t

−∞ 2

M 2 e−2γ(t−u) E β(s) 2 ds

M sup E β(t) 2 < ∞. 2γ t∈R

Second we show that x∗ (t) is mean square exponentially stable. Let x(t) be an arbitrary mild solution of equation (3.1). Then  t  t x(t) = e(t−s)A x(s) + e(t−u)A α(u)du + e(t−u)A β(u)dW (u). (3.5) s

Thus

s

E x(t) − x∗ (t) 2H ≤ e(t−s)A 2H E x(s) − x∗ (s) 2H

≤ M 2 e−2γ(t−s) E x(s) − x∗ (s) 2H , hence the exponential stability follows. Third we prove that x∗ (t) is strongly unique. Let y(t) be a mild solution of equation (3.1) that is different from x∗ (t) and mean square bounded on the whole axis. Then z(t) := x∗ (t) − y(t) is a solution of equation (3.1). Thus E z(t) 2H ≤ M 2 e−2γ(t−τ ) E z(τ ) 2H for all t, τ ∈ R such that t ≥ τ . Set supt∈R E z(t) 2H = a, then we have E z(t) 2H ≤ aM 2 e−2γ(t−τ ). Passing to the limit in the above as τ → −∞, we get E z(t) 2H = 0 for all t ∈ R, whence it follows P{x∗ (t) = y(t)} = 0 for all t ∈ R. Since x∗ (t) and y(t) are continuous, we have % & P sup x∗ (t) − y(t) H > 0 = 0, t∈R

and the uniqueness is proved.

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J. Luo

At last, we prove that x∗ (t) is stationary similar as that in [24]. In fact, since W (t), t ∈ R, has stationary independent increments, and etA is a strongly continuous semigroup, (α(t), β(t)) is stationary, from (3.3) it follows that the joint distribution of x∗ (t) is identical with that of x∗ (t + u) for all u, t ∈ R. 

4. Nonlinear stochastic evolution equations In this section, taking advantage of what has been obtained for the linear equation in the last section 3, it is proved that the infinite-dimensional stochastic differential equation has also a unique non-trivial mild solution which is exponentially stable in mean square. Theorem 4.1. Let (2.2), (2.3) and (2.4) hold. Moreover, suppose that 3M 2 L2F + 3M 2 L2G < γ. γ

(4.1)

Then equation (2.1) has a unique non-trivial stationary mild solution which is exponentially stable in the mean square sense. Proof. We apply the successive approximation method to equation (3.1). We seek its mean square bounded solution as the limit of the sequence {X (m) (t)}, where X (m+1) (t) is defined as a unique mean square bounded mild solution of the following . (m) (m) dX(t) = AX(t)dt + F (Xt )dt + G(Xt )dW (t), t ∈ R, (4.2) (0) X (t) = 0, From the conditions (2.3) and (2.4) we have that (m)

sup E F (Xt t∈R

) 2H < ∞,

(m)

sup E G(Xt

) 2 < ∞,

t∈R

whenever X (m) (t) is mean square bounded. Hence by Theorem 3.1, there is a unique mean square exponentially stable stationary solution of the form  t  t (m+1) (t−u)A (m) X (t) = e F (Xu )du + e(t−u)A G(Xu(m) )dW (u). (4.3) −∞

−∞

It is easy to prove that the solutions {X uniformly on [t1 , t2 ], and such that

(m)

(t)} converge with probability one

E X (m) (t) 2H ≤ C for all t ∈ R and some C > 0. By Fatou’s lemma this implies that E X (∞)(t) 2H ≤ C

Stationary Solutions of Stochastic Equations

175

Passing to the limit in (4.3) as m → +∞ and using the properties of stochastic integrals and the continuity of F and G we have the limit stochastic process X (∞) (t) is such that  t  t (∞) (t−u)A (∞) X (t) = e F (Xu )du + e(t−u)A G(Xu(∞) )dW (u), (4.4) −∞

−∞

which is a mild solution of equation (2.1). The solution obtained this way is exponentially stable in mean square. Indeed, let Y (t) be another solution of equation (2.1). Then  t  t (t−s)A (t−u)A Y (t) = e Y (s) + e F (Yu )du + e(t−u)A G(Yu )dW (u). (4.5) s

s

Hence, from (4.4) and (4.5), similarly as in [10], we have for t ≥ s X (∞) (t) − Y (t) 2H ≤ 3 e(t−s)A 2H X (∞) (s) − Y (s) 2H 8 t 82 8 8 (t−u)A (∞) 8 +38 e [F (X ) − F (Y )]du u u 8 8 s

H

8 t 82 8 8 (t−u)A (∞) 8 +38 e [G(X ) − G(Y )]dW (u) u u 8 8 s

H

2 −2γ(t−s)

≤ 3M e



X

(∞)

(s) − Y

(s) 2H

t

3M 2 L2F e−γ(t−u) X (∞)(u) − Y (u) 2H du γ s−r 8 t 82 8 8 (t−u)A (∞) 8 +38 e [G(Xu ) − G(Yu )]dW (u)8 8 , +

s

H

i.e., eγ(t−s) E X (∞) (t) − Y (t) 2H ≤ 3M 2 e−γ(t−s) E X (∞) (s) − Y (s) 2H

 t 3M 2 L2F + + 3M 2 L2G eγ(u−s) E X (∞) (u) − Y (u) 2H du. γ s−r Hence by the Gronwall-Bellman inequality E X (∞) (t) − Y (t) 2H ≤ Ce(

3M 2 L2 F γ

+3M 2 L2G −γ)(t−s)

,

(4.6)

where C > 0 is a constant, which shows that X (∞) (t) is exponentially stable. Theorem 3.1 implies that X (∞) (t) is the limit of a sequence of stationary processes X (m)(t) defined by (4.3). The existence of a second stationary solution of equation (2.1) contradicts the uniqueness of a mean square bounded solution of this equation, since every stationary solution has bounded second moment. 

176

J. Luo

Remark 4.2. In [6, 8, 9], the authors considered three types of equation (2.1). But they need the restriction that M = 1 in (2.2), that is, etA ≤ e−γt . Remark 4.3. The approach of [20, 21, 22] works only for linear systems with additive noise and is not applicable to the cases of nonlinear systems and linear systems with multiplicative noise. Remark 4.4. In [10], the authors obtained the mean square exponential stability of zero solution of equation (2.1) with variable delays. For example, consider the following equation dX(t) = AX(t)dt + F (X(t − r))dt + G(X(t − r))dW (t), where r > 0 is a constant, by [10], if (2.2), (2.3) (2.4) and

3M 2 L2F + 3M 2 L2G eγr < γ γ

t ∈ R,

(4.7)

(4.8)

hold, then the zero solution of equation (4.7) is mean square exponentially stable. However, by Theorem 4.1 in present paper, if (2.2), (2.3) (2.4) and 3M 2 L2F + 3M 2 L2G < γ (4.9) γ hold, then equation (4.7) has a unique stationary mild solution which is also mean square exponentially stable. Obviously, our condition (4.9), which does not depend on the time delay, is better than the above (4.8). In addition, our conditions are similar as the well-known conditions of the stability for every value of the delay r in the finite dimension setting [14]. Acknowledgement The author would like to thank the referee for the careful reading of the manuscript. The valuable remarks made numerous improvements throughout. This work is partially supported by NNSF of China (Grant No. 10971041), by the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No.20090002110047) and by the Foundation of Guangzhou Education Bureau (Grant No. 08C019).

References [1] Yuri Bakhtin and Jonathan C. Mattingly, Stationary solutions of stochastic differential equation with memory and stochastic partial differential equations, Commun. Contemp. Math., 7(2005), 553–582. [2] Hakima Bessaih, Stationary solutions for the 2D stochastic dissipative Euler equation, Seminar on Stochastic Analysis, Random Fields and Applications V, 23–36, Progr. Probab., 59, Birkh¨ auser, Basel, 2008. [3] Dirk Bl¨ omker and Martin Hairer, Stationary solutions for a model of amorphous thin-film growth, Stochastic Anal. Appl., 22(2004), 903–922.

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[4] Tom´ as Caraballo, Asymptotic exponential stability of stochastic partial differential equations with delay, Stochastics Stochastics Rep., 33(1990), 27–47. [5] Tom´ as Caraballo, M.J. Garrido-Atienza and Jos´e Real, Existence and uniqueness of solutions for delay stochastic evolution equations, Stochastic Anal. Appl., 20(2002), 1225–1256. [6] Tom´ as Caraballo, M.J. Garrido-Atienza and Bj¨ orn Schmalfuß, Existence of exponentially attracting stationary solutions for delay evolution equations, Discrete Contin. Dyn. Syst., 18(2007), 271–293. [7] Tom´ as Caraballo, M.J. Garrido-Atienza and Bj¨ orn Schmalfuß, Asymptotic behaviour of non-trivial stationary solutions of stochastic functional evolution equations, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, 6(2004), 39–42. [8] Tom´ as Caraballo, Peter E. Kloeden and Bj¨ orn Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50(2004),183–207. [9] Tom´ as Caraballo, Peter E. Kloeden and Bj¨ orn Schmalfuß, Stabilization of stationary solutions of evolution equations by noise, Discrete Contin. Dyn. Syst. Ser. B, 6(2006), no. 6, 1199–1212 [10] T. Caraballo and Kai Liu, Exponential stability of mild solutions of stochastic partial differential equations with delays, Stochastic Anal. Appl., 17(1999), 743–763. [11] Tom´ as Caraballo, Kai Liu and A. Truman, Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay property, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 456(2000), 1775–1802. [12] Tom´ as Caraballo, Jos´e Real and Takeshi Taniguchi, The exponential stability of neutral stochastic delay partial partial differential equations, Discrete Contin. Dyn. Syst., 18 (2007), 295–313. [13] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992. [14] Jack Hale, Theory of Functional Differential Equations, Springer-Verlag, 1992. ˘ [15] R.Z. Hasminski ´ 1, Stochastic Stability of Differential Equations, Sijthoff & Nordhoff, Alphen aan den Rijn, The Netherlands; Rockville, Maryland, USA, 1980. [16] U.G. Haussmann, Asymptotic stability of the linear Itˆ o equation in infinite dimensions, J. Math. Anal. Appl., 65(1978), 219–235. [17] A. Ichikawa, Stability of semilinear stochastic evolution equations, J. Math. Anal. Appl., 90(1982), 12–44. [18] Kai Liu, Lyapunov functionals and asymptotic stability of stochastic delay evolution equations, Stochastics Stochastics Rep., 63 (1998), 1–26. [19] Kai Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Monographs and Surveys in Pure and Applied Mathematics; 135, Chapman & Hall/CRC, 2006. [20] Kai Liu, Stationary solutions of retarded Ornstein-Uhlenbeck processes in Hilbert spaces, Statist. Probab. Lett., 78(2008), 1775–1783. [21] Kai Liu, A criterion for stationary solutions of retarded linear equations with additive noise, preprint, 2008.

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[22] Kai Liu, Retarded stationary Ornstein-Uhlenbeck processes driven by L´evy noise and operator self-decomposability, Potential Anal., 33(2010), 291–312. [23] Kai Liu and Xuerong Mao, Exponential stability of nonlinear stochastic evolution equations, Stochastic Process. Appl., 78(1998), 173–193. [24] Jiaowan Luo and Guolie Lan, Stochastically bounded solutions and stationary solutions of stochastic differential equations in Hilbert spaces, Statistics and Probability Letters, 79(2009), 2260–2265. [25] Xuerong Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, New York, 1994. [26] S-.E-.A. Mohammed, Stochastic Functional Differential Equations, Pitman, Boston, 1984. [27] T. Taniguchi, Kai Liu and A. Truman, Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. Differential Equations, 181(2002), 72–91. Jiaowan Luo Department of Probability and Statistics School of Mathematics and Information Sciences Guangzhou University Guangzhou Guangdong 510006, P.R. China e-mail: [email protected]

Progress in Probability, Vol. 65, 179–189 c 2011 Springer Basel AG


Robust Stochastic Control and Equivalent Martingale Measures Bernt Øksendal and Agn`es Sulem Abstract. We study a class of robust, or worst case scenario, optimal control problems for jump diffusions. The scenario is represented by a probability measure equivalent to the initial probability law. We show that if there exists a control that annihilates the noise coefficients in the state equation and a scenario which is an equivalent martingale measure for a specific process which is related to the control-derivative of the state process, then this control and this probability measure are optimal. We apply the result to the problem of consumption and portfolio optimization under model uncertainty in a financial market, where the price process S(t) of the risky asset is modeled as a geometric Itˆ o-L´evy process. In this case the optimal scenario is an equivalent local martingale measure of S(t). We solve this problem explicitly in the case of logarithmic utility functions. Mathematics Subject Classification (2010). 93E20, 60G51, 60H20, 60G44, 91G80. Keywords. Robust stochastic control, worst case scenario, stochastic differential game, equivalent martingale measure, jump diffusion.

1. Introduction During the last decade there has been an increasing awareness of the importance of taking model uncertainty into account when dealing with mathematical models. See, e.g., [HS] and the references therein. A general feature of model uncertainty is the recognition of the uncertainty about the underlying probability law, or scenario, for the model. This leads to the study of robust models, where one seeks an optimal strategy among a family of possible scenarios. A special case is the problem of optimal control in the worst possible scenario. This is the topic of this paper. We consider a class of scenarios which is basically the set of all probability measures Q which are absolutely continuous with respect to a given reference measure P , and we study an optimal control problem for a jump diffusion under the worst possible scenario.

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B. Øksendal and A. Sulem

Mathematically this leads to a stochastic differential game between the controller and the “environment” who chooses the scenario. Assuming the system is Markovian and using the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation, we show that if there exists a control uˆ which annihilates the noise coefficients of the ˆ which is an equivalent martingale measure for a specific system and a scenario Q process which is related to the control-derivative of the state process, then this ˆ is a worst case scenario. control is optimal for the controller and the scenario Q We then apply this result to the problem of optimal consumption and portfolio under model uncertainty in a financial market, where the price process S(t) of the risky asset is modeled as a geometric Itˆ o-L´evy process. In this case the optimal scenario is an equivalent local martingale measure of S(t). We solve this problem explicitly in the case of logarithmic utility functions. Robust control problems and worst case scenario problems have been studied by many researchers. We mention in particular the paper [BMS], where the following approach is used: The authors first fix a strategy and prove the existence of a corresponding optimal scenario Q∗ , and then subsequently use BSDEs to study the optimal strategy problem for a fixed scenario. Robust control problems for possibly non-Markovian systems can also be studied by means of stochastic maximum principles (see [AØ]) and by means of backward stochastic differential equations (see, e.g., [ØS3] and the references therein).

2. Robust optimal control 2.1. Stochastic differential game approach Consider a controlled jump diffusion X(t) = X u (t) in R of the form dX(t) = b(t, X(t), u(t))dt + σ(t, X(t), u(t))dB(t)  ˜ (dt, dz) ; X(0) = x ∈ R ; t ∈ [0, T ] + γ(t, X(t), u(t), z)N

(2.1)

R0

˜ (dt, dz) = N (dt, dz) − ν(dz)dt is the where T > 0 is a fixed constant. Here N compensated Poisson random measure of a L´evy process with jump measure N (·, ·) ˜ (·, ·) and L´evy measure ν(·), and B(t) is an independent Brownian motion. Both N and B(·) live on a filtered probability space (Ω, F, {Ft }, P ). The process u(t) = (u1 (t), . . . , um (t)) ∈ Rm is the control process of the agent and b(s, x, u), σ(s, x, u) and γ(s, x, u, z); s ∈ [0, T ], x ∈ R, u ∈ Rm , z ∈ R0 := R\{0}, are assumed to be C 1 functions with respect to u. The scenario of X(t) is determined by a (positive) measure Qθ of the form dQθ (ω) = K θ (T )dP (ω) on FT ,

(2.2)

Robust Stochastic Control

181

where #

−



dK (t) = −K (t ) θ0 (t)dB(t) + θ

θ

R0

$ ˜ θ1 (t, z)N (dt, dz) ; t ∈ [0, T ]

θ

K (0) = k > 0.

(2.3)

Here θ = θ(t, z) = (θ0 (t), θ1 (t, z)) ∈ R2 is the scenario control, assumed to be Ft -predictable and such that E[K θ (T )] = K θ (0) =: k > 0.

(2.4)

Let V , Θ be two sets such that u(t) ∈ V and θ(t, z) ∈ Θ for all t, z and let U, A be given families of admissible u-controls and θ-controls, respectively. Define the process Y (t) = Y θ,u (t) := (K θ (t), X u (t)). Then Y (t) is a controlled jump diffusion with generator Aθ,u ϕ(t, y) = Aθ,u ϕ(t, k, x)

(2.5) 2

2

∂ϕ 1 2 2 ∂ ϕ 1 2 ∂ ϕ + k θ0 2 + σ (t, x, u) 2 ∂x 2 ∂k 2 ∂x ∂2ϕ − θ0 kσ(t, x, u) ∂k∂x 

= b(t, x, u)

+ R0

{ϕ(t, k − kθ1 (z), x + γ(t, x, u, z)) − ϕ(t, k, x)

+ kθ1 (z)

∂ϕ ∂ϕ (t, k, x) − γ(t, x, u, z) (t, k, x)}ν(dz) ; ϕ ∈ C 1,2,2 (R3 ). ∂k ∂x

We refer to [ØS1] for more information about stochastic control of jump diffusions. Let f : R2 × V → R and g : R → R be functions such that  T

|f (t, X(t), u(t))|dt + |g(X(T ))| < ∞

EQθ 0

for all u ∈ U, θ ∈ A. Define the performance functional by  T

t,y J θ,u (t, y) = EQ θ

f (s, X(s), u(s))ds + g(X(T )) 

=E

t

t,y

T

K θ (s)f (s, X(s), u(s))ds + K θ (T )g(X(T )) ,

(2.6)

t t,y t,y where EQ denotes expectation with respect to Qθ and P , respectively, θ and E given Y (t) = y.

182

B. Øksendal and A. Sulem

We consider the following robust, or worst case scenario, stochastic control problem Problem 2.1. Find θ ∗ ∈ A, u∗ ∈ U and Φ(t, y) such that

∗ ∗ θ,u Φ(t, y) = inf sup J (t, y) = J θ ,u (t, y). θ∈A

(2.7)

u∈U

2.2. The main theorem We now formulate our main result: Theorem 2.2. Suppose there exist a C 1 function ψ(t, x) and feedback controls u ˆ= u ˆ(t, x) ∈ U, θˆ = (θˆ0 (t, x), θˆ1 (t, x, z)) ∈ A such that σ(t, x, u ˆ(t, x)) = γ(t, x, uˆ(t, x), z) = 0 for all t, x, z and

(2.8)

# ∂σ θˆ0 (t, x) (t, x, u ˆ(t, x)) ∂ui $  ∂γ ∂b ∂ψ ˆ + (t, x, z) (t, x, u ˆ (t, x), z)ν(dz) − (t, x, u ˆ (t, x)) (t, x) θ1 ∂u ∂u ∂x i i R0 ∂f = (t, x, uˆ(t, x)) ; for all t, x, i = 1, 2, . . . , m. (2.9) ∂ui

ˆ is an optimal pair for the robust control problem (2.7) and the value Then (ˆ u, θ) function is given by Φ(t, k, x) = kψ(t, x); provided that ψ(t, x) is the solution of the PDE ∂ψ ∂ψ (t, x) + b(t, x, u ˆ(t, x)) (t, x) + f (t, x, u ˆ(t, x)) = 0 ; (t, x) ∈ [0, T ] × R (2.10) ∂t ∂x ψ(T, x) = g(x) ; x ∈ R.

(2.11)

Proof. We apply Theorem 3.2 in [MØ]. Maximizing u → Aθ,u ϕ(t, k, x)+kf (t, x, u) with respect to u gives the following first-order conditions for an optimal u ˆ: ∂b ∂ϕ ∂σ ∂ 2ϕ ∂σ ∂2ϕ (t, x, u ˆ) + σ(t, x, uˆ) (t, x, u ˆ) 2 − θ0 k (t, x, u ˆ) ∂ui ∂x ∂ui ∂x ∂ui ∂k∂x &  % ∂ϕ ∂ϕ ∂γ + (t, k − kθ1 (z), x + γ(t, x, u ˆ, z)) − (t, k, x) (t, x, u ˆ, z)ν(dz) ∂x ∂x ∂u i R0 ∂f +k (t, x, u ˆ) = 0; i = 1, 2, . . . , m. (2.12) ∂ui Minimizing Aθ,ˆu ϕ(t, k, x) + kf (t, x, u ˆ) with respect to θ = (θ0 , θ1 (z)), we get the following first-order conditions for optimal θˆ0 , θˆ1 (z): k 2 θˆ0

∂ 2ϕ ∂ 2ϕ − kσ(t, x, u ˆ) =0 2 ∂k ∂k∂x

(2.13)

Robust Stochastic Control and

 R0

%

& ∂ϕ ∂ϕ ˆ (t, k − k θ1 (z), x + γ(t, x, u ˆ, z)) − (t, k, x) ν(dz) = 0. ∂k ∂k

183

(2.14)

Let us try a value function of the form ϕ(t, k, x) = kψ(t, x).

(2.15)

Then (2.12)–(2.14) get the form ∂b ∂ψ ∂σ ∂2ψ ∂σ ∂ψ (t, x, u ˆ) + σ(t, x, u ˆ) (t, x, u ˆ) 2 − θˆ0 (t, x, u ˆ) ∂ui ∂x ∂ui ∂x ∂ui ∂x &  % ∂ψ ∂ψ ∂γ (t, x + γ(t, x, u ˆ, z)) − (t, x) + (1 − θˆ1 (z)) (t, x, u ˆ, z)ν(dz) ∂x ∂x ∂u i R0 ∂f + (t, x, u ˆ) = 0; i = 1, 2, . . . , m, (2.16) ∂ui σ(t, x, u ˆ) and

∂ψ (t, x) = 0, ∂x

(2.17)

 R0

{ψ(t, x + γ(t, x, u ˆ, z)) − ψ(t, x)}ν(dz) = 0.

(2.18)

Suppose there exists a Markov control u ˆ = uˆ(t, x) such that σ(t, x, u ˆ(t, x)) = γ(t, x, uˆ(t, x), z) = 0 for all z ∈ R0 .

(2.19)

Then (2.17)–(2.18) are satisfied, and (2.16) gets the form $  ∂σ ∂γ ∂b ∂ψ ˆ ˆ θ0 (t, x) (t, x, u ˆ) + θ1 (t, x, z) (t, x, u ˆ, z)ν(dz) − (t, x, u ˆ) (t, x) ∂ui ∂u ∂u ∂x i i R0 ∂f = (t, x, uˆ); i = 1, 2, . . . , m. (2.20) ∂ui Suppose u ˆ, θˆ satisfy (2.19)–(2.20). Then by Theorem 3.2 in [MØ] we are required to have #

∂ϕ ˆ (t, y) + Aθ,ˆu ϕ(t, y) + kf (t, x, u ˆ) = 0 ; t < T. ∂t By (2.5) and (2.19), this gives the equation ∂ψ ∂ψ (t, x) + b(t, x, u ˆ(t, x)) (t, x) + f (t, x, u ˆ(t, x)) = 0 ; t < T, ∂t ∂x with terminal condition ψ(T, x) = g(x) ; x ∈ R. This completes the proof.

(2.21) (2.22) 

ˆ x) might depend on ∂ψ (t, x). Hence equaRemark 2.3. Note that u ˆ(t, x) and θ(t, ∂x tion (2.10) is in general a nonlinear PDE in the unknown function ψ.

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2.3. Equivalent local martingale measures The following definition is motivated by applications in mathematical finance: Definition 2.4. Let S(t) be an Itˆ o-L´evy process of the form  ˜ (dt, dz) dS(t) = α(t)dt + β(t)dB(t) + ζ(t, z)N R0

for predictable processes α(t), β(t), ζ(t, z); t ∈ [0, T ], z ∈ R0 . A probability measure Q on FT is called an equivalent local martingale measure (ELMM) for S(·) if Q ∼ P (i.e., Q  P and P  Q) and {S(t)}t∈[0,T ] is a local martingale with respect to Q. It is well known (see, e.g., [ØS1, Theorem 1.31]) that a measure Qθ of the form (2.2)–(2.4) with k = 1 is an ELMM for S(·) if and only if  θ0 (t)β(t) + θ1 (t, z)ζ(t, z)ν(dz) = α(t) ; t ∈ [0, T ]. (2.23) R0

Suppose that ∂ψ = 0. Then, in view of Definition 2.4, the measure Qθˆ of the ∂x form (2.2)–(2.4) with k = 1, where θˆ is a scenario control satisfying (2.9), is an ELMM for all the processes Gi (t), given by 

−1 ∂ψ ∂f ∂b ˆ ˆ ˆ dGi (t) := (t, X(t), u ˆ(t)) + (t, X(t)) (t, X(t), u ˆ(t)) dt (2.24) ∂ui ∂x ∂ui  ∂σ ∂γ ˆ ˆ ˜ (dt, dz); + (t, X(t), u ˆ(t))dB(t) + (t, X(t), u ˆ(t), z)N ∂ui ∂u i R0 i = 1, 2, . . . , m. ˆ ˆ where u ˆ(t) = u ˆ(t, X(t)) and X(t) = X uˆ (t); t ∈ [0, T ].

3. Example Suppose we have a financial market with a risk free asset with unit price S0 (t) = 1 and a risky asset with unit price S(t) given by # $  ˜ (dt, dz) ; t ∈ [0, T ] dS(t) = S(t− ) b0 (t)dt + σ0 (t)dB(t) + γ0 (t, z)N R0

S(0) > 0,

(3.1)

where b0 (t), σ0 (t) and γ0 (t, z) are bounded deterministic functions, γ0 (t, z) > −1. If we apply a portfolio π(t), representing the proportion of the total wealth X(t) invested in the risky asset at time t and a relative consumption rate λ(t) ≥ 0, the

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corresponding wealth process X(t) = X λ,π (t) will have the dynamics # $  ˜ (dt, dz) dX(t) = π(t)X(t− ) b0 (t)dt + σ0 (t)dB(t) + γ0 (t, z)N R0

− λ(t)X(t)dt ; t ∈ [0, T ], X(0) = x > 0.

(3.2)

We say that the pair u = (λ, π) is an admissible control if λ and π are Fpredictable, λ ≥ 0, π(t)γ0 (t, z) > −1 and  T  2 (π (t) + λ(t) + | log(1 + π(t)γ0 (t, z))|ν(dz))dt < ∞ a.s. R0

0

Note that under these conditions the unique solution X(t) of (3.2) is given by  t  t 1 X(t) = x exp( {π(s)b0 (s) − λ(s) − π 2 (s)σ02 (s)}ds + π(s)σ0 (s)dBs 2 0 0  t + (log(1 + π(s)γ0 (s, z)) − π(s)γ0 (s, z))ν(dz)ds 0

R0

0

R0

 t +

˜ (ds, dz)); t ∈ [0, T ] (log(1 + π(s)γ0 (s, z))N

In particular, X(t) > 0 for all t ∈ [0, T ]. Let U1 : [0, T ] × [0, ∞) → R, U2 : [0, ∞) → R be two given C 1 functions. We assume that c → U1 (t, c) and x → U2 (x) are strictly increasing, concave functions 1 (utility functions) for all t ∈ [0, T ]. We also assume that c → ∂U ∂c (t, c) is strictly ∂U1 1 decreasing and that limc→+∞ ∂c (t, c) = 0 for all t ∈ [0, T ]. Put x0 = ∂U ∂c (t, 0) and define . 0 for x ≥ x0 −1 I(t, x) =  ∂U1 (3.3) (x) for 0 ≤ x < x0 ∂c (t, ·) Define the absolute consumption rate at time t by c(t) = λ(t)X(t) ; t ∈ [0, T ].

(3.4)

Suppose the performance functional is given by  T

t,y J θ,λ,π (t, y) = EQ θ

U1 (s, c(s))ds + U2 (X(T )) 

t

= E t,y

T

K θ (s)U1 (s, c(s))ds + K θ (T )U2 (X(T )) .

(3.5)

t

To solve the problem



Φ(t, y) = inf

θ∈A

 sup J (λ,π)∈U

θ,λ,π

(t, y)

= Jθ

∗

,λ∗ ,π ∗

(t, y)

(3.6)

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we apply Theorem 2.2. Thus we search for a solution cˆ, π ˆ , θˆ = (θˆ0 , θˆ1 ) and ψ(t, x) such that (2.8) and (2.9) hold, when b(t, x, u) = πxb0 (t) − λx, σ(t, x, u) = πxσ0 (t), γ(t, x, u, z) = πxγ0 (t, x, z) and f (t, x, u) = U1 (t, c), g(x) = U2 (x), u = (λ, π), c = λx. We see that (2.8) holds with π ˆ = 0, for all cˆ. Writing



∂ ∂ ∂ ∂ , = , , ∂u1 ∂u2 ∂λ ∂π equation (2.9) becomes

and

∂ψ ∂U1 (t, x) = (t, cˆ(t, x)) ∂x ∂c  ˆ θ0 (t, x)σ0 (t) + θ1 (t, x, z)γ0 (t, z)ν(dz) = b0 (t).

(3.7) (3.8)

R0

The equation (2.10)–(2.11) for ψ gets the form ∂ψ ∂U1 (t, x) − cˆ(t, x) (t, cˆ(t, x)) + U1 (t, cˆ(t, x)) = 0 ; t < T ∂t ∂c ψ(T, x) = U2 (x) which has the solution



T

{U1 (s, cˆ(s, x)) − cˆ(s, x)

ψ(t, x) = U2 (x) + t

∂U1 (s, cˆ(s, x))}ds. ∂c

(3.9)

(3.10)

In this case the processes Gi defined in (2.24) are 

−1 ∂ψ ∂U1 ˆ ˆ ˆ dG1 (t) = X(t) −1 + (t, X(t)) (t, cˆ(t, X(t))) dt = 0 ∂x ∂c # $  − ˆ ˜ dG2 (t) = X(t ) b0 (t)dt + σ0 (t)dB(t) + γ0 (t, z)N (dt, dz) R0

ˆ −) X(t = dS(t). S(t− ) We conclude that the optimal scenario control is to choose θˆ = (θˆ0 , θˆ1 ) such that (3.8) holds, i.e., such that Qθˆ is an ELMM for S(t). The corresponding optimal portfolio is to choose π ˆ = 0 (no money in the risky asset). This is intuitively reasonable, because if the price process is a martingale, there is no money to be gained by investing in this asset. Finally, to find the optimal consumption rate cˆ(t, x) we combine (3.7) and (3.10) : From (3.7) and (3.3) we have

∂ψ (t, x) . (3.11) cˆ(t, x) = I t, ∂x Differentiating (3.10) we therefore get



∂ψ ∂ψ ∂ψ ∂ψ (t, x) = −U1 t, I t, (t, x) + I t, (t, x) (t, x) ; t < T. (3.12) ∂t ∂x ∂x ∂x

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This is a nonlinear first-order partial differential equation in ψ(t, x). Together with the terminal value (obtained from (3.10)) ψ(T, x) = U2 (x)

(3.13)

this determines ψ(t, x) uniquely. To summarize, we have proved Theorem 3.1. The optimal (i.e., worst case) scenario control for the problem (3.6) is to choose θ∗ = (θˆ0 , θˆ1 ) such that (3.8) holds, which is equivalent to saying that the measure Qθˆ is an ELMM for the price process S(t) given by (3.1). The optimal portfolio under this scenario is to choose π ˆ = 0 (no money in the risky asset). The optimal consumption rate cˆ(t, x) under this scenario is given by (3.11), i.e., ∂U1 ∂ψ (t, cˆ(t, x)) = (t, x) (3.14) ∂x ∂x where ψ(t, x) is the solution of (3.12)–(3.13). The corresponding value function is, by (2.15), Φ(t, k, x) = kψ(t, x). (3.15) This is an extension of the result in [ØS2]. A special case. To illustrate the content of Theorem 3.1, we consider the special case when U1 and U2 are logarithmic utility functions, i.e., U1 (s, c) = U1 (c) = ln c; U2 (x) = a ln x where a > 0 is constant. Then equation (3.12) gets the form

U1 (c)

∂ψ (t, x) = ln ∂t

=

1 c

and, by (3.3) I(x) =

∂ψ (t, x) + 1. ∂x

(3.16) 1 . x

Therefore

(3.17)

Set h(t, x) = ψ(t, x) − t. Then ∂h (t, x) = ln ∂t Set H(t, x) =



∂h (t, x) . ∂x

∂h (t, x). ∂x

Then H satisfies the nonlinear PDE: ∂H ∂H H(t, x) (t, x) = (t, x); t ≤ T ∂t ∂x We assign the terminal condition a H(T, x) = x and try to solve this equation by setting 1 H(t, x) = H1 (t) . x

(3.18)

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Substituting into (3.18) gives H1 (t) = −1. Since H1 (T ) = λ, this gives the solution H(t, x) =

T −t+a x

and hence h(t, x) = (T − t + a) ln x + C(t) for some function C(t). This gives ψ(t, x) = (T − t + a) ln x + t + C(t) and hence ∂ψ (t, x) = − ln x + C  (t) + 1 ∂t while

∂ψ (t, x) = ln(T − t + a) − ln x. ln ∂x Using (3.17), we get C  (t) = ln(T − t + a) − 1 or C(t) = −(T − t + a) ln(T − t + a) + (T − t + a) − t + C0 Hence ψ(t, x) = (T − t + a)[ln x − ln(T − t + a) + 1] + C0 . Requiring ψ(T, x) = U2 (x) = a ln x leads to the condition C0 = a ln a − a. Hence the solution is ψ(t, x) = (T − t + a)[ln x − ln(T − t + a)] + a ln a + T − t

(3.19)

and the optimal consumption rate is then x . (3.20) T −t+a In particular, we see that the consumption increases with time, which makes sense, because a large early consumption reduces the growth for the whole remaining time period. cˆ(t, x) =

Acknowledgment The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. [228087].

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References [AØ]

[BMS]

[HS] [MØ] [ØS1] [ØS2]

[ØS3]

T.T.K. An and B. Øksendal: A maximum principle for stochastic differential games with partial information. J. Optim. Theory and Appl. 139 (2008), 463– 483. G. Bordigoni, A. Matoussi and M. Schweizer: A stochastic control approach to a robust utility maximization problem. In F.E. Benth et al. (editors): Stochastic Analysis and Applications. Proceedings of the Second Abel Symposium, Oslo 2005. Springer 2007, pp. 135–151. L.P. Hansen and T.J. Sargent: Robust control and model uncertainty. The American Economic Review, Vol. 91, no. 2, (2001), pp. 60–66. S. Mataramvura and B. Øksendal: Risk minimizing portfolios and HJBI equations for stochastic differential games. Stochastics 80 (2008), 317–337. B. Øksendal and A. Sulem: Applied Stochastic Control of Jump Diffusions. Second Edition, Springer 2007. B. Øksendal and A. Sulem: A game theoretic approach to martingale measures in incomplete markets. Eprint, Dept. of Mathematics, University of Oslo 24/2006. Survey of Applied and Industrial Mathematics (TVP Publishers, Moscow), 15, (2008), 18–24. B. Øksendal and A. Sulem: Portfolio optimization under model uncertainty and BSDE games. Eprint, University of Oslo, Dept. of Mathematics, No. 1, 2011. Submitted to Quantitative Finance.

Bernt Øksendal Center of Mathematics for Applications (CMA) Dept. of Mathematics University of Oslo P.O. Box 1053 Blindern N–0316 Oslo, Norway and Norwegian School of Economics and Business Administration Helleveien 30 N–5045 Bergen, Norway e-mail: [email protected] Agn`es Sulem INRIA Paris-Rocquencourt Domaine de Voluceau BP 105 F-78153 Le Chesnay Cedex, France e-mail: [email protected]

Progress in Probability, Vol. 65, 191–205 c 2011 Springer Basel AG


Multi-valued Stochastic Differential Equations Driven by Poisson Point Processes Jiagang Ren and Jing Wu Abstract. We prove the existence and uniqueness of solutions of multi-valued stochastic differential equations driven by Poisson point processes when the domain of the multi-valued maximal monotone operator is the whole space Rd . Mathematics Subject Classification (2000). 60H10, 34F05,47H04. Keywords. Multi-valued stochastic differential equation; Skorohod problem; L´evy processes; Maximal monotone operator; random time change; Helly’s first theorem.

1. Introduction Multi-valued stochastic differential equations driven by Brownian motions have been the object of several works (see, e.g., [5, 6, 13, 11]). In this paper we consider the following multi-valued stochastic differential equation (MSDE in short) of jump type: dXt ∈ − A(Xt )dt + b(Xt )dt + σ(Xt )dBt   ˜ + f (Xt− , u)N (dt, du) + g(Xt− , u)N (dt, du) U0

(1)

U\U0

where A is a multi-valued maximal monotone operator defined on (some domain of) Rd and B is an r-dimensional Brownian motion, N is the counting measure of a stationary Poisson point process with characteristic measure ν on some mea˜ (dt, du) is the compensator of N and U0 ∈ BU with surable space (U, BU ), N ν(U \ U0 ) < ∞. If no point process is involved, the existence and uniqueness of a solution to (1) are proved in [5, 6, 13]. Note that for a multi-valued operator A whose domain is not necessarily the whole space, the continuity of the sample paths Research supported by NSFC (Grant no. 10871215), Doctor Fund of Ministry of Education (Grant no. 20100171110038), and by the Plan of Yat-sen Innovative Talents (2009).

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of the processes under consideration plays an indispensable role in proving the existence of solutions. In the jump case, however, it is easy to image and a trivial example shows that the equation may have no solution if the domain of A is not the whole space (see the remark after Proposition 3.10). So, for simplicity and for putting the emphasis on the general principle we shall assume throughout the paper that A is defined on the whole space, leaving the troublesome problem of determining the minimal domain to a forthcoming paper.

2. Preliminaries and main result Let (U, BU , ν) be a σ-finite measure space and (Ω, F, Ft , P ) a filtered probability space on which an r-dimensional Brownian motion B = (B(t)) and a stationary Poisson point process p = (p(t)) on (U, BU ) with L´evy measure ν are defined. Let N (dt, du) be the counting measure of p and set ˜ (dt, du) := N (dt, du) − dtν(du). N Let Rd be the d-dimensional Euclidean space. Definition 2.1. By a multi-valued operator on Rd we mean an operator A from Rd d to 2R . We set D(A) := {x ∈ Rd : A(x) = ∅}, Gr(A) := {(x, y) ∈ R2d : x ∈ Rd , y ∈ A(x)}. (1) A multi-valued operator A is called monotone if y1 − y2 , x1 − x2 
0,

∀(x1 , y1 ), (x2 , y2 ) ∈ Gr(A).

(2) A monotone operator A is called maximal monotone if and only if (x1 , y1 ) ∈ Gr(A) ⇔ y1 − y2 , x1 − x2 
0,

∀(x2 , y2 ) ∈ Gr(A).

We collect here some facts about the maximal monotone operator which will be used in the paper. For more details, we refer to [2]. Proposition 2.2. Let A be a maximal monotone operator on Rd , then: (i) Int(D(A)) and D(A) are convex subsets of Rd and Int(D(A)) = Int(D(A)). (ii) For each x ∈ D(A), A(x) is a closed and convex subset of Rd . Let A◦ (x) := projA(x) (0) be the minimal section of A, where projD is designated as the projection on every closed and convex subset D on Rd and proj∅ (0) = ∞. Then x ∈ D(A) ⇔ |A◦ (x)| < +∞. (iii) The resolvent operator Jn := (I + n1 A)−1 is a single-valued and contractive operator defined on Rd and valued in D(A). Moreover, as n → ∞, Jn (x) → projD(A) (x).

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(iv) The Yosida approximation An := n(I −Jn ) is a single-valued, maximal monotone and Lipschitz continuous function with Lipschitz constant n. Moreover, for every x ∈ D(A), as n ∞, An (x) → A◦ (x) and |An (x)|

|A◦ (x)|

if

x ∈ D(A),

+∞

if

x∈ / D(A).

|An (x)|

The following definition is taken from [5, 6]. Definition 2.3. If X, K are (Ft )-adapted processes, X is c`adl` ag and K is continuous. Then the pair (X, K) is called a strong solution of (1) if (i) X0 = x0 and X(t) ∈ D(A) for every 0  t < +∞; (ii) K is of locally finite variation and K0 = 0 a.s.;
˜ (dt, du) (iii) dXt = b(Xt )dt + σ(Xt )dBt + U0 f (Xt− , u)dtN
+ U\U0 g(Xt− , u)N (dt, du) − dKt , 0  t < ∞, a.s.; (iv) For any c`adl` ag and (Ft )-adapted functions (α, β) with (αt , βt ) ∈ Gr(A),

∀t ∈ [0, +∞),

the measure Xt − αt , dKt − βt dt
0

a.s.

Remark. The attribute “strong” is used in the usual sense signifying the solution is an adapted function of the driving L´evy process. We make the following assumptions: (H1) There exists a U0 ∈ BU such that ν(U \ U0 ) < ∞. (H2) (Lipschitz continuity) There exists a constant C1 such that for any y1 , y 2 ∈ Rd , y1 − y2 , b(y1 ) − b(y2 ) + y1 − y2 , σ(y1 ) − σ(y2 ) < =  + y 1 − y2 , (f (y1 , u) − f (y2 , u))ν(du)  C1 |y1 − y2 |2 . U0

(H3) (Linear growth) There exists C2 > 0 such that for any y ∈ Rd ,  |b(y)|2 + σ(y) 2 + |f (y, u)|2 ν(du)  C2 (1 + |y|2 ). U0

(H4) A is a maximal monotone operator with D(A) = Rd . We can now state our main result. Theorem 2.4. Under the above assumptions, Equation (1) admits a unique solution. The next section is devoted to the proof of this result.

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3. Proof of Theorem 2.4 The following result is a slight improvement of [6, Lemma 4.1]. Proposition 3.1. Let A be a multi-valued maximal monotone operator, (X, K) and (X  , K  ) be c` adl` ag functions with X, X  ∈ D(A), K, K  of finite variation. If for any c` adl` ag functions α ∈ D(A) and βt := A0 (α(t)) satisfying  t |βs |ds < ∞, ∀t
0, 0

it holds that Xt − αt , dKt − βt dt
0, Xt − αt , dKt − βt dt
0, then Xt − Xt , dKt − dKt 
0. Note that if α(v) is a c`adl` ag function and α(v) ∈ D(A) for all v ∈ R+ , then it is easy to see that β(v) := A0 (α(v)) is Lebesgue measurable. In fact, this can be seen from the facts that A0 (α(v)) = lim An (α(v)), ∀v
0 n→∞

and v → An (α(v)) is c` adl`ag, hence Lebesgue measurable. Hence the integral in the above proposition makes sense. Note also that any c` adl` ag function is bounded on bounded intervals. Proof. The proof is based on the following two properties of maximal monotone operators: (i) n1 An (x) → 0 (n → ∞), ∀x ∈ D(A); (ii) y, A0 (x)
|A0 (x)|2 , ∀(x, y) ∈ Gr(A). For (ii) see [2, p. 77]; (i) holds since An = n(I − Jn ) and Jn (x) → projD(A) (x) for all x. Set Xv + Xv αn (v) = Jn ( ), β n (v) = A0 (αn (v)). 2 Then αn is c` adl` ag. Thus for any s < t,  t Xv − αn (v), dKv − β n (v)dv
0, s  t Xv − αn (v), dKv − β n (v)dv
0. s

Note that Jn = I − we have

=  t< Xv − Xv 1 Xv + Xv n + An , dKv − β (v)dv
0; 2 n 2 s 1 A , n n

Multi-valued Stochastic Differential Equations  t< s

Xv − Xv 1 + An 2 n



Xv + Xv 2



, dKv

195

= − β (v)dv

Summing them we get  t  < 1 1 t Xv Xv − Xv , dKv − dKv 
− An n s s 2  < 2 t Xv + An n s

n

+ Xv 2


0.



, dKv + dKv

= + Xv , β n (v) dv. 2

=

By (ii) the second term is positive since An (x) ∈ AJn (x) for all x. Since |An (x)|  cn|x| for some constant c independent of n, the first term goes to 0 as n → ∞ in virtue of (i) and the dominated convergence theorem. Hence we get the result.  We can deduce from this immediately the following corollary. Corollary 3.2. In Definition 2.3, (iv) can be replaced by the following equivalent condition (iv ) For any c` adl` ag and (Ft )-adapted function α with αt ∈ D(A), and



∀t ∈ [0, +∞),

t

|A0 (αs )|ds < ∞, ∀t
0, 0

we have Xt − αt , dKt − A0 (αt )dt
0

a.s.

The rest of the section is devoted to the proof of Theorem 2.4. We first consider the following equation:    dZt ∈ −A(Zt )dt + b(Zt )dt + σ(Zt )dBt + ˜ (dt, du), f (Zt− , u)N (2) U0  Z 0 = x0 . where U0 ∈ B(U) such that ν(U \ U0 ) < ∞. We have Theorem 3.3. Under (H1)–(H4), Equation (2) has a unique strong solution. 3.1. Proof of Theorem 3.3: Uniqueness Suppose (Z, K) and (Z  , K  ) are solutions. By Itˆ o’s formula,  t  t |Zt − Zt |2 = 2 Zs − Zs , b(Zs ) − b(Zs )ds + 2 Zs − Zs , σ(Zs ) − σ(Zs )dBs 0 0  t  t+   ˜ (ds, du) −2 Zs − Zs , dKs − dKs  + (f (Zs− , u) − f (Zs− , u))2 N 0

 +2

0

t+

 Zs− − Zs− ,

0



U0

 ˜ (ds, du) (f (Zs− , u) − f (Zs− , u))N U0

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t

σ(Zs ) −

+

σ(Zs ) 2 ds

 t

 (f (Zs− , u) − f (Zs− , u))2 dsν(du)

+

0

0

U0

Taking expectations on both sides, we have by (H1) and (H2)  t  t E|Zt − Zt |2  E Zs − Zs , b(Zs ) − b(Zs )ds + E σ(Zs ) − σ(Zs ) 2 ds 0

0+

 t

 (f (Zs− , u) − f (Zs− , u))2 ν(du)ds

+E 0  t

 CE

U0

|Zs − Zs |2 ds.

0

Gronwall’s lemma yields that E|Zt − Zt |2 = 0 and therefore we get the uniqueness. 3.2. Proof of Theorem 3.3: Existence We shall denote by D the space of c` adl` ag increasing functions f on [0, ∞) such that f (0) = 0. We equip it with the topology of convergence at the continuity points of the target function. Then it is a Polish space with the metric  ∞ dD (f, g) := e−t [1 ∧ |f (t) − g(t)|]dt. 0

By Helly’s first theorem, for any positive increasing function C(t), the set {f : |f (t)|  C(t), ∀t > 0} is compact in D. Hence we have Lemma 3.4. If (ξn ) is a family of D-valued random variables with sup E|ξn (t)| < ∞, ∀t > 0, n

then (ξn ) is tight on D. Let D([0, +∞); Rd ) (resp. C([0, +∞); Rd )) denote the space of c`adl` ag (resp. continuous) functions from [0, +∞) to Rd equipped with the Skorohod (resp. compact uniform convergence) topology. Denote by Ts [0, ∞) (resp. Tc [0, ∞)) the space of right-continuous and strictly increasing (resp. continuous and nondecreasing) functions a on [0, ∞) with a(0) = 0 and a(∞) = ∞. The following proposition is due to [9]. Proposition 3.5. Suppose {xn } ⊂ D([0, +∞); Rd ), {an } ⊂ Ts [0, ∞) and an (0) = 0, an (t) → ∞ when t → ∞. Let yn (t) = xn (a−1 n (t)). Suppose there exist y ∈ D([0, +∞); Rd ) and a ∈ Ts [0, ∞) such that yn → y,

n → ∞,

in D([0, +∞); R ) and an (t) → a(t) at each continuity point of a. Define x(t) = y(a(t)). If y is constant except for at most one jump on any interval [s, t] on which a−1 is constant, then xn → x in the Skorohod topology. d

Multi-valued Stochastic Differential Equations

197

Consider the equation: dZtn = − An (Ztn )dt + b(Ztn )dt + σ(Ztn )dBt  n ˜ (dt, du); Z0n = x0 . + f (Zt− , u)N

(3)

U0

Since An is Lipschitz by Proposition 2.2, there exists a unique solution Z n to (3) under (H1)–(H4) by [7, Ch. 4, Th. 9.1]. Now we set  t n at = |An (Zvn )|dv + t; θtn = (ant )−1 ; 0  t  t n n n ˜ (dv, du); Mt = σ(Zv )dBv + f (Zv− , u)N 0



0

U0

t

Ktn =

An (Zvn )dv; 0

Ytn = Zθntn ;

Htn = Kθntn .

We need the lemma below from C´epa [5]: Lemma 3.6. ∃a0 ∈ Rd , r > 0, µ
0 such that for all n ∈ N, x ∈ Rd An (x), x − a0 
r|An (x)| − µ|x − a0 | − rµ. Proposition 3.7. (an , M n , Y n , H n , θn )n is tight in D × D(R2d ) × C(Rd+1 ). Proof. (i) (H n , θn )n is tight in C(Rd+1 ) since for any n |θtn − θsn |  |t − s| and

(4)

 n     θt   t A (Y n )   n v n n n   |Ht − Hs | =  An (Zv − )dv  =  dv  n  θsn  s 1 + |An (Yv )|   t n  An (Yv )    dv  |t − s|.   n  s 1 + |An (Yv )|

(5)

n (ii) Define TNn = inf{t
0 : |Ztn |
N or |Zt− |
N }. By Itˆ o’s formula and Lemma 3.6,  t∧TNn +  1 n 1 n n ˜ (ds, du) |Zt∧T n − a0 |2 = |x0 − a0 |2 + Zs− − a0 , f (Zs− , u)N N 2 2 0 U0 n   t∧TNn  1 t∧TN + n 2 ˜ + (f (Zs− , u)) N (ds, du) + Zsn − a0 , b(Zsn )ds 2 0 U0 0  t∧TNn  t∧TNn + Zsn − a0 , σ(Zsn )dBs − Zsn − a0 , An (Zsn )ds 0

1 + 2

 0

n t∧TN



0

(f (Zsn , u))2 dsν(du) U0

1 + 2



n t∧TN

0

tr(σσ ∗ )(Zun )du

198

J. Ren and J. Wu  t∧TNn +  1 2 n n ˜ (ds, du)  |x0 − a0 | + Zs− − a0 , f (Zs− , u)N 2 0 U0 n    t∧TNn 1 t∧TN + n 2 ˜ + (f (Zs− , u)) N (ds, du) + Zsn − a0 , b(Zsn )ds 2 0 U0 0  t∧TNn + Zsn − a0 , σ(Zsn )dBs 0

+

1 2



n t∧TN

 (f (Zsn , u))2 dsν(du) +

0



U0 n t∧TN

−C



n t∧TN

|An (a0 )|2 ds + C

0

1 2



n t∧TN

tr(σσ ∗ )(Zsn )ds

0

|Zsn − a0 |2 ds.

0

By the BDG inequality :  .  s∧TNn +    n n ˜ (dv, du) E sup  Zv− − a0 , f (Zv− , u)N  0
s
t  0 U0 . : 1/2 n  t∧TN

 CE .

1 E 4

%

f (Zsn , u) 2 ν(du)ds U0

n t∧TN

 CE 

|Zsn − a0 |2

0

|Zsn

− a0 | (1 + 2

0

:1/2 |Zsn |2 )ds

&

n 2 sup |Zs∧T n − a0 | N

0
s
t



n t∧TN

+ CE

|Zvn − a0 |2 dv + Ct,

0

where we used the inequality |xy|  Similarly

1 2 |x| + 9|y|2 . 9

 :  s∧TNn    E sup  Zvn − a0 , σ(Zvn )dBv    0
s
t 0 % &  t∧TNn 1 n 2  E sup |Zs∧TNn − a0 | + CE |Zvn − a0 |2 dv + Ct. 4 0
s
t 0  : .  s∧TNn +     n 2 ˜ E sup  (f (Zv− , u)) N (dv, du)  0
s
t  0 U0 % &  t∧TNn 1 n 2  E sup |Zs∧T + CE |Zsn − a0 |2 ds + Ct. n − a0 | N 4 0
s
t 0 .

It is easy to prove that the expectation of the other terms are bounded by  t∧TNn CE |Zsn − a0 |2 ds. 0

(6)

Multi-valued Stochastic Differential Equations

199

Combining all these inequalities we obtain finally % &

 t n 2 n 2 E sup |Zs∧T n − a0 |  C(t + 1) + C E sup |Z n − a | ds. 0 v∧TN N 0
s
t

0
v
s

0

Then Gronwall’s Lemma yields that & % n 2 E sup |Zs∧TNn − a0 |  C(t)eCt . 0
s
t

Letting N → ∞ we have

% E

& sup

0
s
t

|Zsn

− a0 |

 C(t).

2

(7)

Substituting this into (6) we obtain  t∧TNn n 2 E |An (Zvn )|dv  CE|Zt∧T n − a0 | . N 0

Letting N → ∞ gives



t

|An (Zvn )|dv  C(t).

E 0

Hence for all n, E(ant )  C(t) < ∞. This proves that (an )n is tight in D by Lemma 3.4. (iii) By (H2) and (ii),  t  t+  n ˜ (dv, du)|2 E|Mtn − Msn |2 = E| σ(Zvn )dBv + f (Zv− , u)N s



s



t

 E|

U0 t+



σ(Zvn )dBv |2 + E|



s t

E

n ˜ (dv, du)|2 f (Zv− , u)N s

 t σ(Zvn ) 2 dv + E

s

 E

(8)

U0

f (Zvn , u) 2 ν(du)dv s

U0

t

(1 + Zvn 2 )dv s

 C(t)|t − s|. Hence for 0  t1 < t < t2 < ∞, we have by the Markov property E{|Mtn − Mtn1 |2 |Mtn2 − Mtn |2 } = E{E(|Mtn − Mtn1 |2 |Mtn2 − Mtn |2 |Ft )} = E{|Mtn − Mtn1 |2 E(|Mtn2 − Mtn |2 |Ft )} = E{|Mtn − Mtn1 |2 E(|Mtn2 − Mtn |2 |Mtn )}  CE{|Mtn − Mtn1 |2 |t2 − t|}  C|t − t1 ||t2 − t|  C|t2 − t1 |2 .

(9)

200

J. Ren and J. Wu For s < t we have by the BDG inequality

E[|Zθntn − Zθnsn |2 |Fθsn ]   n =E |

θt

θsn

  +E |

b(Zvn )dv − Htn + Hsn |2 |Fθsn θtn

θsn

  +E |

σ(Zvn )dBv |2 |Fθsn

θtn +

θsn

 U0

n ˜ (dv, du)|2 |Fθn f (Zv− , u)N s

 CE[|θsn − θtn ||Fθsn ]  C|t − s|. Consequently for t1 < t < t2 we have E{|Ytn2 − Ytn |2 |Ytn − Ytn1 |2 } = E[|Zθntn − Zθntn |2 |Zθntn − Zθntn |2 ] 2

1

= E[|Zθntn − Zθntn |2 E[|Zθntn − Zθntn |2 |Fθtn ]] 1

2

 C|t2 − t|E[|Zθntn − Zθntn |2 ] 1

 C|t2 − t||t − t1 |  C|t2 − t1 |2 .

(10)

Hence the tightness of (M n , Y n )n in D(R2d ) follows from (9) and (10) by [3, Ch. 3] (or [10, Theorem 6.23]). The proof is now complete.  Denote the law of (an , M n , Y n , H n , θn ) by P n . Then by the proposition above, there exists a subsequence nk such that P nk → P,

k → ∞.

˜ F, ˜ P˜ ) on By the Skorohod realization theorem, there exists a probability space (Ω, n ˜ n ˜ n ˜ n ˜n ˜ ˜ ˜ ˜ which (˜ a , M , Y , H , θ ) and (˜ a, M , Y , H, θ) are defined such that for all n ˜ n , Y˜ n , H ˜ n , θ˜n ) (˜ an , M has the same distribution as (an , M n , Y n , H n , θn )

and

˜ ˜ ˜ ˜ P˜ (˜a,M,Y ,H,θ) = P

and P˜ -almost surely ˜ ˜ n , Y˜ n , H ˜ n , θ˜n ) → (˜ ˜ , Y˜ , H, ˜ θ). (˜ an , M a, M

(11)

˜ n and θ˜n are almost surely Lipschitz continuous with a In particular, since H ˜ ˜ and θ. common Lipschitz constant 1, so are H Denote Z˜ n = Y˜ nn , Z˜t = Y˜a˜t ; t

a ˜t

˜n = H ˜ nn , K ˜t = H ˜ a˜t . K t a ˜t

Multi-valued Stochastic Differential Equations

201

˜ 1 ∈ F˜ with P˜ (Ω ˜ 1 ) = 1 such that if ω ˜ 1 , and Proposition 3.8. There exists Ω ˜ ∈Ω d ˜ ˜ there exists 0  s  t < ∞ such that θs (˜ ω ) = θt (˜ ω ), then for all x ∈ R ,  t ˜ u (˜ Y˜u (˜ ω ) − x, dH ω )
0. s

Proof. The proof is the same as the continuous case given in [6] and we give the short proof here for completeness. ˜1 ⊂ Ω ˜ such that P˜ (Ω ˜ 1 ) = 1 and the convergence (11) holds a.s. on Choose Ω it. Since An is monotone,  θ˜tn ˜ vn − An (x)dv
0. Z˜vn − x, dK θ˜sn

Thus





t

t

Y˜vn − x, An (x)dθ˜vn .

˜ n
Y˜vn − x, dH v s

Now

s

 t     Y˜vn − x, An (x)dθ˜vn   C( sup |Y˜vn | + |x|)|A◦ (x)||θ˜tn − θ˜sn |   s
v
t s

 C|θ˜tn − θ˜sn |, and the result follows by letting n → ∞ and from (11).



˜ 2 ∈ F, ˜ P˜ (Ω ˜ 2 ) = 1 such that for all ω ˜ 2 , if there Proposition 3.9. There exists Ω ˜∈Ω exists 0  s  t < ∞ such that θ˜s = θ˜t , then ˜ s (˜ ˜ v (˜ Y˜v (˜ ω ) − Y˜s (˜ ω) = H ω) − H ω ), ∀v ∈ [s, t]. ˜ n the subspace of Ω ˜ such that P˜ (Ω ˜ n ) = 1 on which the equation Proof. Denote by Ω below holds:  n ˜n + ˜ (dt, du) − dK ˜ n. dZ˜tn = b(Z˜tn)dt + σ(Z˜tn )dB f (Z˜t− , u)N t t U0

Set ˜ 2 = ∩n Ω ˜ n, Ω ˜ 2 ) = 1 and on Ω ˜ 2 we have for v ∈ (s, t] then P˜ (Ω  θ˜vn  θ˜vn n n n ˜ ˜ ˜ ˜hn Yv − Ys = b(Zh )dh + σ(Z˜hn )dB θ˜sn



θ˜vn +



n ˜ (dh, du) − (H ˜ vn − H ˜ sn ) f (Z˜h− , u)N

+ 

θ˜sn θ˜vn

= θ˜sn

θ˜sn

U0

˜ ˜nn − M ˜ ˜nn ) − (H ˜n − H ˜ n ). b(Z˜hn )dh + (M v s θ θ v

s

202

J. Ren and J. Wu

Since  ˜n   ˜n  θ˜vn  θv  θv   n b(Z˜h )dh  |b(Z˜hn )|dh  C2 |(1 + |Z˜hn |)dh  C2 |θ˜vn − θ˜sn |,   θ˜sn  n n ˜ ˜ θs θs thus if θ˜t = θ˜s , by the convergence (11),  ˜n   θv    b(Z˜hn )dh  C|θ˜sn − θ˜vn | → 0.   θ˜sn  ˜ ˜ is constant on [s, t] it is continuous at least on (s, t]. Hence, by the Since v → M θv ˜n →M ˜ in D(Rd ) one knows that convergence M ˜ ˜nn → M ˜˜ M θv

θv

for v ∈ (s, t]. Thus ˜ ˜nn − M ˜ ˜nn → 0, ∀v ∈ (s, t]. M θ θ t

s

Taking limit yields for all v ∈ (s, t], ˜s − H ˜v ; Y˜v − Y˜s = H note that the equality holds of course for v = s and the proof is therefore complete.  ˜  ∈ F˜ , P˜ (Ω ˜  ) = 1 such that for all ω ˜  , there exists Proposition 3.10. ∃Ω ˜ ∈ Ω 0  s  t < ∞ such that θ˜s = θ˜t . Then Y˜v = Y˜t for v ∈ [s, t]. Proof. Take

˜ = Ω ˜1 ∩ Ω ˜ 2, Ω ˜  ) = 1. If θ˜s = θ˜t , since θ˜· is increasing, thus θ˜· is constant on [s, t]. then P˜ (Ω ˜ h is Lipschitz continuous hence absolutely continuous, by Noticing that h → H Proposition 3.8 and 3.9 we have for v ∈ [s, t],  t 2 2 ˜ ˜ ˜ ˜ ˜h − H ˜ v , dH ˜ h |Yt − Yv | = |Ht − Hv |  2 H v  t ˜ h   0. Y˜h − Y˜v , dH = −2 v

Thus Y˜v = Y˜t .



Remark. With (H4), we do not need to prove Y (t) ∈ D(A)-a.s. for all t ∈ [0, ∞). Otherwise it would be a sticky job since after time change, the c`adl` ag function Y (t) may jump out of the domain. ˜ n ) → (Z, ˜ K) ˜ in By Proposition 3.5 and Proposition 3.10, we know (Z˜ n , K the Skorohod topology. Since relativized to the space continuous functions C the Skorohod topology coincides with the compact uniform topology we have ˜ n) → (Z, ˜ K) ˜ in D(Rd ) × C(Rd ). Proposition 3.11. (Z˜ n , K

Multi-valued Stochastic Differential Equations

203

To proceed we need the following lemma (see [8, Lemma 2]): Lemma 3.12. Let f n, g n ∈ D(Rd ) and supn |g n|T < +∞, (f n , g n) → (f, g) in D(R2d ), then



 .  . n n n n f ,g , fs dgs → f, g, fs dgs in D(R2d+1 ). 0

0

˜ F˜ , P˜ ), (Z, ˜ K) ˜ solves the following We need to prove that on the space (Ω, equation:    ˜t + ˜ (dt, du),  dZ˜t ∈ −A(Z˜t )dt + b(Z˜t )dt + σ(Z˜t )dB f (Z˜t− , u)N U0 (12)   Z˜ = x . 0 0 Proposition 3.13. Z˜ satisfies the following equation:  ˜ ˜ ˜ ˜ ˜ (dt, du) − dK ˜t . dZt = b(Zt)dt + σ(Zt )dBt + f (Z˜t− , u)N

(13)

U0

Proof. Since Z˜ n solves:



˜n + dZ˜tn = b(Z˜tn)dt + σ(Z˜tn )dB t

n ˜ (dt, du) − dK˜n , f (Z˜t− , u)N t U0

˜ n ) → (Z, ˜ K) ˜ on D(Rd ) × C(Rd ). Then the result follows by letting and (Z˜ n , K n → ∞ and using the same argument as in [12] by using Lemma 3.12.  ˜ K) ˜ is a solution, it remains to verify (iv) in DefiniFinally, to prove that (Z, tion 2.3. To this end, by Corollary 3.2, it suffices to prove Proposition 3.14. For any adapted c` adl` ag function αt such that αs ∈ D(A), and



∀s ∈ [0, +∞)

t

|A0 (αs )|ds < ∞, ∀t
0, 0

we have ˜ s − A0 (αs )ds
0. Z˜s − αs , dK Proof. Since An is monotone,  t ˜ v − An (αv )dv
0. Z˜vn − αv , dK s

From the proof of Proposition 3.7 we know that ˜n − H ˜ n |  |t − s|. an  C(t) a.s.; |H t

t

s

Thus ˜ n |t  |H n |an  ant  C(t, ω |K ˜ ) < ∞. t

(14)

204

J. Ren and J. Wu

By Lemma 3.6, Proposition 3.11 we have by letting n → ∞,  t  t ˜ n → ˜ v ; Z˜vn − αv , dK Z˜v − αv , dK v s s  t  t Z˜vn − αv , An (αv )dv → Z˜v − αv , A◦ (αv )dv. s

(15) (16)

s

Combining (14) and (15) gives  t ˜ v − A◦ (αv )dv
0. Z˜v − αv , dK



s

˜ K) ˜ is a weak solution of Equation (2). By Subsection 3.1, the Hence (Z, pathwise uniqueness for this equation holds. Consequently, there is a unique strong solution to Equation (2) by Yamada-Watanabe theorem and the proof is complete. 3.3. Proof of Theorem 1.1 Now that Theorem 3.3 is proved, to prove Theorem 2.4 it suffices to patch up together the unique solutions in different intervals between big jumps. The procedure is standard as in, e.g., [7, 1] and we omit the details. Acknowledgment We thank Xicheng Zhang heartily for his very carefully reading the manuscript and for helpful discussions. We also thank the anonymous referees for their useful suggestions.

References [1] Applebaum, D.: Levy Processes and Stochastic Calculus. Cambridge University Press, U.K., 2004. [2] Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Leyden Netherland, Noord-Hoff Internet Publishing, 1976. [3] Billingsley, P.: Convergence of Probability Measures. John Wiley and Sons, Inc., New York, 1968. [4] Carothers, N.L.: Real Analysis. Cambridge Univ. Press, 2000. ´ [5] C´epa, E.: Equations diff´erentielles stochastiques multivoques. Lect. Notes in Math. S´em. Prob. XXIX 86–107 (1995). [6] C´epa, E.: Probl`eme de skorohod multivoque. Ann. Prob. V. 26, No. 2, 500–532 (1998). [7] Ikeda, N. and Watanabe S.: Stochastic Differential Equations and Diffusion Processes. 2nd ed., Kodansha, Tokyo/North-Holland, Amsterdam, 1989. [8] Jakubowski, A., M´emin, J. and Page, G.: Convergence en loi des suite d’int´egrales stochastiques sur l’espace D1 de Skorohod, Probab. Th. Rel. Fields V. 81, No. 2, 111–137 (1989). [9] Kurtz, T.: Random time change and convergence in distribution under the MeyerZheng conditions. Ann. Probab. 19, No. 3, 1010–1034 (1991).

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[10] Qian, M. and Gong, G.: Theory of Stochastic Processes. Beijing Univ. Press, 1997 (in Chinese). [11] Ren, J. and Xu, S.: A transfer principle for multivalued stochastic differential equations. J. Funct. Anal. 256, No. 9, 2780–2814 (2009). [12] Skorohod, A.V.: Studies in the Theory of Random Processes. Addison-Wesley, Reading, Massachusetts, 1965. [13] Zhang, X.: Skorohod problem and multivalued stochastic evolution equations in Banach spaces. Bull. Sci. Math. 131, 175–217 (2007). Jiagang Ren and Jing Wu School of Mathematics and Computational Science Sun Yat-sen University Guangzhou Guangdong 510275, P.R. China e-mail: [email protected] [email protected]

Progress in Probability, Vol. 65, 207–219 c 2011 Springer Basel AG


Sensitivity Analysis for Jump Processes Atsushi Takeuchi Abstract. Consider stochastic differential equations with jumps. The goal in this paper is to study the sensitivity of the solution with respect to the initial point, under the conditions on the L´evy measure and the uniformly elliptic condition on the coefficients. The key tool is the martingale property based upon the Kolmogorov backward equation for the infinitesimal generator associated with the equation. Mathematics Subject Classification (2000). 60H30, 60J75, 60H07. Keywords. Heat kernel, Jump processes, Logarithmic derivatives, Stochastic differential equations.

1. Introduction The Malliavin calculus helps us to study the existence of smooth densities for solutions to stochastic differential equations (SDEs). It is well known that the H¨ ormander type conditions on the coefficients imply the negative-moment integrability of the Malliavin covariance matrix, and that the Sobolev inequality leads to the existence of the smooth density. See [16, 17, 18]. In this paper, we shall focus on the sensitivity analysis for the solution to the jump-type SDE with respect to the initial point, which is equivalent to the Greeks (Delta) computations for payoff functions of asset price models in mathematical finance. There have been a lot of studies in case of the process without any jumps, by using the Malliavin calculus on the Wiener space ([12, 17]), the Girsanov transform on Brownian motions ([3]), and the Kolmogorov backward equation for the infinitesimal generator associated with the diffusion process ([11]). On the other hand, a similar problem has been discussed in case of jump-type financial models. Cass and Friz [7], and Davis and Johansson [8] studied for jump diffusion processes, but their approach does not take any effects from the jumps. Bally et al. [2], Elkhatib and Privault [9], and Elliott and Tsoi [10] applied the calculus focused on the jump term effects to the Greeks computations. The author in [19] studied the same problem in case of solutions to SDEs with jumps via the martingale method

208

A. Takeuchi

by using the Kolmogorov backward equation for the integro-differential operator associated with the equation. The results obtained in [19] reflect the effects from the diffusion and the jump coefficients. Moreover, the process can be of pure jump type, and of infinitely activity type. This paper is organized as follows: In Section 2, we shall prepare some notations and introduce our framework. Furthermore, the main result on the sensitivity with respect to the initial point will be also stated there. Section 3 is devoted to give the sketch of the proof, and we shall give some examples in Section 4.

2. Preliminaries and main result At the beginning, we shall prepare some notations. Fix T > 0, and let (Ω, F , P) be our underlying probability space. Denote by E[ · ] the expectation with respect to the measure dP, and by IK the indicator function for K ⊂ R. The script “ 0 ” of C02 (R), etc. indicates the compact support, and the script “ b ” of Cb (R), etc. indi∞ cates the boundedness. Let C1+,b (R) be the family of C ∞ (R)-functions such that all derivatives of any orders more than 1 are bounded. For an R-valued function f (t, y, θ), denote the derivative in t by ∂t f (t, y, θ), the one in y by f  (t, y, θ), and the one in θ by ∂θ f (t, y, θ). Denote by ci (i = 0, 1, 2, . . . ) different positive finite constants. Write R0 = R\ {0}, and let dν be a L´evy measure over R0 . Throughout the paper, suppose that (i) (the moment condition): for any p ≥ 1,   |θ| dν + |θ|p dν < +∞,

(2.1)

(ii) (the order condition): there exists α > 0 such that    lim inf ρα (θ/ρ)2 ∧ 1 dν > 0,

(2.2)

|θ|≤1

|θ|>1

ρ0

R0

(iii) the measure dν has a C 1 -density g (θ) with respect to the Lebesgue measure over R0 such that lim

|θ|+∞

θ2 g(θ) = 0.

(2.3)

Let W = {Wt ; t ∈ [0, T ]} be a one-dimensional Brownian motion with W0 = 0, and dJ a Poisson random measure on [0, T ] × R0 with the intensity dJˆ = dt dν. Denote by {Ft ; t ∈ [0, T ]} the augmented filtration generated by W and dJ. In order to avoid the lengthy expression, we shall use the notations: dJ˜ = dJ − dJˆ,

dJ = I(|θ|≤1) dJ˜ + I(|θ|>1) dJ.

Sensitivity Analysis for Jump Processes

209

∞,∞ ∞ Let Ai (y) ∈ C1+,b (R) (i = 0, 1), and Bθ (y) in C1+,b (R × R0 ) as a function of (y, θ). Suppose that the functions A1 (y) and Bθ (y) satisfy

inf inf |1 + Bθ (y)| > 0,

lim Bθ (y) = 0,

|θ|0

y∈R θ∈R0

2

inf (A1 (y)) > 0,

y∈R

2

inf inf (∂θ Bθ (y)) > 0.

y∈R θ∈R0

(2.4) (2.5)

For x ∈ R, consider the jump process X = {Xt ; t ∈ [0, T ]} determined by the SDE of the form: X0 = x and  dXt = A0 (Xt ) dt + A1 (Xt ) ◦ dWt + Bθ (Xt− ) dJ , (2.6) R0

where the symbol “ ◦dWt ” indicates the Stratonovich stochastic integral. From the conditions on the coefficients, there exists a unique solution to the equation (2.6). The process X is Markovian, and the associated infinitesimal generator is the integro-differential operator of the form:  1 Lf (y) = A0 f (y) + A1 A1 f (y) + {Bθ f (y) − Bθ f (y)} dν 2 R0 for f ∈ C02 (R), where Ai f (y) = Ai (y) f  (y) (i = 0, 1) and Bθ f (y) = Bθ (y) f  (y) are vector fields over R, A1 A1 f (y) = A1 (y) (A1 (y) f  (y)) and Bθ f (y) = f (y + Bθ (y)) − f (y). The Kolmogorov continuity criterion on random fields implies that, for each t ∈ [0, T ], the function R  x −→ Xt ∈ R has a C 1 -modification (cf. [13]). Moreover, the derivative Zt = ∂x Xt is invertible a.s., and the process Z = {Zt ; t ∈ [0, T ]} satisfies the linear SDE: Z0 = 1 and  dZt = A0 (Xt ) Zt dt + A1 (Xt ) Zt ◦ dWt + Bθ (Xt− ) Zt− dJ. (2.7) R0

Now, we shall briefly introduce the criterion on the existence of the smooth density for the random variable XT stated in [16, 18]. Write   ˜θ (y) = 1 + B  (y) −1 ∂θ Bθ (y) θ, B θ and let 0 < ρ < 1. Since       ˜θ (y)/ρ 2 ≥ c0 ∂θ Bθ (y) θ/ρ 2 ≥ c1 θ/ρ 2 B ∞,∞ from Bθ (y) ∈ C1+,b (R × R0 ) and the condition (2.5), we see that  ! 2    " ˜θ (y)/ρ 2 ∧ 1 dν inf A1 (y)/ρ + B y∈R R  0 !  2  " ≥ c2 ρ−2 + θ/ρ ∧ 1 dν R0

−γ

≥ c3 ρ

by the condition (2.2), where γ = 2 ∧ α. Then, using the result in [16, 18], there exists a smooth density pT (x; y) for the probability law of the random variable XT

210

A. Takeuchi

with respect to the Lebesgue measure on R. The goal in the present paper is to study the sensitivity for the solution XT in the initial point x ∈ R. In order to state the main result on the sensitivity for XT with respect to x, we shall prepare the notations. Write  T A˜T = θ2 dJ, 0

R0

and define the family of R-valued functions by !n " F= αk fk IAk ; αk ∈ R, fk ∈ CLG (R), Ak ⊂ R : interval , k=1

where CLG (R) is the set of continuous functions ψ with the linear growth condition: |ψ(y)| ≤ c4 (1 + |y|). Theorem 1. For ϕ ∈ F, it holds that #

$ LT − VT KT ∂x (E [ϕ(XT )]) = E ϕ(XT ) + 2 , AT AT

(2.8)

where AT = T + A˜T and  T  T Zt 2Zt− θ4 LT = dWt , KT = dJ, ˜ A1 (Xt ) 0 0 R0 Bθ (Xt− )

 T 1 ∂ g(θ) Zt− θ 3 ˜ VT = dJ. ˜θ (Xt− ) B 0 R0 g(θ) ∂θ Remark 2.1. The class F includes typical functions which often appear in mathematical finance, such as continuous, or discontinuous European payoffs (e.g., call options, put options, digital options, etc.).  Remark 2.2. Let λ > 1 be sufficiently large, and write  T   λ ˆ NT = exp(−λθ2 ) − 1 dJ. 0

R0

Since AT is non-negative and   2  T λ NT ≤ − (λθ ) ∧ 1 dν ≤ −c5 λα/2 T 2 R0 from the condition (2.2), we see that, for any p > 0,  +∞ ' ( ' ( 1 = λp−1 E exp(−λ AT ) dλ E A−p T Γ(p) 0  +∞ 3  4 1 = λp−1 E exp − λA˜T − NTλ exp(−λT + NTλ ) dλ Γ(p) 0  +∞ ) * 1 ≤ λp−1 exp −λT − c5 λα/2 T dλ < +∞ Γ(p) 0 from the Fubini theorem.



Sensitivity Analysis for Jump Processes

211

3. Proof of Theorem 1 We shall give the proof of Theorem 1 briefly. Suppose that ϕ ∈ C02 (R) for a while. Define  ' ( u(t, x) = E ϕ(XT −t )X0 = x (t ∈ [0, T )) . Since ϕ is bounded, u belongs to Cb1,2 ([0, T )×R). Moreover, the function u satisfies the Kolmogorov backward equation (cf. [14]): ∂t u + Lu = 0,

lim u(t, x) = ϕ(x).

tT

The following lemma plays a crucial role in our discussion. Lemma 3.1. For ϕ ∈ C02 (R), it holds that  T ϕ(XT ) = E[ϕ(XT )] + u (s, Xs ) A1 (Xs ) dWs 0

 T

(3.9)

+ R0

0

Proof. Since u ∈

˜ Bθ u(s, Xs− ) dJ.

Cb1,2 ([0, T )

× R), the Itˆo formula yields that  t u(t, Xt ) = u(0, x) + u (s, Xs ) A1 (Xs ) dWs 0  t + Bθ u(s, Xs− ) dJ˜

(3.10)

R0

0

for t ∈ [0, T ). Both sides of (3.10) converge to the ones of (3.9) as t ϕ ∈ C02 (R) and the boundedness of u .

T , by 

Lemma 3.2. For ϕ ∈ C02 (R), it holds that   T ∂x E [ϕ(XT )] = E [ϕ(XT ) LT ] .

(3.11)

Proof. Since   T R0

0

  ˜ Bθ u(t, Xt− ) dJ LT = 0

 t

T



0

T



+ 0

R0

˜ Bθ u(s, Xs− ) dJ dLt

R0

Bθ u(t, Xt− ) Lt− dJ˜

from the Itˆ o formula, we have     T Bθ u(t, Xt− ) dJ˜ LT E 0

R0



T

 t

=E 0

= 0.

0

R0



Bθ u(s, Xs− ) dJ˜ dLt + E 0

T

 R0

Bθ u(t, Xt− ) Lt− dJ˜

212

A. Takeuchi

Multiplying (3.9) in Lemma 3.1 by LT yields that  T

E [ϕ(XT ) LT ] = E[ϕ(XT )] E[LT ] + E

0



T

u (t, Xt ) A1 (Xt ) dWt

LT

0

  T +E

 

R0

 ˜ Bθ u(t, Xt− ) dJ LT

u (t, Xt ) Zt dt

=E 0

= T ∂x (E [ϕ (XT )]) . The last equality can be obtained by the commutativity between the derivative ∂x and the integral with respect to dP ⊗ dt, which comes from the dominated convergence theorem and ϕ ∈ C02 (R).  Remark 3.1. As seen above, only the uniformly elliptic condition on the diffusion coefficient A1 (y) is used in Lemma 3.2. The condition on the L´evy measure dν and the uniformly elliptic condition on the jump term coefficient are not necessary.  Lemma 3.3. For ϕ ∈ C02 (R), it holds that  ' ( ∂x E ϕ(XT ) A˜T = −E [ϕ(XT ) VT ] . Proof. Write J˜T = 

T


T
0

(3.12)

˜ Since θ2 dJ.

R0

 



T



t

J˜T =

u (s, Xs ) A1 (Xs ) dWs 0





u (s, Xs ) A1 (Xs ) dWs dJ˜t 0



+

0 T

u (t, Xt ) A1 (Xt ) J˜t dWt

0

from the Itˆ o formula, we have   T





E

u (s, Xs ) A1 (Xs ) dWs 0



T



=E





u (s, Xs ) A1 (Xs ) dWs dJ˜t 0

0



T

+E 0

= 0.

t

J˜T

u (t, Xt ) A1 (Xt ) J˜t dWt

Sensitivity Analysis for Jump Processes

213

Multiplying (3.9) in Lemma 3.1 by J˜T yields that   3 4 T ' (  ˜ ˜ E ϕ(XT ) JT = E[ϕ(XT )] E JT + E u (t, Xt ) A1 (Xt ) dWt J˜T 0

  T +E

R0

  T

0

0

R0

=E

 Bθ u(t, Xt− ) dJ˜ J˜T

Bθ u(t, Xt ) θ 2 dJˆ .

Hence, the Fubini theorem enables us to see that   T 4 3 ˜ E ϕ(XT ) AT = E u(t, Xt + Bθ (Xt )) θ2 dJˆ . 0

(3.13)

R0

On the other hand, since   T  u (t, Xt ) A1 (Xt ) dWt VT 0



=

T

 T 



t

u (t, Xt ) A1 (Xt ) Vt dWt + 0





u (s, Xs ) A1 (Xs ) dWs dVt 0

0

from the Itˆ o formula, we have    T  E u (t, Xt ) A1 (Xt ) dWt VT 0



T

u (t, Xt ) A1 (Xt ) Vt dWt

=E 0

  

T t +E u (s, Xs ) A1 (Xs ) dWs dVt 0

0

= 0. Multiplying (3.9) in Lemma 3.1 by VT yields that   T E [ϕ(XT ) VT ] = E[ϕ(XT )] E[VT ] + E u (t, Xt ) A1 (Xt ) dWt VT 0

  T +E   T =E

R0

0

R0

0

Bθ u(t, Xt )

  T = −E 0

 ˜ Bθ u(t, Xt− ) dJ VT

R0

1 ∂ g(θ) ∂θ



g(θ) Zt θ 3 ˜θ (Xt ) B

dJˆ

u (t, Xt + Bθ (Xt )) (1 + Bθ (Xt )) Zt θ 2 dJˆ

214

A. Takeuchi    T = −∂x E 0

 u(t, Xt + Bθ (Xt )) θ dJˆ 2

R0

) ' (* = −∂x E ϕ(XT ) A˜T . Here, the third equality holds by the condition (2.3), and the forth equality can be justified from the commutativity between the derivative ∂x and the integral with respect to dP ⊗ dJˆ, which comes from the dominated convergence theorem and ϕ ∈ C02 (R). The last equality can be justified by the equality (3.13).  Remark 3.2. The uniformly elliptic condition on the function Bθ (y) and the conditions on the L´evy measure dν are essential in Lemma 3.3.  Summing up Lemma 3.2 and Lemma 3.3, we have Corollary 3.1. For ϕ ∈ C02 (R), it holds that   ' ( ∂x E[ϕ(XT ) AT ] = E ϕ(XT ) (LT − VT ) .

(3.14)

Proof of Theorem 1. Since the random variable XT admits a smooth density with respect to the Lebesgue measure on R as stated in Section 2, it is sufficient to prove the assertion of Theorem 1 in case of ϕ ∈ C02 (R), via the standard density argument. See [15, 19] for details. Since we have already obtained in Corollary 3.1 that   ' ( ∂x E[ϕ(XT ) AT ] = E ϕ(XT ) (LT − VT ) for ϕ ∈ C02 (R), it is sufficient to discuss how to get rid of AT from the left-hand side of the above equality. Write MTλ = exp(−λT + NTλ ), and introduce the new probability measure dPλ over our underlying measurable space (Ω, F) defined by    dPλ  = exp − λA˜T − NTλ .  dP FT From the Girsanov theorem (cf. [1]), we see that under the probability measure dPλ , the process W = {Wt ; t ∈ [0, T ]} is also the one-dimensional Brownian motion, and that dJ is the Poisson random measure on [0, T ] × R with the 2 ˆ Moreover, remark that the random measure dJ˜λ defined by intensity e−λθ dJ. −λθ 2 ˆ ˜ dJλ = dJ − e dJ generates the martingale under the measure dPλ . We shall rewrite the equation (2.6) as follows:  dXt = A0 (Xt ) dt + A1 (Xt ) ◦ dWt + Bθ (Xt− ) dJ R  0 (3.15) ˜ Bθ (Xt− ) dJ λ , = A0 (Xt ) dt + A1 (Xt ) ◦ dWt + R0

Sensitivity Analysis for Jump Processes where

 A˜0 (y) = A0 (y) +

|θ|≤1

215

  2 Bθ (y) e−λθ − 1 dν,

dJ λ = I(|θ|≤1) dJ˜λ + I(|θ|>1) dJ. Denote by Eλ [ · ] the expectation with respect to the measure dPλ . Then, similarly to Corollary 3.1 on the original probability space (Ω, F , P), it holds that   ' ( ∂x Eλ [ϕ(XT ) AT ] = Eλ ϕ(XT ) (LT − VTλ ) (3.16) 2

on the new probability space (Ω, F , Pλ ), where gλ (θ) = e−λθ g(θ) and

 T 1 ∂ gλ (θ) Zt− θ3 λ VT = dJ˜λ . ˜ θ (Xt− ) B 0 R0 gλ (θ) ∂θ Since 2 ˆ dJ˜λ = dJ˜ + (1 − e−λθ ) dJ,



1 ∂ gλ (θ) Zt θ3 2λZt θ 4 1 ∂ g(θ) Zt θ3 =− + , ˜θ (Xt ) ˜θ (Xt ) ˜θ (Xt ) gλ (θ) ∂θ g(θ) ∂θ B B B

and the condition (2.3) implies that

 T  T 2 1 ∂ g(θ) Zt θ 3 2Zt θ 4 −λθ 2 ˆ (1 − e−λθ ) dJˆ = −λ e dJ ˜θ (Xt ) ˜ B 0 R0 g(θ) ∂θ 0 R0 Bθ (Xt ) from the integration by parts, we can get VTλ = VT − λKT . Then, it holds that ∂x (E [ϕ(XT )]) #  = ∂x E ϕ(XT ) 

+∞

e−λAT dλ AT

0 +∞

=

$

 = ∂x

+∞

MTλ Eλ [ϕ(XT ) AT ] dλ

0

  MTλ ∂x Eλ [ϕ(XT ) AT ] dλ

0



+∞

MTλ Eλ [ϕ(XT ) (LT − VT + λKT )] dλ 0 # $ # $  +∞  +∞ −λAT −λAT = E ϕ(XT ) (LT − VT ) e dλ + E ϕ(XT ) KT λe dλ 0 0 #

$ LT − VT KT = E ϕ(XT ) + 2 . AT AT =

Here, we have used the Fubini theorem in the second equality, the third equality can be obtained by the commutativity between the derivative ∂x and the integral with respect to dPλ ⊗ dλ, which comes from the dominated convergence theorem, ϕ ∈ C02 (R) and the independence of MTλ and AT on the variable x. In the forth equality, we have used (3.16). The proof of Theorem 1 is complete. 

216

A. Takeuchi

Remark 3.3. Consider the equation (2.6) where only the condition 2

inf (A1 (y)) > 0

(3.17)

y∈R

is satisfied. Then, we can get from Lemma 3.2 that # $   LT ∂x E[ϕ(XT )] = E ϕ(XT ) T

(3.18)

for ϕ ∈ C02 (R), and the equality (3.18) can be extended to the case of ϕ ∈ F via the standard density argument because of the existence of the smooth density for XT under (3.17). Remark that the conditions (2.1), (2.2) and (2.3) on the measure dν are not necessary. In case of the diffusion process given by dXt = A0 (Xt ) dt + A1 (Xt ) ◦ dWt with the uniform ellipticity on A1 (y), the formula (3.18) is exactly the same as the Bismut-Elworthy-Li formula (cf. [3, 5, 11]).  Remark 3.4. Consider the equation (2.6) where only the condition 2

inf inf (∂θ Bθ (y)) > 0

(3.19)

y∈R θ∈R0

is satisfied. We have already obtained in Lemma 3.3 that ) 3 4* ∂x E ϕ(XT ) A˜T = −E [ϕ(XT ) VT ]

(3.20)

for ϕ ∈ C02 (R), and the equality (3.20) can be extended to the case of ϕ ∈ F via the standard density argument because of the existence of the smooth density for XT under (3.19). By a similar argument to the proof of Theorem 1, we can get rid of the effect of A˜T from the left-hand side of (3.20). Thus, we have    VT KT ∂x (E[ϕ(XT )]) = E ϕ(XT ) − + 2 A˜T A˜ T

for ϕ ∈ F. In this case, the assumptions (2.1), (2.2) and (2.3) on the measure dν are essential.  Remark 3.5. As seen in Theorem 1, Remark 3.3 and Remark 3.4, we have already obtained the sensitivity formula for ϕ ∈ F in the three forms below: #

$ LT − VT KT ∂x (E[ϕ(XT )]) = E ϕ(XT ) + 2 , AT AT # $ LT ∂x (E[ϕ(XT )]) = E ϕ(XT ) , T    VT KT ∂x (E[ϕ(XT )]) = E ϕ(XT ) − + 2 . A˜T A˜T

Sensitivity Analysis for Jump Processes

217

It seems to be a natural question which minimizes the variance under the condition (2.5). In order to avoid lengthy expression, write αT =

LT − VT KT + 2 , AT AT

βT =

LT , T

γT = −

VT KT + 2 . ˜ AT A˜T

We shall consider the minimization problem of V [aαT + bβT + cγT ] under a, b, c ∈ R with a + b + c = 1. It is a routine work to see that V [aαT + bβT + cγT ] ≥ V [a0 αT + b0 βT + c0 γT ] , where C[X, Y ] = E [(X − E[X])(Y − E[Y ])], a0 = b0 =

C[αT − γT , βT − γT ] C[βT − γT , γT ] − V[βT − γT ] C[αT − γT , γT ] 2

,

2

,

V[αT − γT ] V[βT − γT ] − {C[αT − γT , βT − γT ]} C[αT − γT , βT − γT ] C[αT − γT , γT ] − V[αT − γT ] C[βT − γT , γT ] V[αT − γT ] V[βT − γT ] − {C[αT − γT , βT − γT ]}

and c0 = 1 − a0 − b0 . Thus, we have ∂x (E [ϕ (XT )]) = E [ϕ(XT ) (a0 αT + b0 βT + c0 γT )] , 

which minimizes the variance.

Remark 3.6. Our method using the martingale property based on the Kolmogorov backward equation can be also applied to the sensitivity analysis on the secondorder derivative (Gamma) in the initial point, and the other parameters (e.g., Vega) governing the process {Xt ; t ∈ [0, T ]} (cf. [19]). 

4. Examples Example 1 (L´evy measures). Let a, b, c > 0 and 0 < β < 1. Define , dν = a exp(bθ) I(θ<0) + exp(−cθ) I(θ>0) |θ|−1−β dθ, which is a special case of CGMY processes ([6]). It is easy to see that the measure dν satisfies the conditions (2.1), (2.2) and (2.3). In particular, the L´evy measures of inverse Gaussian processes (b = +∞, β = 1/2) and tempered stable processes (b = +∞, 0 < β < 1) are in our position.  Example 2 (L´evy processes). Let X = {Xt ; t ∈ [0, T ]} be the R-valued L´evy process given by  t Xt = x + γ t + σ Wt + δ θ dJ, 0

where x, γ ∈ R and (σ, δ) ∈ [0, +∞) × [0, +∞). (1) If σ > 0 and δ > 0, then we have   T  g(θ) θ2 WT ˜ LT = , VT = dJ, σ δ g(θ) 0 R0

R0

 T KT = 0

R0

2θ3 dJ. δ

218

A. Takeuchi Hence, the sensitivity formula in this case is as follows: #

$ LT − VT KT ∂x (E [ϕ (XT )]) = E ϕ(XT ) + 2 . AT AT

(2) If σ > 0, then we have

# $ WT ∂x (E [ϕ (XT )]) = E ϕ(XT ) . σT

(3) If δ > 0, then we have





∂x (E [ϕ (XT )]) = E ϕ(XT )

VT KT − + 2 A˜T A˜T

 .



Example 3 (geometric L´evy processes). Let (x, γ, σ, δ) ∈ (0, +∞) × R × [0, +∞) × [0, +∞). Define the R-valued L´evy process {xt ; t ∈ [0, T ]} by  t xt = γ t + σ Wt + δ θ dJ. 0

R0

The geometric L´evy process X = {Xt ; t ∈ [0, T ]} is defined by Xt = x exp(xt ). From the Itˆo formula, we can easily see that . :    dXt = γ + exp(δθ) − 1 − δθ dν Xt dt |θ|≤1

+ σ Xt ◦ dWt +



 R0

 exp(δθ) − 1 Xt− dJ.

Thus, the coefficients of the above equation doesn’t satisfy the condition (2.5) any more. However, since   Xt 1  ∂x (ϕ(Xt )) = ϕ (Xt ) = ∂X ((ϕ ◦ h)(X + xt ))  , x x X=log x

where h(y) = exp(y), we can compute the sensitivity formula for bounded functions ϕ ∈ F by using the result stated in Example 2. 

References [1] D. Applebaum: L´evy processes and stochastic calculus, 2nd ed. Cambridge Univ. Press (2009). [2] V. Bally, M.-P. Bavouzet, M. Messaoud: Integration by parts formula for locally smooth laws and applications to sensitivity computations, Ann. Appl. Probab. 17, 33–66 (2007). [3] J.M. Bismut: Martingales, the Malliavin calculus and hypoellipticity under general H¨ ormander’s conditions, Z. Wahrsch. Verw. Gebiete 56, 469–505 (1981).

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[4] J.M. Bismut: Calcul des variations stochastique et processus de sauts, Z. Wahrsch. Verw. Gebiete 63, 147–235 (1983). [5] J.M. Bismut: Large deviations and the Malliavin calculus, Birkh¨ auser (1984). [6] P. Carr, H. Geman, D.B. Madan, M. Yor: The fine structure of asset returns: an empirical investigation, Journal of Business 75, 305–332 (2002). [7] T.R. Cass, P.K. Friz: The Bismut-Elworthy-Li formula for jump-diffusions and applications to Monte Carlo methods in finance, available at arXiv:math/0604311v3 (2007). [8] M.H. A. Davis, M.P. Johansson: Malliavin Monte Carlo Greeks for jump diffusions, Stoch. Processes Appl. 116, 101–129 (2006). [9] Y. El-Khatib, N. Privault: Computations of Greeks in a market with jumps via the Malliavin calculus, Finance Stoch. 8, 161–179 (2004). [10] R. Elliott, A.H. Tsoi: Integration by parts for Poisson processes, J. Multivariate Anal. 44, 179–190 (1993). [11] K.D. Elworthy, X.M. Li: Formulae for the derivatives of heat semigroups, J. Funct. Anal. 125, 252–286 (1994). [12] E. Fourni´e, J.M. Lasry, J. Lebuchoux, P.L. Lions, N. Touzi: Applications of Malliavin calculus to Monte Carlo methods in finance, Finance Stoch. 3, 391–412 (1999). [13] T. Fujiwara, H. Kunita: Stochastic differential equations of jump type and L´evy processes in diffeomorphisms group, J. Math. Kyoto Univ. 25, 71–106 (1985). [14] I.I. Gihman, A.V. Skorokhod: Stochastic differential equations, Springer-Verlag (1972). [15] R. Kawai, A. Takeuchi: Greeks formulae for an asset price model with gamma processes, to appear in Math. Finance (2011). [16] T. Komatsu, A. Takeuchi: Generalized H¨ ormander theorem for non-local operators, In Recent Developments in Stochastic Analysis and Related Topics; S. Albeverio et al. Eds.; World Scientific, Singapore (2004), 234–245. [17] D. Nualart: The Malliavin calculus and related topics, 2nd edition, Springer-Verlag (2006). [18] A. Takeuchi: The Malliavin calculus for SDE with jumps and the partially hypoelliptic problem, Osaka J. Math. 39, 523–559 (2002). [19] A. Takeuchi: The Bismut-Elworthy-Li type formulae for stochastic differential equations with jumps, J. Theoret. Probab. 23, 576–604 (2010). Atsushi Takeuchi Department of Mathematics Osaka City University Sugimoto 3-3-138 Osaka, 558-8585, Japan e-mail: [email protected]

Progress in Probability, Vol. 65, 221–252 c 2011 Springer Basel AG


Quantifying Model Uncertainties in Complex Systems Jiarui Yang and Jinqiao Duan Abstract. Uncertainties are abundant in complex systems. Appropriate mathematical models for these systems thus contain random effects or noises. The models are often in the form of stochastic differential equations, with some parameters to be determined by observations. The stochastic differential equations may be driven by Brownian motion, fractional Brownian motion, or L´evy motion. After a brief overview of recent advances in estimating parameters in stochastic differential equations, various numerical algorithms for computing parameters are implemented. The numerical simulation results are shown to be consistent with theoretical analysis. Moreover, for fractional Brownian motion and α-stable L´evy motion, several algorithms are reviewed and implemented to numerically estimate the Hurst parameter H and characteristic exponent α. Mathematics Subject Classification (2000). Primary: 37L55, 35R60; Secondary: 58B99, 35L20. Keywords. Model uncertainty, parameter estimation, Brownian motion (BM), fractional Brownian motion (fBM), L´evy motion (LM), Hurst parameter, characteristic exponent, stochastic differential equations (SDEs).

1. Introduction Since random fluctuations are common in the real world, mathematical models for complex systems are often subject to uncertainties, such as fluctuating forces, uncertain parameters, or random boundary conditions [89, 55, 44, 121, 122, 125]. Uncertainties may also be caused by the lack of knowledge of some physical, chemical or biological mechanisms that are not well understood, and thus are This work was partially supported by NSF grants 0620539 and 0731201, and by an open research grant of the State Key Laboratory for Nonlinear Mechanics, China.

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not appropriately represented (or missed completely) in the mathematical models [19, 65, 97, 123, 124]. Although these fluctuations and unrepresented mechanisms may be very small or very fast, their long-term impact on the system evolution may be delicate [7, 55, 44] or even profound. This kind of delicate impact on the overall evolution of dynamical systems has been observed in, for example, stochastic bifurcation [25, 17, 55], stochastic resonance [59], and noise-induced pattern formation [44, 14]. Thus taking stochastic effects into account is of central importance for mathematical modeling of complex systems under uncertainty. Stochastic differential equations (SDEs) are appropriate models for many of these systems [7, 27, 108, 122]. For example, the Langevin type models are stochastic differential equations describing various phenomena in physics, biology, and other fields. SDEs are used to model various price processes, exchange rates, and interest rates, among others, in finance. Noises in these SDEs may be modeled as a generalized time derivative of some distinguished stochastic processes, such as Brownian motion (BM), L´evy motion (LM) or fractional Brownian motion (fBM) [36]. Usually we choose different noise processes according to the statistical property of the observational data. For example, if the data has long-range dependence, we consider fractional Brownian motion rather than Brownian motion. If the data has considerable discrepancy with Gaussianity or normality, L´evy motion may be an appropriate choice. In building these SDE models, some parameters appear, as we do not know certain quantities exactly. Based on the choice of noise processes, different mathematical techniques are needed in estimating the parameters in SDEs with Brownian motion, L´evy motion, or fractional Brownain motion. In this article, we are interested in estimating and computing parameters contained in stochastic differential equations, so that we obtain computational models useful for investigating complex dynamics under uncertainty. We first review recent theoretical results in estimating parameters in SDEs, including statistical properties and convergence of various estimates. Then we develop and implement numerical algorithms in approximating these parameters. Theoretical results on parameter estimations for SDEs driven by Brownian motion are relatively well developed ([5, 28, 48, 99]), and various numerical simulations for these parameter estimates ([1, 3, 99, 62]) are implemented. So, in Section 2 below, we do not present such numerical results. Instead, we will concentrate on numerical algorithms for parameter estimations in SDEs driven by fractional Brownian motion and L´evy motion in Sections 3 and 4, respectively. This paper is organized as follows. In Section 2, we consider parameter estimation for SDEs with Brownian motion Bt . We present a brief overview of some available techniques on estimating parameters in these stochastic differential equations with continuous-time or discrete-time observations. In fact, we present results about how to estimate parameters in diffusion terms and drift terms, given continuous observations and discrete observations, respectively.

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In Section 3, we consider parameter estimation for SDEs driven by fractional Brownian motion BtH with Hurst parameter H. After discussing basic properties of fBM, we consider parameter estimation methods for Hurst parameter H from given fBM data. Then, we compare the convergence rate of each method by comparing estimates computed with hypothetic data. Unlike the case of SDEs with Brownian motion, there is no general formula for the estimate of the parameter in the drift (or diffusion) coefficient of a stochastic differential equation driven by fBM. We discuss different estimates associated with different models and discuss the statistical properties respectively. We develop and implement numerical simulation methods for these estimates. Finally, in Section 4, for stochastic differential equation with (non-Gaussian) α-stable L´evy motion Lα t , we consider estimates and their numerical implementation for parameter α and other parameters in the drift or diffusion coefficients.

2. Quantifying uncertainties in SDEs driven by Brownian motion In this section, we consider a scalar diffusion process Xt ∈ Rd , 0 ≤ t ≤ T satisfying the following stochastic differential equation dXt = µ(θ, t, Xt )dt + σ(ϑ, t, Xt )dBt , X0 = ζ

(1)

where Bt is an m-dimensional Brownian motion, θ ∈ Θ a compact subset of Rp and ϑ ∈ Ξ a compact subset of Rq are unknown parameters which are to be estimated on the basis of observations. Here µ : Θ×[0, T ]×Rd → Rd , the drift coefficient, and σ : Ξ × [0, T ] × Rd → Rd×m , the diffusion coefficient, are usually known functions but with unknown parameters θ and ϑ. Some remarks are in order here. • Under local Lipschitz and the sub-linear growth conditions on the coefficients µ and σ, there exists a unique strong solution of the above stochastic differential equation (see [77] or [85]) and this is a universal assumption for all results we discuss below. • The diffusion coefficient σ is almost surely determined by the process, i.e., it can be estimated without any error if observed continuously throughout a time interval (see [47] and [30]). • The diffusion matrix defined by Σ(ϑ, t, Xt ) ≡ σ(ϑ, t, Xt )σ(ϑ, t, Xt )T plays an important role on parameter estimation problems. 2.1. How to estimate parameters given continuous observation Since it is not easy to estimate parameters θ and ϑ at the same time, usually we simplify our model by assuming one of those parameters is known and then estimate the other. Moreover, instead of representing the results of all types of diffusion processes, we choose to present the conclusion of the most general one, such as, we prefer the nonhomogeneous case rather than the homogeneous one or the nonlinear one rather than the linear one.

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2.1.1. Parameter estimation of diffusion terms with continuous observation. We assume that the unknown parameter θ in the drift coefficient is known. Then our model can be simplified as dXt = µ(t, X)dt + σ(ϑ, t, Xt )dBt , X0 = ζ

(2)

Remarks • Different with the model (1), the drift coefficient µ(t, X) in model (2) is possibly unknown and maybe related to the whole past of process X instead of Xt . In this case, our model can be easily extended to the non-Markovian case which is more general than case (1). • If µ is depending on the unknown parameter ϑ in model (2), we can also prove the local asymptotic mixed normality property for the maximum likelihood estimate (MLE) when µ(t, X) = µ(ϑ, Xt ) and σ(ϑ, t, Xt ) = σ(ϑ, Xt ) (see [38]). If the diffusion matrix Σ(ϑ, t, Xt ) is invertible, then define a family of contrasts by n 4 1 3 U n (ϑ) = log det Σ(ϑ, tni−1 , Xtni−1 ) + (Xin)T Σ(ϑ, tni−1 , Xtni−1 )−1 Xin , (3) n i=1 where

1 (Xtni − Xtni−1 ), δin = tni − tni−1 , for 1 ≤ i ≤ n, δin and tni is an appropriate partition of [0, T ]. However, this assumption does not always hold. So, we consider a more general class of contrasts of the form Xin =

1 f (Σ(ϑ, tni−1 , Xtni−1 ), Xin ), n i=1 n

U n (ϑ) =

(4)

where f should satisfy certain conditions to obtain the asymptotic property and consistency property for the estimate generated by these contrasts below (see [45]). Let ϑ?n be a minimum contrast estimate associated with U n , i.e., ϑ?n satisfies the following equation U n (ϑ?n ) = min U n (ϑ). ϑ∈Ξ

Under some smoothness assumptions on the coefficient µ and θ, empirical sampling measure assumption on the sample times tni , and identifiability assumption on the law of the solution of (2), Genon-Catalot and Jacod [47] have proved that the √ ? estimate ϑ?n has a local asymptotic mixed normality property, i.e., n(ϑ n − ϑ0 ) where ϑ0 is the true value of the parameter converges in law to N(0, S). Remarks • We do not include the drift coefficient µ in the contrast U n (ϑ) because it is possibly unknown. Even if it is known, we still do not want it involved since it is a function of the whole past of X and thus is not observable.

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• If the diffusion matrix Σ is invertible, it can be proven that the contrast of form (3) is optimal in the class of contrasts of type (4). 2.1.2. Parameter estimation of drift terms with continuous observations. We assume that the unknown parameter ϑ in the diffusion coefficient is known. Then the model (1) can be simplified as dXt = µ(θ, t, Xt )dt + σ(t, Xt )dBt , X0 = ζ.

(5)

Since no good result for above general model exists, we introduce the result for the following nonhomogeneous diffusion process instead. Consider a real-valued diffusion process {Xt , t ≥ 0} satisfying the following stochastic differential equation: dXt = µ(θ, t, Xt )dt + dBt , X0 = ζ,

(6)

where the drift coefficient function µ is assumed to be nonanticipative. Denote the observation of the process by X0T := {Xt , 0 ≤ t ≤ T } and let PθT be the measure generated by the process X0T . Then the Radon-Nicodym derivative (likelihood function) of PθT with respect to PθT0 where θ0 is the true value of the parameter θ is given by (see [80]) LT (θ) := (dPθT /dPθT0 )(X0T ) % T &  1 T 2 = exp [µ(θ, t, Xt ) − µ(θ0 , t, Xt)]dXt − [µ (θ, t, Xt ) − µ2 (θ0 , t, Xt )]dt . 2 0 0 So we can get the Maximal Likelihood Estimate (MLE) defined by θ?T := argsupθ∈Θ LT (θ). Then we can show that the MLE is strongly consistent, i.e., θ?T → θ0 Pθ0 − a.s. as T → ∞, and converge to a normal distribution (see Chapter 4 in [13] for more details). Remarks • In [13], Bishwal also proves that the MLE and a regular class of Bayes estimates (BE) are asymptotically equivalent. • By applying an increasing transformation as described in [1],  X du Yt = g(X) ≡ , (7) σ(u) we can transform the diffusion process Xt defined by dXt = µ(θ, Xt )dt + σ(Xt )dBt into another diffusion process Yt defined by dY˜t = µ ˜(θ, Y˜t )dt + dBt , where µ ˜ (θ, y) ≡

µ(g −1 (y), θ) 1 ∂σ(g −1 (y)) − . σ(g −1 (y)) 2 ∂y

(8)

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J. Yang and J. Duan Then, we can get the MLE of process Xt by calculating the MLE of process Yt according to what we learned in this section (see [1] or [2] for more details).

2.2. How to estimate parameters given discrete observation Given the practical difficulty in obtaining a complete continuous observation, we now discuss parameter estimations with discrete observation. 2.2.1. Parameter estimation of drift terms with discrete time. In this section, we assume that the unknown parameter ϑ in the diffusion coefficient σ is known. Then the model (1) can be simplified as dXt = µ(θ, t, Xt )dt + σ(t, Xt )dBt , X0 = ζ.

(9)

Ideally, when the transition densities p(s, x, t, y; θ) of X are known, we can use the log likelihood function ln (θ) =

n 

log p(ti−1 , Xti−1 , ti , Xti ; θ),

i=1

to compute the LME θ? which is strongly consistent and asymptotically normally distributed. (see [12] and [26], [79] and [109]). If the transition densities of X are unknown, instead of computing the log likelihood function ln (θ), we would like to use approximate log-likelihood function which, under some regularity conditions (see [56]), is given by  T  µ(θ, t, Xt ) 1 T µ2 (θ, t, Xt ) lT (θ) = dX − dt t σ 2 (t, Xt ) 2 0 σ 2 (t, Xt ) 0 to approximate the log-likelihood function based on continuous observations (see [103]). Then, using an Itˆ o type approximation for the stochastic integral we can obtain ˜ln (θ) =

n  µ(θ, ti−1 , Xti−1 ) i=1

σ 2 (ti−1 , Xti−1 )

(Xti − Xti−1 )

1  µ2 (θ, ti−1 , Xti−1 ) (ti − ti−1 ). 2 i=1 σ 2 (ti−1 , Xti−1 ) n

−

Thus, the maximizer of ˜ln (θ) provides an approximate maximum likelihood estimate (AMLE). In 1992, Yoshida [130] proved that the AMLE is weakly consistent and asymptotically normally distributed when the diffusion is homogeneous and ergodic. In [13], Bishwal got the similar result for the nonhomogeneous case with drift function µ(θ, t, X) = θf (t, Xt ) for some smooth functions f(t,x). Moreover, he measured the loss of information using several AMLEs according to different approximations to lT (θ).

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2.2.2. Parameter estimation of diffusion terms (and/or drift terms) with discrete observation. In previous sections, we always assume one of those parameters is known and then estimate the other one. In this section, I want to include the situation when both θ and ϑ are unknown and how to estimate them based on the discrete observation of the diffusion process at the same time. Suppose we are considering the real-valued diffusion process Xt satisfying the following stochastic differential equation dXt = µ(θ, Xt )dt + σ(ϑ, Xt )dBt .

(10)

Denote the observation times by t0 = 0, t1 , t2 , . . . , tNT , where NT is the smallest integer such that τNT +1 > T . In this section, we mainly consider three cases of estimating β = (θ, ϑ), jointly, β = θ with ϑ known and β = ϑ with θ known. ˆ ¯ In √ regular circumstances, the estimate β converges in probability to some β and ¯ converges in law to N (0, Ωβ ) as T tends to infinity, where β0 is the T (βˆ − β) true value of the parameter. For simplicity, we set the law of the sampling intervals ∆n = τn − τn−1 as ∆ = ∆0 ,

(11)

where ∆0 has a given finite distribution and  is deterministic. Remark. We are not only studying the case when the sampling interval is fixed, i.e., V ar[∆0 ] = 0, but also the continuous observation case, i.e.,  = 0 and the random sampling case. Let h(y1 , y0 , δ, β, ) denote an r-dimensional vector function which consists of r moment conditions of the discretized stochastic differential equation (10) (please see [51] or [54] for more details). Moreover, this function also satisfies E∆n ,Yn ,Yn−1 [h(Yn , Yn−1 , ∆n , β, )] = 0, where the expectation is taken with respect to the joint law of (∆n , Yn , Yn−1 ). By the Law of Large Numbers, E[h(Yn , Yn−1 , ∆n , β, )] may be estimated by the sample average defined by mT (β) ≡ NT−1

N T −1

h(Yn , Yn−1 , ∆n .β, ).

(12)

n=1

Then we can obtain an estimate βˆ by minimizing a quadratic function QT (β) ≡ mT (β) WT mT (β),

(13)

where WT is a r×r positive definite weight matrix and this method is called Generalized Method of Moments (GMM). In [51], Hansen proved the strong consistency and asymptotic normality of GMM estimate, i.e., √ N (θˆ − θ) → N (0, V ), when θ = ϑ and WT satisfied certain conditions. Mykland used this technique to obtain the closed form for the asymptotic bias but sacrificed the consistency of the estimate.

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3. Quantifying uncertainties in SDEs driven by fractional Brownian motion Colored noise, or noise with non-zero correlation in time, are common in physical, biological and engineering sciences. One candidate for modeling colored noise is fractional Brownian motion [36]. 3.1. Fractional Brownian motion Fractional Brownian motion (fBM) was introduced within a Hilbert space framework by Kolmogorov in 1940 in [73], where it was called Wiener Helix. It was further studied by Yaglom in [127]. The name fractional Brownian motion is due to Mandelbrot and Van Ness, who in 1968 provided in [84] a stochastic integral representation of this process in terms of a standard Brownian motion. Definition 3.1 (Fractional Brownian motion [96]). Let H be a constant belonging to (0, 1). A fractional Brownian motion (fBM) (B H (t))t≥0 of Hurst index H is a continuous and centered Gaussian process with covariance function E[B H (t)B H (s)] =

1 2H (t + s2H − |t − s|2H ). 2

By the above definition, we see that a standard fBM B H has the following properties: 1. B H (0) = 0 and E[B H (t)] = 0 for all t ≥ 0. 2. B H has homogeneous increments, i.e., B H (t + s) − B H (s) has the same law of B H (t) for s, t ≥ 0. 3. B H is a Gaussian process and E[B H (t)2 ] = t2H , t ≤ 0, for all H ∈ (0, 1). 4. B H has continuous trajectories. Using the method presented in [23, 24], we can simulate sample paths of fractional Brwonian motion with different Hurst parameters (see Figure 1). 2

2

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Figure 1. Three sample paths of fBM with Hurst parameter H = 0.25, 0.5, 0.9 For H = 1/2, the fBM is then a standard Brownian motion. Hence, in this case the increments of the process are independent. On the contrary, for H = 1/2 the increments are not independent. More precisely, by the definition of fBM, we

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know that the covariance between B H (t + h) − B H (t) and B H (s + h) − B H (s) with s + h ≤ t and t − s = nh is ' ( 1 ρH (n) = h2H (n + 1)2H + (n − 1)2H − 2n2H . 2 In particular, we obtain that the two increments of the form B H (t + h) − B H (t) and B H (t + 2h) − B H (t + h) are positively correlated for H > 1/2, while they are negatively correlated for H < 1/2. In the first case the process presents an aggregation behavior and this property can be used in order to describe “cluster” phenomena (systems with memory and persistence). In the second case it can be used to model sequences with intermittency and antipersistence. From the above description, we can get a general ideal that the Hurst parameter H plays an important role on how respective fBM behaves. So, it should be considered as an extra parameter when we estimate others in the coefficients of the SDE driven by fBM. Considering the further computation, we would like to introduce one more useful property of fBM. Definition 3.2 (Self-similarity). A stochastic process X = {Xt , t ∈ R} is called b-self-similar or satisfies the property of self-similarity if for every a > 0 there exists b > 0 such that Law(Xat , t ≥ 0) = Law(ab Xt , t ≥ 0). Note that “Law = Law” means that the two processes Xat and ab Xt have the same finite-dimensional distribution functions, i.e., for every choice t0 , . . . , tn in R, P (Xat0 ≤ x0 , . . . , Xatn ≤ xn ) = P (ab Xt0 ≤ x0 , . . . , ab Xtn ≤ xn ). for every x0 , . . . , xn in R. Since the covariance function of the fBM is homogeneous of order 2H, we obtain that B H is a self-similar process with Hurst index H, i.e., for any constant a > 0, the processes B H (at) and aH B H (t) have the same distribution law. 3.2. How to estimate Hurst parameter H Let’s start with the simplest case: dXt = dB H (t),

i.e., Xt = B H (t) t ≥ 0,

where {B H (t), t ≥ 0} is a fBM with Hurst parameter H ∈ (0, 1). Now, our question is how to estimate Hurst parameter H given observation data X0 , X1 , . . . , XN . For a parameter estimation of Hurst parameter H, we need an extra ingredient, fractional Gaussian noise (fGn). Definition 3.3 (Fractional Gaussian noise, [110]). Fractional Gaussian noise (fGn) {Yi , i ≥ 1} is the increment of fractional Brownian motion, namely Yi = B H (i + 1) − B H (i), i ≥ 1.

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Remark. It is a mean zero, stationary Gaussian time series whose autocovariance function is given by 1 ρ(h) = E(Yi Yi+h ) = {(h + 1)2H − 2h2H + |h − 1|2H }, h ≥ 0. 2 An important point about ρ(h) is ρ(h) ∼ H(2H − 1)h2H−2 , as h → ∞, when H = 1/2. Since ρ(h) = 0 for h ≥ 1 when H = 1/2, the Xi ’s are white noise in this case. The Xi ’s, however, are positively correlated when 12 < H < 1, and we say that they display long-range dependence (LRD) or long memory. From the expression of fGn, we know it is the same to estimate the Hurst parameter of fBM as to estimate the Hurst parameter of the respective fGn. Here, we introduce 4 different methods for measuring the Hurst parameter. Measurements are given on artificial data and the results of each method are compared in the end. However, the measurement techniques used in this paper can only be described briefly but references to fuller descriptions with mathematical details are given. 3.2.1. R/S method. The R/S method is one of the oldest and best known techniques for estimating H. It is discussed in detail in [83] and [10], p. 83–87. n For a time series {Yt : t = 1, 2, . . . , N } with partial sums given by Z(n) = i=1 Yi and the sample variance given by 1  2 1 Yi − Z(n)2 , n−1 n(n − 1) n

S 2 (n) =

i=1

then the R/S statistic, or the rescaled adjusted range, is given by: #



$ 1 R t t (n) = max Z(t) − Z(n) − min Z(t) − Z(n) 1≤t≤n S S(n) 1≤t≤n n n For fractional Gaussian noise, E[R/S(n)] ∼ CH nH , as n → ∞, where CH is another positive, finite constant not dependent on n. The procedure to estimate H is therefore as follows. For a time series of length N , subdivide the series into K blocks with each of size n = N/K. Then, for each lag n, compute R/S(ki , n), starting at points ki = iN/K + 1, i = 1, 2, . . . , K − 1. In this way, a number of estimates of R/S(n) are obtained for each value of n. For values of n approaching N , one gets fewer values, as few as 1 when n ≥ N − N/K. Choosing logarithmically spaced values of n, plot log[R/S(ki , n)] versus log n and get, for each n, several points on the plot. This plot is sometimes called the pox plot for the R/S statistic. The parameter H can be estimated by fitting a line to the points in the pox plot. There are several disadvantages with this technique. Most notably, there are more estimates of the statistic for low values of n where the statistic is affected most

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heavily by short range correlation behavior. On the other hand, for high values of n there are too few points for a reliable estimate. The values between these high and low cut off points should be used to estimate H but, in practice, often it is the case that widely differing values of H can be found by this method depending on the high and low cut off points chosen. To modify the R/S statistic, we can use a weighted sum of autocovariance instead of the sample variance. Details can be found in [82]. 3.2.2. Aggregated variance. Given a time series {Yt : t = 1, 2, . . . , N }, divide this into blocks of length m and aggregate the series over each block. Y (m) (k) :=

1 m

km 

Yi ,

k = 1, 2, . . . , [N/m].

i=(k−1)m+1

We compute its sample variance,
(m) = V arY

N/m 1  (m) (Y (k) − Y )2 . N/m k=1

where

N Y =

i=1

Yi

. N is the sample mean. The sample variance should be asymptotically proportional to m2H−2 for large N/m and m. Then, for successive values of m, the sample variance of the aggregated series is plotted versus m on a log-log plot. So we can get the estimate of H by computing the gradient of that log-log plot. However, jumps in the mean and slowly decaying trends can severely affect this statistic. One technique to combat this is to difference the aggregated variance and work instead with

(m+1) − V arY (m) . V arY 3.2.3. Variance of residuals. This method is described in more detail in [101]. Take the series {Yt : t = 1, 2, . . . , N } and divide it into blocks of length m. Within each (k−1)m+t block calculate partial sums: Zk (t) = i=(k−1)m+1 Yi , k = 1 . . . N/m, t = 1 . . . m. For each block make a least squares fit to a line ak + bk t. Subtract this line from the samples in the block to obtain the residuals and then calculate their variance 1  (Zk (t) − ak − bk t)2 . m t=1 m

Vk =

The variance of residuals is proportional to m2H . For the proof in the Gaussian case, see [118]. This variance of residuals is computed for each block, and the median (or average) is computed over the blocks. A log-log plot versus m should follow a straight line with a slope of 2H.

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3.2.4. Periodogram. The periodogram is a frequency domain technique described in [49]. For a time series {Yt : t = 1, 2, . . . , N }, it is defined by N 2  1  ijλ  I(λ) = Yj e  ,  2πN j=1

where λ is the frequency. In the finite variance case, I(λ) is an estimate of the spectral density of Y , and a series with long-range dependence will have a spectral density proportional to |λ|1−2H for frequencies close to the origin. Therefore, the log-log plot of the periodogram versus the frequency displays a straight line with a slope of 1 − 2H. 3.2.5. Results on simulated data. In this subsection, we would like to use artificial data to check the robustness of above techniques and compare the result in the end. For each of the simulation methods chosen, traces have been generated. Each trace is 10,000 points of data. Hurst parameters of 0.65 and 0.95 have been chosen to represent a low and a high level of long-range dependence in data. From Figure 2 and Figure 3, we can see that the Variance of Residual Method and R/S have the most accurate result. The Modified Aggregated Variance Method improved a little bit over the original one, but both of them still fluctuate too much. 3.3. How to estimate parameters in SDEs driven by fBM After we discuss how to estimate the Hurst parameter of a series of artificial fBM data, now we want to concern how to estimate the parameters of the linear/nonlinear stochastic differential equation(s) driven by fBM. The coefficients in the stochastic differential equation could be deterministic or random, linear or nonlinear. No general results are available. So some specific statistical results will be discussed below according to what kind of specified models we deal with. 3.3.1. Preparation. The main difficulty in dealing with a fBm is that it is not a semimartingale when H = 12 and hence the results from the classical stochastic integration theory for semimartingales can not be applied. So, we would like to introduce the following integral transformation which can transform fBM to martingale firstly and it will be a key point in our development below. For 0 < s < t ≤ T , denote (1/2)−H kH (t, s) = κ−1 (t − s)(1/2)−H , H s

(14)

κH = 2HΓ(3/2 − H)Γ(H + 1/2),

(15)

2−2H wtH = λ−1 ; H t

 MtH =

λH =

(16)

t

kH (t, s)dBsH . 0

2HΓ(3 − 2H)Γ(H + 1/2) , Γ(3/2 − H)

(17)

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Figure 2. Numerical estimation of the Hurst parameter H of fBM: Actual value H = 0.65 Then the process M H is a Gaussian martingale (see [78] and [92]), called the fundamental martingale with variance function wH . 3.3.2. Parameter estimation for a fractional Langevin equation. Suppose {Xt, t ≥ 0} satisfies the following stochastic differential equation  t Xt = θ Xs ds + σBtH ; 0 ≤ t ≤ T, 0 2

where θ and σ are unknown constant parameters, BtH is a fBM with Hurst parameter H ∈ [1/2, 1]. Denote the process Z=(Zt , t ∈ [0, T ]) by  t Zt = kH (t, s)dXs .

(18)

0

Then we can prove that process Z is a semimartingale associated to X with following decomposition (see [69])  t Zt = θ Q(s)dwsH + σMtH , (19) 0

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Figure 3. Numerical estimation of the Hurst parameter H of fBM: Actual value H = 0.95 where Q(t) =

d dwtH



t

kH (t, s)X(s)ds,

(20)

0

and MtH is the Gaussian martingale defined by (17). From the representation (19), we know the quadratic variation of Z on the interval [0, t] is nothing but Zt = σ 2 wtH ,

a.s.

Hence the parameter σ 2 can be obtained by 42 3 Ztni+1 − Ztni = σ 2 , [wtH ]−1 lim n

a.s.

i

where tni is an appropriate partition of [0,t] such that supi |tni+1 − tni | → 0 as n → ∞. So, the variance parameter can be computed with probability 1 on any finite time interval. As for the parameter θ, by applying the Girsanov type formula for fBM which is proved in [69], we can define the following maximum likelihood estimate of θ

Quantifying Uncertainties in Complex Systems based on the observation on the interval [0, t] by . :−1  T 2 H θT = Q (s)dws 0

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Q(s)dZs ,

(21)

0

where processes Q, Z and wtH are defined by (20), (18) and (16), respectively. For this estimate, strong consistency is proven and explicit formulas for the asymptotic bias and mean square error are derived by Kleptsyna and Le Breton [70]. Remarks 1/2 • When H = 1/2, since Q = Z = X and dωs = ds, the formula (21) reduces to the result of [80] for a usual Ornstein-Uhlenbeck process. • For an arbitrary H ∈ [1/2, 1], we could derive the following alternative expression for θT : .  :−1 . :  T T λ H θT = 2 Q2 (s)dwsH ZT s2H−1 dZs − t . 2 − 2H 0 0 Example 3.4. Consider a special Ornstein-Uhlenbeck model dXt = θXt dt + 2dBtH . Then, according to the above approximation scheme, we can numerically estimate θ = 1 and the results are shown in Figure 4. 600 Num Estimator Actual value

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Figure 4. Numerical estimation of drift parameter θ in dXt = θXt dt + 2dBtH with Hurst parameter H = 0.75: Actual value θ0 = 1

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3.3.3. Parameter estimation in linear deterministic regression. Suppose Xt satisfies the following stochastic differential equation  t  t Xt = θ A(s)ds + C(s)dBsH , 0 ≤ t ≤ T, 0

0

where A and C are deterministic measurable functions on [0,T], BtH is a fBM with Hurst parameter H ∈ [1/2, 1]. Let qt be defined by  t d A qt = kH (t, s) (s)ds, C dwtH 0 where wtH and kH (t, s) are defined by (16) and (14). Then, from Theorem 3 in [69], we obtain the maximum likelihood estimate of θ defined by . :−1  T T 2 H θT = qt dwt qt dZt , 0

0

where Zt is defined by (18). Remark. This result can be extended to an arbitrary H in (0, 1) (see [78]) and θT is also the best linear unbiased estimate of θ. Example 3.5. Consider a special Linear Deterministic Regression dXt = −θdt + tdBtH . Then, using the above estimate, we can do numerical simulation with result shown in Figure 5. 3.3.4. Parameter estimation in linear random regression. Let us consider a stochastic differential equation dX(t) = [A(t, X(t)) + θC(t, X(t))]dt + σ(t)dBtH , t ≥ 0, where B = {BtH , t ≥ 0} is a fractional Brownian motion with Hurst parameter H and σ(t) is a positive nonvanishing function on [0, ∞). According to [105], the Maximum Likelihood Estimate θˆT of θ is given by
T
T J2 (t)dZt + 0 J1 (t)J2 (t)dwtH 0 θT = ,
T 2 H 0 J2 (t)dwt where the processes Zt , J1 , J2 are defined by  t kH (t, s) Zt = dXs , t ≥ 0, σ(s) 0  t d A(s, X(s)) J1 (t) = kH (t, s) ds, σ(s) dwtH 0  t d C(s, X(s)) J2 (t) = kH (t, s) ds, σ(s) dwtH 0

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1.2

1 Num Estimator Actual value 0.8



0.6

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Figure 5. Numerical estimation of drift parameter θ in a Linear Deterministic Regression dXt = −θdt + tdBtH with Hurst parameter H = 0.75: Actual value θ=1 and wtH , kH (t, s) are defined by (16) and (14). Also in the same paper, they proved that θT is strongly consistent for the true value θ. Example 3.6. Consider a special Linear Random Regression dXt = (t + θXt )dt + tdBtH . A numerical estimation of the parameter θ is shown in Figure 6.

4. Parameter estimation for SDE driven by α-stable L´evy motion Brownian motion, as a Gaussian process, has been widely used to model fluctuations in engineering and science. For a particle in Brownian motion, its sample paths are continuous in time almost surely (i.e., no jumps), its mean square displacement increases linearly in time (i.e., normal diffusion), and its probability density function decays exponentially in space (i.e., light tail or exponential relaxation) [95]. However some complex phenomena involve non-Gaussian fluctuations, with properties such as anomalous diffusion (mean square displacement is a nonlinear power law of time) [15] and heavy tail (non-exponential relaxation) [129]. For instance, it has been argued that diffusion in a case of geophysical turbulence [114] is anomalous. Loosely speaking, the diffusion process consists of a series of “pauses”, when the particle is trapped by a coherent structure, and “flights” or “jumps” or other extreme events, when the particle moves in a jet flow. Moreover,

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20 Num. Estimator Actual value 15



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Figure 6. Numerical estimation of drift parameter θ in a Linear Random Regression dXt = (t + θXt )dt + tdBtH with Hurst parameter H = 0.75: Actual value θ=1

anomalous electrical transport properties have been observed in some amorphous materials such as insulators, semiconductors and polymers, where transient current is asymptotically a power law function of time [112, 53]. Finally, some paleoclimatic data [29] indicates heavy tail distributions and some DNA data [114] shows long range power law decay for spatial correlation. L´evy motions are thought to be appropriate models for non-Gaussian processes with jumps [111]. Here we consider a special non-Gaussian process, the α-stable L´evy motion, which arise in many complex systems [126]. 4.1. α-stable L´evy motion There are several reasons for using a stable distribution to model a fluctuation process in a dynamical system. Firstly, there are theoretical reasons for expecting a non-Gaussian stable model, e.g., hitting times for a Brownian motion yielding a L´evy distribution, and reflection off a rotating mirror yielding a Cauchy distribution. The second reason is the Generalized Central Limit Theorem which states that the only possible non-trivial limit of normalized sums of i.i.d. terms is stable. The third argument for modeling with stable distributions is empirical: Many large data sets exhibit heavy tails and skewness. In this section, we consider one-dimensional α-stable distributions defined as follows.

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Definition 4.1 ([64], Chapter 2.4). The Characteristic Function ϕ(u) of an α-stable random variable is given by ϕ(u) = exp((−σ α )|u|α {1 − iβ sgn(u) tan(απ/2)} + iµu) where α ∈ (0, 1) ∪ (1, 2), β ∈ [−1, 1], σ ∈ R+ , µ ∈ R, and by % &

2 ϕ(u) = exp −σ|u| 1 + iβ sgn(u) log(|u|) + iµu π

(22)

(23)

when α = 1, it gives a very well-known symmetric Cauchy distribution and

1 2 ϕ(u) = exp − σ|u| + iµu , (24) 2 when α = 2, it gives the well-known Gaussian distribution. For the random variable X distributed according to the rule described above we use the notation X ∼ Sα (σ, β, µ). Especially when µ = β = 0, i.e., X is a symmetric α-stable random variable, we will denote it as X ∼ SαS. Also, from above definition, it is easy to see that the full stable class is characterized by four parameters, usually designated α, β, σ, and µ. The shift parameter µ simply shifts the distribution to the left or right. The scale parameter σ compresses or extends the distribution about µ in proportion to σ which means, if the variable x has the stable distribution X ∼ Sα (σ, β, µ), the transformed variable z = (x − µ)/σ will have the same shaped distribution, but with location parameter 0 and scale parameter 1. The two remaining parameters completely determine the distribution’s shape. The characteristic exponent α lies in the range (0, 2] and determines the rate at which the tails of the distribution taper off. When α = 2, a normal distribution results. When α < 2, the variance is infinite. When α > 1, the mean of the distribution exists and is equal to µ. However, when α ≤ 1, the tails are so heavy that even the mean does not exist. The fourth parameter β determines the skewness of the distribution and lies in the range [−1, 1]. Now let us introduce α-stable L´evy motions. Definition 4.2 (α-stable L´evy motion [64]). A stochastic process {X(t) : t ≥ 0} is called the (standard) α-stable L´evy motion if 1. X(0)=0 a.s.; 2. {X(t) : t ≥ 0} has independent increments; 3. X(t)-X(s)∼ Sα ((t − s)1/α , β, 0) for any 0 ≤ s < t < ∞. So, from the third condition, we can simulate all α-stable L´evy motion if we know how to simulate X ∼ Sα (σ, β, 0). Especially, it is enough to simulate X ∼ Sα (σ, 0, 0) if we want to get the trajectories of symmetric α-stable L´evy motions. We recall an important property of α-Stable random variables giving us the following result: It is enough to know how to simulate X ∼ Sα (1, 0, 0) in order to get any X ∼ Sα (σ, 0, 0), ∀σ ∈ R+ .

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0.05

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Figure 7. Three sample paths of symmetric α-stable L´evy motion with α = 0.4, 1.2, 1.9, respectively Proposition 4.3. If we have X1 , X2 ∼ Sα (σ, β, µ) and A, B are real positive constants and C is a real constant, then AX1 + BX2 + C ∼ Sα (σ(Aα + B α )1/α , β, µ(Aα + B α )1/α + C) Proposition 4.4. Let X ∼ Sα (σ, β, 0), with 0 < α < 2, Then E|X|p < ∞ for any 0 < p < α, E|X|p = ∞ for any p ≥ α. Figure 7 shows sample paths of the α-stable L´evy motion with different α. As we can see in Figure 7, the bigger the parameter α is, the more the path looks like Brownian motion. Generally speaking, when we deal with concrete data, we have to choose α-stable processes very carefully to get the best estimation. We now discuss how to estimate α. 4.2. How to estimate the characteristic exponent α Five different methods about how to estimate the characteristic exponent α of αstable distribution are considered: Characteristic Function Method(CFM), Quantile Method, Maximum Likelihood Method, Extreme Value Method and Moment Method. As in the last section, measurements are given on artificial data and the results of each method are compared in the end of this section. 4.2.1. Characteristic function method. Since α-stable distributions are uniquely determined by their Characteristic Function (CF), it is natural to consider how to estimate parameter by studying their CF. Press [106] introduced a parameter estimation method based on CF, which gets estimations of parameters by minimizing differences between values of sample CF and the real ones. But this method is only applicable to standard distributions. Another method which uses the linearity of logarithm of CF was developed by Koutrouvelis [74] and it can be applied to general α-stable cases. This method is denoted as Kou-CFM. The idea is as follows: On the one hand, taking the logarithm of real part of CF gives ln[−Re(ϕ(u))] = α ln |u| + α ln σ.

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On the other hand, the sample characteristic function of ϕ(θ) is given by Φ(θ) = N ( k=1 eiθyk ) where yk ’s are n independent observations. In [74], a regression technique is applied to gain estimates for all parameters of an observed α stable distribution. In [72], Kogon improved this method by replacing a linear regression fit by a linear least square fit which gave a more accurate estimate and its computational complexity became lower. 4.2.2. Quantile method. Quantiles are points taken at regular intervals from the cumulative distribution function of a random variable. Suppose we have n independent symmetric α-stable random variables with the stable distribution Sα (σ, β, µ), whose parameters are to be estimated. Let xp be the pth quantile, so that Sα (xp ; σ, β, µ) = p. Let xˆp be the corresponding sample quantile, then xˆp is a consistent estimate of xp . In 1971, Fama and Roll [41] discovered that, for some large p (for example, p = 0.95), ˆ1−p x ˆp − x ˆ1−p x ˆp − x zˆp = = (0.827) 2σ x ˆ0.72 − x ˆ0.28 is an estimate of the p-quantile of the standardized symmetric stable distribution with exponent α. According to this, they proposed an estimate (QM) for symmetric α-stable distributions. However, the serious disadvantage of this method is that its estimations are asymptotically biased. Later on, McCulloch [87] improved and extended this result to general α-stable distributions, denoted as McCulloch-QM. Firstly, he defined vα = (x−0.95 − x−0.05 )/(x−0.75 − x−0.25 ) vβ = (x−0.95 + x−0.05 − 2x0.5 )/(x−0.95 − x−0.05 ) and let vˆα and vˆβ be the corresponding sample value: vˆα = (ˆ x−0.95 − x ˆ−0.05 )/(ˆ x−0.75 − x ˆ−0.25 ) vˆβ = (ˆ x−0.95 + x ˆ−0.05 − 2ˆ x0.5 )/(ˆ x−0.95 − x ˆ−0.05 ) which are the consistent estimates of the index vα and vβ . Then, he illustrated that estimates of α can be expressed by a function of vˆα and vˆβ . Compared with QM, McCulloch-QM could get consistent and unbiased estimations for the general αstable distribution, and extend the estimation range of parameter α to 0.6 ≤ α ≤ 2. Despite its computational simplicity, this method has a number of drawbacks, such as, there are no analytical expressions for the value of the fraction, and the evaluation of the tables implies that it is highly dependent on the value of α in a nonlinear way. This technique does not provide any closed-form solutions. 4.2.3. Extreme value method. In 1996, based on asymptotic extreme value theory, order statistics and fractional lower-order moments, Tsihrintzis and Nikias [119] proposed a new estimate which can be computed fast for symmetric α stable distribution from a set of i.i.d. observations. Five years later, Kuruoglu [76] extended

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it to the general α stable distributions. The general idea of this method is as follows. Given a data series {Xi : i = 1, 2, . . . , N }, divide this into L nonoverlapping blocks of length K such that K = N/L. Then the logarithms of the maximum and minimum samples of each segment are computed as follows Yl = log(max{XlK−K+i |i = 1, 2, . . . , K}), Yl = log(− min{XlK−K+i |i = 1, 2, . . . , K}). The sample means and variances of Yl and Yl are calculated as 1 Yl , L L

Y =

s2 =

l=1

Y =

1 L

L  l=1

Yl , s2 =

L 1  (Yl − Y )2 , L−1 l=1

L 1  (Yl − Y )2 . L−1 l=1

Finally, an estimate for α is given by sample variance as follows

π 1 1 α ˆ= √ + . 2 6 s s Even though the accuracy and computational complexity decrease, there is now a closed form for the block size which means we have to look-up table to determine the segment size K. 4.2.4. Moment estimation method. Another way to estimate parameters of the general α-stable distribution is the Logarithmic Moments Methods which was also introduced by Kuruoglu [76]. The advantage of this method relative to the Fractional Lower-Order Method is that it does not require the inversion of a sinc function or the choice of a moment exponent p. The main feature is that the estimate of α can be expressed by a function of the second-order moment of the skewed process, i.e.,

−1/2 L2 1 α ˆ= − , ψ1 2 where ψ1 =

π2 6

and, for any X ∼ Sα (σ, β, 0), L2 is defined as follows

1 1 θ2 L2 = E[(log |X| − E[log |X|])2 ] = ψ1 + 2 − 2. 2 α α

4.2.5. Results on simulated data. In this subsection, we would like to use artificial data to check the robustness of the above techniques and compare the results. For each of the simulation methods chosen, estimates of α have been generated respectively and each trace is 1,000 points of data. Characteristic exponents of 0.95 and 1.70 have been chosen to represent a low and a high level of the rate at which the tails of the distribution taper off. From Figures 8 and 9, we can see that the Characteristic Function Method and the Moment Estimate Method have the most accurate result. The Quantile

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Figure 8. Numerical estimation of the characteristic exponent α in the α-stable L´evy motion Lα t by 4 different methods: Actual value α = 0.95

Method behaved a little better than Extreme Value Method, but both of them still fluctuate too much when α is small. As to the convergence, we can see that all the methods get closer and closer to the real value when the points of data increase except for the Extreme Value Method. 4.3. How to Estimate Parameters in SDEs Driven by L´evy Motion After we discussed how to estimate the characteristic exponent of α-stable L´evy motions, now we consider how to estimate the parameters in stochastic differential equations driven by general L´evy motion. Just as what we discussed about fBM, no general results about the parameter estimation for general cases are available at this time. Some special results will be listed below for different equations. We consider parameter estimation of the L´evy-driven stationary OrnsteinUhlenbeck process. Recently, Brockdwell, Davis and Yang [16] studied parameter estimation problems for L´evy-driven Langevin equation (whose solution is called an Ornstein-Uhlenceck process) based on observations made at uniformly and closelyspaced times. The idea is to obtain a highly efficient estimate of the L´evy-driven Ornstein-Uhlenceck process coefficient by estimating the corresponding coefficient of the sampled process. The main feature is discussed below. Consider a stochastic differential equation driven by the L´evy motion {L(t), t ≥ 0} dY (t) = −θY (t)dt + σdL(t).

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Figure 9. Numerical estimation of the characteristic exponent α in the α-stable L´evy motion Lα t by 4 different methods: Actual value α = 1.70 When L(t) is Brownian motion, the solution of above equation can be expressed as  t −θt Y (t) = e Y (0) + σ e−θ(t−u) dL(t). (25) 0

For any second-order driving L´evy motion, the process {Y (t)} can be defined in the same way, and if {L(t)} is non-decreasing, {Y (t)} can also be defined pathwise as a Riemann-Stieltjes integral by (25). For the convenience of the simulation, we rewrite solution as follows  t −θ(t−s) Y (t) = e Y (s) + σ e−θ(t−u)dL(u), for all t > s ≥ 0. (26) s

Now we collect all information corresponding to the sampled process in order to get the estimate. Set t = nh and s = (n − 1)h in equation (26). Then the sampled (h) process {Yn , n = 0, 1, 2, . . .} (or the discrete-time AR(1) process) satisfies (h)

Yn(h) = ϕYn−1 + Zn , where ϕ=e

−θh

 ,

nh

e−θ(nh−u) dL(u).

and Zn = σ (n−1)h

Then, using the highly efficient Davis-McCormick estimate of ϕ, namely (h)

(h)

ϕˆN = min

1≤n≤N

Yn

(h)

Yn−1

,

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we can get the estimate of θ and σ as follows (h) (h) θˆN = −h−1 log ϕˆN , (2)

σ ˆN =

(h) N 2θˆN  (h) (h) (Y − Y N )2 . N i=0 i

Example 4.5. Consider a L´evy-driven Ornstein-Uhlenbeck process satisfying the following SDE dXt = −Xt dt + σdLα (27) t. A numerical estimation of the diffusion parameter σ is shown in Figure 10. 12

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Figure 10. Numerical estimation of the diffusion parameter σ in L´evydriven Ornstein-Uhlenbeck process dXt = −Xt dt+σdLα t with α = 0.95: Actual value σ = 2

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Part II Financial Applications

Progress in Probability, Vol. 65, 255–298 c 2011 Springer Basel AG


Convertible Bonds in a Defaultable Diffusion Model Tomasz R. Bielecki, St´ephane Cr´epey, Monique Jeanblanc and Marek Rutkowski Abstract. In this paper, we study convertible securities (CS) in a primary market model consisting of: a savings account, a stock underlying a CS, and an associated CDS contract (or, alternatively to the latter, a rolling CDS more realistically used as an hedging instrument). We model the dynamics of these three securities in terms of Markovian diffusion set-up with default. In this model, we show that a doubly reflected Backward Stochastic Differential Equation associated with a CS has a solution, meaning that super-hedging of the arbitrage value of a convertible security is feasible in the present setup for both issuer and holder at the same initial cost, and we provide the related (super-)hedging strategies. Moreover, we characterize the price of a CS in terms of viscosity solutions of associated variational inequalities and we prove the convergence of suitable approximation schemes. We finally specify these results to convertible bonds and their straight bond and game exchange option components, and provide numerical results. Mathematics Subject Classification (2000). 62L15, 91G40, 60G40, 49J40. Keywords. BSDE, Superhedging, Markov models, Variational inequalities.

1. Introduction The goal of this work is a detailed and rigorous examination of convertible securities (CS) in a financial market model endowed with the following primary traded assets: a savings account, a stock underlying the CS and an associated credit default swap The research of T.R. Bielecki was supported by NSF Grant DMS-0604789 and NSF Grant DMS0908099. The research of S. Cr´ epey benefited from the support of the ‘Chaire Risque de cr´edit’, F´ ed´ eration Bancaire Fran¸caise, and of Ito33. The research of M. Jeanblanc benefited from the support of the ‘Chaire Risque de cr´edit’, F´ ed´ eration Bancaire Fran¸caise, and of Moody’s Corporation grant 5-55411. The research of M. Rutkowski was supported by the ARC Discovery Project DP0881460.

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(CDS) contract or, alternatively to the latter, a rolling CDS. Let us stress that we deal here not only with the valuation, but also, even more crucially, with the issue of hedging convertible securities that are subject to credit risk. Special emphasis is put on the properties of convertible bonds (CB) with credit risk, which constitute an important class of actively traded convertible securities. It should be acknowledged that convertible bonds were already extensively studied in the past by, among others, Andersen and Buffum [1], Ayache et al. [2], Brennan and Schwartz [12, 13], Davis and Lischka [22], Kallsen and K¨ uhn [31], Kwok and Lau [35], Lvov et al. [37], Sˆırbu et al. [42], Takahashi et al. [43], Tsiveriotis and Fernandes [45], to mention just a few. Of course, it is not possible to give here even a brief overview of models, methods and results from the above-mentioned papers (for a discussion of some of them and further references, we refer to [4]– [6]). Despite the existence of these papers, it was nevertheless our feeling that a rigorous, systematic and fully consistent approach to hedging-based valuation of convertible securities with credit risk (as opposed to a formal risk-neutral valuation approach) was not available in the literature, and thus we decided to make an attempt to fill this gap in a series of papers for which this work can be seen as the final episode. We strive to provide here the most explicit valuation and hedging techniques, including numerical analysis of specific features of convertible bonds with call protection and call notice periods. The main original contributions of the present paper, in which we apply and make concrete several results of previous works, can be summarized as follows: • we make a judicious choice of primary traded instruments used for hedging of a convertible security, specifically, the underlying stock and the rolling credit default swap written on the same credit name, • the completeness of the model until the default time of the underlying name in terms of uniqueness of a martingale measure is studied, • a detailed specification of the model assumptions that subsequently allow us to apply in the present framework our general results from the preceding papers [4]–[6] is provided, • it is shown that super-hedging of the arbitrage value of a convertible security is feasible in the present set-up for both issuer and holder at the same initial cost, • sufficient regularity conditions for the validity of the aggregation property for the value of a convertible bond at call time in the case of positive call notice period are given, • numerical results for the decomposition of the value of a convertible bond into straight bond and embedded option components are provided, • the precise definitions of the implied spread and implied volatility of a convertible bond are stated and some numerical analysis for both quantities is conducted. Before commenting further on this work, let us first describe very briefly the results of our preceding papers. In [4], working in an abstract set-up, we characterized arbitrage prices of generic convertible securities (CS), such as convertible

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bonds (CB), and we provided a rigorous decomposition of a CB into a straight bond component and a game option component, in order to give a definite meaning to commonly used terms of ‘CB spread’ and ‘CB implied volatility.’ Subsequently, in [5], we showed that in the hazard process set-up, the theoretical problem of pricing and hedging CS can essentially be reduced to a problem of solving an associated doubly reflected Backward Stochastic Differential Equation (BSDE for short). Finally, in [6], we established a formal connection between this BSDE and the corresponding variational inequalities with double obstacles in a generic Markovian intensity model. The related mathematical issues are dealt with in companion papers by Cr´epey [18] and Cr´epey and Matoussi [19]. In the present paper, we focus on a detailed study of convertible securities in a specific market set-up with the following traded assets: a savings account, a stock underlying a convertible security, and an associated rolling credit default swap. In Section 2, the dynamics of these three securities are formally introduced in terms of Markovian diffusion set-up with default. We also study there the arbitragefree property of this model, as well as its completeness. The model considered in this work appears as the simplest equity-to-credit reduced-form model, in which the connection between equity and credit is reflected by the fact that the default intensity γ depends on the stock level S. To the best of our knowledge, it is widely used by the financial industry for dealing with convertible bonds with credit risk. This specific model’s choice was the first rationale for the present study. Our second motivation was to show that all assumptions that were postulated in our previous theoretical works [4]–[6] are indeed satisfied within this set-up; in this sense, the model can be seen as a practical implementation of the general theory of arbitrage pricing and hedging of convertible securities. Section 3 is devoted to the study of convertible securities. We first provide a general result on the valuation of a convertible security within the present framework (see Proposition 3.1). Next, we address the issue of valuation and hedging through a study of the associated doubly reflected BSDE. Proposition 3.3 provides a set of explicit conditions, obtained by applying general results of Cr´epey [18], which ensure that the BSDE associated with a convertible security has a unique solution. This allows us to establish in Proposition 3.2 the form of the (super-)hedging strategy for a convertible security. Subsequently, we characterize in Proposition 3.4 the pricing function of a convertible security in terms of the viscosity solution to associated variational inequalities and we prove in Proposition 3.5 the convergence of suitable approximation schemes for the pricing function. In Section 4, we further specify these results to the special case of a convertible bond. In [4, 6] we worked under the postulate that the value Utcb of a convertible bond upon a call at time t yields, as a function of time, a well-defined process satisfying some natural conditions. In the specific framework considered here, using the uniqueness of arbitrage prices established in Propositions 2.1 and 3.1 and the continuous aggregation property for the value Utcb of a convertible bond upon a call at time t furnished by Proposition 4.7, we actually prove that this assumption is satisfied and we subsequently discuss in Propositions 4.6 and 4.8 the methods

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for computation of Utcb. We also examine in some detail the decomposition into straight bond and embedded game option components, which is practically relevant, since it provides a formal way of defining the implied volatility of a convertible bond. We conclude the paper by illustrating some results through numerical computations of relevant quantities in a simple example of an equity-to-credit model.

2. Markovian equity-to-credit framework We first introduce a generic Markovian default intensity set-up. More precisely, we consider a defaultable diffusion model with time- and stock-dependent local default
t intensity and local volatility (see [1, 2, 6, 14, 22, 24, 34]). We denote by 0 the integral over (0, t]. 2.1. Default time and pre-default equity dynamics Let us be given a standard stochastic basis (Ω, G, F, Q), over [0, Θ] for some fixed Θ ∈ R+ , endowed with a standard Brownian motion (Wt )t∈[0,Θ] . We assume that F is the filtration generated by W . The underlying probability measure Q is aimed to represent a risk-neutral probability measure (or ‘pricing probability’) in a financial market model that we are now going to construct. In the first step, we define the pre-default factor process (S+t )t∈[0,Θ] (to be interpreted later as the pre-default stock price of the firm underlying a convertible security) as the diffusion process with the initial condition S+0 and the dynamics over [0, Θ] given by the stochastic differential equation (SDE) ) *  dS+t = S+t r(t) − q(t) + ηγ(t, S+t ) dt + σ(t, S+t ) dWt (1) with a strictly positive initial value S+0 . We denote by L the infinitesimal generator + that is, the differential operator given by the formula of S,   σ 2 (t, S)S 2 2 ∂S 2 . L = ∂t + r(t) − q(t) + ηγ(t, S) S ∂S + 2

(2)

Assumption 2.1. (i) The riskless short interest rate r(t), the equity dividend yield q(t), and the local default intensity γ(t, S) ≥ 0 are bounded, Borel-measurable functions and η ∈ [0, 1] is a real constant, to be interpreted later as the fractional loss upon default on the stock price. (ii) The local volatility σ(t, S) is a positively bounded, Borel-measurable function, so, in particular, we have that σ(t, S) ≥ σ > 0 for some constant σ. (iii) The functions γ(t, S)S and σ(t, S)S are Lipschitz continuous in S, uniformly in t. Note that we allow for negative values of r and q in order, for instance, to possibly account for repo rates in the model. Under Assumption 2.1, SDE (1) is known

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+ which is non-negative over [0, Θ]. Moreover, to admit a unique strong solution S, the following (standard) a priori estimate is available, for any p ∈ [2, +∞) ) * ) * EQ sup |S+t |p ≤ C 1 + |S+0 |p . (3) t∈[0,Θ]

In the next step, we define the [0, Θ]∪{+∞}-valued default time τd , using the so-called canonical construction [8]. Specifically, we set (by convention, inf ∅ = ∞)  t ! " τd = inf t ∈ [0, Θ]; γ(u, S+u ) du ≥ ε , (4) 0

where ε is a random variable on (Ω, G, F, Q) with the unit exponential distribution and independent of F. Because of our construction of τd , the process Gt := Q(τ > t | Ft ) satisfies, for every t ∈ [0, Θ], Gt = e−




t 0

+u ) du γ(u,S

and thus it has continuous and non-increasing sample paths. This also means that the process γ(t, S+t ) is the F-hazard rate of τd (see, e.g., [8, 30]). The fact that the hazard rate γ may depend on S+ is crucial, since this dependence actually conveys all the ‘equity-to-credit’ information in the model. A natural choice for γ is a decreasing (e.g., negative power) function of S+ capped when S+ is close to zero. A possible further refinement would be to put a positive floor on the function γ. The lower bound on γ would then reflect the perceived level the systemic default risk, as opposed to firm-specific default risk. Let Ht = 1{τd ≤t} be the default indicator process and let the process d (Mt )t∈[0,Θ] be given by the formula  t Mtd = Ht − (1 − Hu )γ(u, S+u ) du. 0

We denote by H the filtration generated by the process H and by G the enlarged filtration given as F∨H. Then the process M d is known to be a G-martingale, called the compensated jump martingale. Moreover, the filtration F is immersed in G, in the sense that all F-martingales are G-martingales; this property is also frequently referred to as Hypothesis (H). It implies, in particular, that the F-Brownian motion W remains a Brownian motion with respect to the enlarged filtration G under Q. 2.2. Market model We are now in a position to define the prices of primary traded assets in our market model. Assuming that τd is the default time of a reference entity (firm), we consider a continuous-time market on the time interval [0, Θ] composed of three primary assets: • the savings account evolving according to the deterministic short-term interest rate r; we denote by β the discount factor (the inverse of the savings
t account), so that βt = e− 0 r(u) du ;

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• the stock of the reference entity with the pre-default price process S+ given by (1) and the fractional loss upon default determined by a constant η ≤ 1; • a CDS contract written at time 0 on the reference entity, with maturity Θ, the protection payment given by a Borel-measurable, bounded function ν : [0, Θ] → R and the fixed CDS spread ν¯. Remarks 2.1. It is worth noting that the choice of a fixed-maturity CDS as a primary traded asset is only temporary and it is made here mainly for the sake of expositional simplicity. In Section 2.3 below, we will replace this asset by a more practical concept of a rolling CDS, which essentially is a self-financing trading strategy in market CDSs. The stock price process (St )t∈[0,Θ] is formally defined by setting ) *  dSt = St− r(t) − q(t) dt + σ(t, St ) dWt − η dMtd , S0 = S+0 ,

(5)

so that, as required, the equality (1 − Ht )St = (1 − Ht )S+t holds for every t ∈ [0, Θ]. Note that estimate (3) enforces the following moment condition on the process S ) * EQ sup St < ∞, a.s. (6) t∈[0,τd ∧Θ]

We define the discounted cumulative stock price β S? stopped at τd by setting, for every t ∈ [0, Θ],  t∧τd   ? + βt St = βt (1 − Ht )St + βu (1 − η)S+u dHu + q(u)S+u du 0

or, equivalently, in terms of S βt S?t = βt∧τd St∧τd +



t∧τd

βu q(u)Su du. 0

Note that we deliberately stopped β S? at default time τd , since we will not need to consider the behavior of the stock price strictly after default. Indeed, it will be enough to work under the assumption that all trading activities are stopped no later than at the random time τd ∧ Θ. Let us now examine the valuation in the present model of a CDS written on the reference entity. We take the perspective of the credit protection buyer. Consistently with the no-arbitrage requirements (cf. [7]), we assume that the pre+t )t∈[0,Θ] is given as B +t = B(t, + S+t ), where the pre-default CDS default CDS price (B + S) is the unique (classical) solution on [0, Θ] × R+ to the pricing function B(t, following parabolic PDE + S) + δ(t, S) − µ(t, S)B(t, + S) = 0, B(Θ, + LB(t, S) = 0, (7) where • the differential operator L is given by (2), • δ(t, S) = ν(t)γ(t, S) − ν¯ is the pre-default dividend function of the CDS, • µ(t, S) = r(t) + γ(t, S) is the credit-risk adjusted interest rate.

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? equals, for every t ∈ [0, Θ], The discounted cumulative CDS price β B ?t = βt (1 − Ht )B +t + βt B



t∧τd

  βu ν(u) dHu − ν¯ du .

(8)

0

Remarks 2.2. (i) It is worth noting that as soon as the risk-neutral parameters in the dynamics of the stock price S are given by (5), the dynamics (8) of the CDS price are derived from the dynamics of S and our postulate that Q is the ‘pricing probability’ for a CDS. This procedure resembles the standard method of completing a stochastic volatility model by taking a particular option as an additional primary traded asset (see, e.g., Romano and Touzi [41]). We will sometimes refer to dynamics (5) as the model; it will be implicitly assumed that this model is actually completed either by trading a fixed-maturity CDS (as in Section 2.2.1) or by trading a rolling CDS (see Section 2.3). Given the interest rate r, dividend yield q, the parameter η, and the covenants of a (rolling) CDS, the model calibration will then reduce to a specification of the local intensity γ and the local volatility σ only. We refer, in particular, to Section 4.3.6 in which the concepts of the implied spread and the implied volatility of a convertible bond are examined. (ii) Assuming dependence of the default intensity on the stock level is, of course, not the only way to model the equity-to-credit relationship. It is even fair to say that the above set-up is in fact a very stylized model of equity-to-credit. More precisely, it is not realistic to model default by only having equity and convertible bonds in the structure of the firm. Rather than modeling default on equity, it would be more realistic to model default as the inability to pay senior debt, or, in case no senior debt is outstanding, as the inability to repay the face value of the convertible bond (that would happen at maturity only, though). However the simple equity-to-credit framework of this paper is customary in the credit risk literature. From an economic point of view it is motivated by the empirical evidence of a negative correlation between equity levels and CDS spreads. More pragmatically this model is justified by good calibrability properties to CDS and equity option data. ? are man2.2.1. Risk-neutral measures and model completeness. Since β S? and β B ifestly locally bounded processes, a risk-neutral measure for the market model is + equivalent to Q such that the discounted defined as any probability measure Q + ? ? cumulative prices β S and β B are (G, Q)-local martingales (see, for instance, Page 234 in Bj¨ork [9]). In particular, we note that the underlying probability measure Q is a risk-neutral measure for the market model. The following lemma can be easily proved using the Itˆ o formula.

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?t = Lemma 2.1. Let us denote X  ?t ) = d d(βt X

S?t ?t B

βt S?t ?t βt B

. We have, for every t ∈ [0, Θ], # = 1{t≤τd } βt Σt d

Wt Mtd

$ ,

where the F-predictable, matrix-valued process Σ is given by the formula  σ(t, S+t )S+t −η S+t Σt = + S+t ) ν(t) − B(t, + S+t ) . σ(t, S+t )S+t ∂S B(t,

(9)

(10)

In what follows, we work under the following standing assumption. Assumption 2.2. The matrix-valued process Σ is invertible on [0, Θ]. The next proposition suggests that, under Assumption 2.2, our market model is complete with respect to defaultable claims maturing at τd ∧ Θ. + Proposition 2.1. For any risk-neutral measure )  Q *for the market model, we have +  dQ that the Radon-Nikodym density Zt := EQ dQ  Gt = 1 on [0, τd ∧ Θ]. + equivalent to Q on (Ω, GΘ ), the RadonProof. For any probability measure Q Nikodym density process Zt , t ∈ [0, Θ], is a strictly positive (G, Q)-martingale. Therefore, by the predictable representation theorem due to Kusuoka [33], there exist two G-predictable processes, ϕ and ϕd say, such that   dZt = Zt− ϕt dWt + ϕdt dMtd , t ∈ [0, Θ]. (11) + is then a risk-neutral measure whenever the process A probability measure Q + ? is a (G, Q)-local ? is a βX martingale or, equivalently, whenever the process β XZ (G, Q)-local martingale. The latter condition is satisfied if and only if # $ ϕt Σt = 0. (12) γ(t, S+t )ϕdt The unique solution to (12) on [0, τd ∧ Θ] is ϕ = ϕd = 0 and thus Z = 1 on [0, τd ∧ Θ].  2.3. Modified market model In market practice, traders would typically prefer to use for hedging purposes the rolling CDS, rather than a fixed-maturity CDS considered in Section 2.2. Formally, the rolling CDS is defined as the wealth process of a self-financing trading strategy that amounts to continuously rolling one unit of long CDS contracts indexed by their inception date t ∈ [0, Θ], with respective maturities θ(t), where θ : [0, Θ] → [0, Θ] is an increasing and piecewise constant function satisfying θ(t) ≥ t (in particular, θ(Θ) = Θ). We shall denote such contracts as CDS(t, θ(t)). Intuitively, the above-mentioned strategy amounts to holding at every time t ∈ [0, Θ] one unit of the CDS(t, θ(t)) combined with the margin account, that is, either positive or negative positions in the savings account. At time t + dt

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the unit position in the CDS(t, θ(t)) is unwound (or offset) and the net markto-market proceeds, which may be either positive or negative depending on the evolution of the CDS market spread between the dates t and t + dt, are reinvested in the savings account. Simultaneously, a freshly issued unit credit default swap CDS(t + dt, θ(t + dt)) is entered into at no cost. This procedure is carried on in continuous time (in practice, on a daily basis) until the hedging horizon. In the ? in (9) is meant to represent the discounted case of the rolling CDS, the entry β B cumulative wealth process of this trading strategy. The next result shows that the only modification with respect to the case of a fixed-maturity CDS is that the matrix-valued process Σ, which was given previously by (10), should now be adjusted to Σ given by (13). ? represents the rolling CDS, Lemma 2.1 Lemma 2.2. Under the assumption that B holds with the F-predictable, matrix-valued process Σ given by the expression  σ(t, S+t )S+t −ηS+t Σt = (13) + + + + + + + + + σ(t, St )St ∂S Pθ(t) (t, St ) − ν¯(t, St )σ(t, St )St ∂S Fθ(t) (t, St ) ν(t) where the functions P+θ(t) and F+θ(t) are the pre-default pricing functions of the protection and fee legs of the CDS(t, θ(t)), respectively, and the quantity P+θ(t) (t, S+t ) ν¯(t, S+t ) = F+θ(t) (t, S+t ) represents the related CDS spread. Proof. Of course, it suffices to focus on the second row in matrix Σ. We start by noting that Lemma 2.4 in [7], when specified to the present set-up, yields the ? of the rolling CDS following dynamics for the discounted cumulative wealth β B between the deterministic times representing the jump times of the function θ   ?t ) = (1 − Ht )βt α−1 dpt − ν¯(t, S+t ) dft + βt ν(t) dM d , d(βt B (14) t t where we denote ) pt = EQ

θ(t)

 *  αu ν(u)γ(u, S+u ) du  Ft ,

0

ft = EQ

)

θ(t)

 *  αu du  Ft ,

0

and where in turn the process α is given by αt = e −




t 0

+u ) du µ(u,S

= e−




t +u )) du (r(u)+γ(u,S 0

. ? is necessarily conIn addition, being a (G, Q)-local martingale, the process β B tinuous prior to default time τd (this follows, for instance, from Kusuoka [33]). It is therefore justified to use (14) for the computation of a diffusion term in the ? dynamics of β B. To establish (13), it remains to compute explicitly the diffusion term in (14). Since the function θ is piecewise constant, it suffices in fact to examine the stochastic differentials dpt and dft for a fixed value θ = θ(t) over each interval of

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constancy of θ. By the standard valuation formulae in an intensity-based framework, the pre-default price of a protection payment ν with a fixed horizon θ is given by, for t ∈ [0, θ],  * ) θ  P+θ (t, S+t ) = α−1 E αu ν(u)γ(u, S+u ) du  Ft . Q t t

Therefore, by the definition of p, we have that, for t ∈ [0, θ],  t pt = αu ν(u)γ(u, S+u ) du + αt P+θ (t, S+t ).

(15)

0

Since p is manifestly a (G, Q)-martingale, an application of the Itˆ o formula to (15) yields, in view of (1), dpt = αt σ(t, S+t )S+t ∂S P+θ (t, S+t ) dWt . Likewise, the pre-default price of a unit rate fee payment with a fixed horizon θ is given by  * ) θ  F+θ (t, S+t ) = α−1 E α du  Ft . Q u t t

By the definition of f , we obtain, for t ∈ [0, θ],  t ft = αu du + αt F+θ (t, S+t ) 0

and thus, noting that f is a (G, Q)-martingale, we conclude easily that dft = αt σ(t, S+t )S+t ∂S F+θ (t, S+t ) dWt . By inserting dpt and dft into (14), we complete the derivation of (13).



Remarks 2.3. It is worth noting that for a fixed u the pricing functions P+θ(u) and F+θ(u) can be characterized as solutions of the PDE of the form (7) on [u, θ(u)] × R+ with the function δ therein given by δ 1 (t, S) = ν(t)γ(t, S) and δ 2 (t, S) = 1, respectively. Hence the use of the Itˆ o formula in the proof of Lemma 2.2 can indeed be justified. Note also that, under the standing Assumption 2.2, a suitable form of completeness of the modified market model will follow from Proposition 2.1.

3. Convertible securities In this section, we first recall the concept of a convertible security (CS). Subsequently, we establish, or specify to the present situation, the fundamental results related to its valuation and hedging. We start by providing a formal specification in the present set-up of the notion of a convertible security. Let 0 (resp. T ≤ Θ) stand for the inception date (resp. the maturity date) of a CS with the underlying asset S. For any t ∈ [0, T ], we write FTt (resp. GTt ) to denote the set of all F-stopping times (resp. G-stopping times) with values in [t, T ]. Given the time of lifting of a call protection of a CS,

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which is modeled by a stopping time τ¯ belonging to GT0 , we denote by G¯Tt the following class of stopping times , G¯Tt = ϑ ∈ GTt ; ϑ ∧ τd ≥ τ¯ ∧ τd . We will frequently use τ as a shorthand notation for τp ∧ τc , for any choice of (τp , τc ) ∈ GTt × G¯Tt . For the definition of the game option, we refer to Kallsen and K¨ uhn [31] and Kiefer [32]. Definition 3.1. A convertible security with the underlying S is a game option with the ex-dividend cumulative discounted cash flows π(t; τp , τc ) given by the following expression, for any t ∈ [0, T ] and (τp , τc ) ∈ GTt × G¯Tt ,  τ ) * βt π(t; τp , τc ) = βu dDu + 1{τd >τ } βτ 1{τ =τp
where: • the dividend process D = (Dt )t∈[0,T ] equals   Dt = (1 − Hu ) dCu + [0,t]

• • • •

Ru dHu [0,t]

for some coupon process C = (Ct )t∈[0,T ] , which is a G-adapted, real-valued, c`adl`ag process with bounded variation, and a G-adapted, real-valued, c` adl` ag recovery process R = (Rt )t∈[0,T ] , the put/conversion payment L is given as a G-adapted, real-valued, c` adl` ag process on [0, T ], the call payment U is a G-adapted, real-valued, c` adl` ag process on [0, T ], such that Lt ≤ Ut on [τd ∧ τ¯, τd ∧ T ), the payment at maturity ξ is a GT -measurable, real-valued random variable, the processes R, L and the random variable ξ are assumed to satisfy the following inequalities, for a positive constant c, −c ≤ Rt ≤ c (1 ∨ St ) , t ∈ [0, T ], −c ≤ Lt ≤ c (1 ∨ St ) , t ∈ [0, T ],

(16)

−c ≤ ξ ≤ c (1 ∨ ST ) . 3.1. Arbitrage valuation of a convertible security We are in a position to recall and specify to the present set-up a general valuation result for a convertible security. Let us mention that the notion of an arbitrage price of a convertible security, referred to in what follows, is a suitable extension to game options (see Definition 2.6 in Kallsen and K¨ uhn [31]) of the No Free Lunch with Vanishing Risk (NFLVR) condition of Delbaen and Schachermayer [23]. We also use here the well known connection between Dynkin games and the valuation of game options (see Kiefer [32]).

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Proposition 3.1. If the Dynkin game related to a convertible security admits a value Π, in the sense that    esssupτp ∈GTt essinfτc ∈G¯Tt EQ π(t; τp , τc )  Gt = Πt    (17) = essinfτ ∈G¯t esssupτ ∈Gt EQ π(t; τp , τc )  Gt , t ∈ [0, T ], c

T

p

T

and Π is a G-semimartingale, then Π is the unique arbitrage (ex-dividend) price of the CS. Proof. Except for the uniqueness statement, this follows by applying the general results in [4]. To verify the uniqueness property, )  *we first note that for any risk+  + neutral measure Q, we have that Zt = EQ dQ  Gt = 1 on [0, τd ∧ T ], by ProposidQ

tion 2.1. In view of the estimate (6) on supt∈[0,T ∧τd ] St , and since supt∈[0,T ∧τd ] St is a Gτd ∧T -measurable random variable, this implies that, for any risk-neutral mea+ sure Q, ) * ) * EQ sup St = EQ sup St < ∞. (18) + t∈[0,T ∧τd ]

t∈[0,T ∧τd ]

+ we Obviously, for the supremum over the set M of all risk-neutral measures Q thus have that ) * sup EQ+ sup St < ∞. (19) + Q∈M

t∈[0,T ∧τd ]

Under condition (19), any arbitrage price of a CS with underlying S is then given + by by the value of the related Dynkin game for some risk-neutral measure Q, the general results of [4]. Furthermore, π(t; τp , τc ) is a Gτd ∧T -measurable random variable. Therefore, for any t ∈ [0, T ], τp ∈ GTt , τc ∈ G¯Tt ,       EQ+ π(t; τp , τc )  Gt = EQ π(t; τp , τc )  Gt . (20) + We conclude that the Q-Dynkin game has the value Π for any risk-neutral mea+ sure Q.  We now define two special cases of CSs that correspond to American- and European-style CSs. Definition 3.2. A puttable security (as opposed to puttable and callable, in the case of a general convertible security) is a convertible security with τ¯ = T . An elementary security is a puttable security with a bounded variation dividend process D over [0, T ], a bounded payment at maturity ξ, and such that   βu dDu + 1{τd >t} βt Lt ≤ βu dDu + 1{τd >T } βT ξ, t ∈ [0, T ). (21) [0,t]

[0,T ]

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3.2. Doubly reflected BSDEs approach We will now apply to convertible securities the method proposed by El Karoui et al. [27] for American options and extended by Cvitani´c and Karatzas [20] to the case of stochastic games. In order to effectively deal with the doubly reflected BSDE associated with a convertible security, which is introduced in Definition 3.3 below, we need to impose some technical assumptions. We refer the reader to Section 4 for concrete examples in which all these assumptions are indeed satisfied. Assumption 3.1. We postulate that: • the coupon process C satisfies



Ct = C(t) :=

t

c(u) du + 0



ci ,

(22)

0≤Ti ≤t

for a bounded, Borel-measurable continuous-time coupon rate function c(·) and deterministic discrete times and coupons Ti and ci , respectively; we take the tenor of the discrete coupons as T0 = 0 < T1 < · · · < TI−1 < TI with TI−1 < T ≤ TI ; • the recovery process (Rt )t∈[0,T ] is of the form R(t, St− ) for a Borel-measurable function R; • Lt = L(t, St), Ut = U (t, St ), ξ = ξ(ST ) for some Borel-measurable functions L, U and ξ such that, for any t, S, we have L(t, S) ≤ U (t, S),

L(T, S) ≤ ξ(S) ≤ U (T, S);

• the call protection time τ¯ ∈ FT0 . The accrued interest at time t is given by t − Tit −1 it c , A(t) = Tit − Tit −1

(23)

where it is the integer satisfying Tit −1 ≤ t < Tit . On open intervals between the cit discrete coupon dates we thus have dA(t) = a(t) dt with a(t) = Ti −T . it −1 t To a CS with data (functions) C, R, ξ, L, U and lifting time of call protection τ¯, we associate the Borel-measurable functions f (t, S, x) (for x real), g(S), (t, S) and h(t, S) defined by g(S) = ξ(S) − A(T ),

(t, S) = L(t, S) − A(t),

h(t, S) = U (t, S) − A(t), (24)

and (recall that µ(t, S) = r(t) + γ(t, S)) f (t, S, x) = γ(t, S)R(t, S) + Γ(t, S) − µ(t, S)x,

(25)

Γ(t, S) = c(t) + a(t) − µ(t, S)A(t).

(26)

where we set Remarks 3.1. In the case of a puttable security, the process U is not relevant and thus we may and do set h(t, S) = +∞. Moreover, in the case of an elementary security, the process L plays no role either, and we redefine further (t, S) = −∞.

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We define the quadruplet (f, g, , h) associated to a CS (parameterized by x ∈ R, regarding f ) as ft(x) = f (t, S+t , x),

g = g(S+T ),

t = (t, S+t ),

ht = 1{t<¯τ } ∞ + 1{t≥¯τ } h(t, S+t )

(27)

with the convention that 0 × ∞ = 0 in the last equality. Let us also write γt = γ(t, S+t ) µt = µ(t, S+t ) αt = e−




t 0

µu du

.

(28)

It is well known that game options (in particular, convertible securities) can be studied by analyzing the corresponding doubly reflected Backward Stochastic Differential Equations (cf. [20]). In our set-up, this connection is formalized through the following definition. Definition 3.3. Consider a convertible security with data C, R, ξ, L, U, τ¯ and the associated quadruplet (f, g, , h) given by (27). The associated doubly reflected Backward Stochastic Differential Equation has the form, for t ∈ [0, T ),  ? t = ft (Π ? t ) dt + dKt − Zt dWt ,  − dΠ   ? t ≤ ht , (E) t ≤ Π    ? + − ? (Πt − t ) dKt = (ht − Πt ) dKt = 0, ? T = g. with the terminal condition Π To define a solution of the doubly reflected BSDE (E), we need to introduce the following spaces: H2 the set of real-valued, F-predictable processes X such that ) T * EQ Xt2 dt < ∞, 0

S

2

the set of real-valued, F-adapted, continuous processes X such that ) * EQ sup Xt2 < ∞, t∈[0,T ]

A

2

the space of continuous processes of finite variation K with (continuous and non-decreasing) Jordan components K ± ∈ S 2 null at time 0, 2 Ai the space of non-decreasing, continuous processes null at 0 and belonging to S 2 . For any K ∈ A2 , we thus have that K = K + − K − , where K ± ∈ A2i define mutually singular measures on R+ . Definition 3.4. By a solution to the doubly reflected BSDE (E) with data (f, g, , h), ? Z, K) ∈ S 2 × H2 × A2 satisfying all conditions we mean a triplet of processes (Π, ? have to be in (E). In particular, the process K, and thus also the process Π, continuous.

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Remarks 3.2. (i) For a puttable security, we have that τ¯ = T and thus K − = 0 in any solution ? Z, K) to (E). Therefore, the doubly reflected BSDE (E) reduces to the (Π, reflected BSDE (E.1) with data (f, g, ) and K = K + ∈ A2i in any solution; specifically,  + ? ?   − dΠt = ft (Πt ) dt + dKt − Zt dWt ,  ? t, (E.1) t ≤ Π    ? (Πt − t ) dKt+ = 0, ? T = g. with the terminal condition Π ? Z, K) to (E). (ii) For an elementary security, we have K = 0 in any solution (Π, Consequently, the doubly reflected BSDE (E) becomes the standard BSDE (E.2) with data (f, g), that is, ? t = ft ( Π ? t ) dt − Zt dWt dΠ

(E.2)

? T = g. with the terminal condition Π In order to establish the well-posedness of the doubly reflected BSDE, as well as its connection with the related obstacles problem examined in the next section, we will work henceforth under the following additional assumption. Assumption 3.2. The functions r, q, γ, σ, c, R, g, h,  are continuous. 3.2.1. Super-hedging strategies for a convertible security. The following definition of a self-financing trading strategy is standard. Definition 3.5. By a self-financing strategy over the time interval [0, T ], we mean a pair (V0 , ζ) such that: • V0 is a real number representing the initial wealth, ? • (ζt )t∈[0,T ] is an R1⊗2 -valued (bi-dimensional row vector), G-predictable, β Xintegrable process (cf. (9)) representing holdings (number of units held) in primary risky assets. The wealth process V of a self-financing strategy (V0 , ζ) is given by  t ?u ), t ∈ [0, T ]. βt Vt = V0 + ζu d(βu X (29) 0

? we can take in Definition 3.5 Remarks 3.3. It should be emphasized that as β B either the dynamics of the discounted wealth a fixed-maturity CDS, given by (8), or of a rolling CDS, given by (14). Consequently, in view of Lemmas 2.1 and 2.2, equality (29) becomes  # $  t  t Wu βu S?u 1 2 1 2 βt Vt = V0 + [ζu , ζu ] d ?u = V0 + 0 1{u≤τd } βu [ζu , ζu ] Σu d Mud , βu B 0 (30)

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where the matrix-valued process Σ is given by (10) in the case of a fixed-maturity CDS, and it is given by (13) in the case of a rolling CDS. Formula (30) makes it clear that the wealth process V is stopped at time τd ; this property reflects the fact that we are only interested in trading on the stochastic interval [0, τd ∧ T ], where T is the maturity date of a considered convertible security. In the set-up of this paper, the notions of the issuer’s and holder’s (super-) hedges take the following form. Recall that we denote τ = τp ∧ τc . Definition 3.6. (i) An issuer’s hedge for a convertible security is represented by a triplet (V0 , ζ, τc ) such that: • (V0 , ζ) is a self-financing strategy with the wealth process V , • the call time τc belongs to G¯T0 , • the following inequality is valid, for every put time τp ∈ GT0 , βτ Vτ ≥ β0 π(0; τp , τc ),

a.s.

(31)

(ii) A holder’s hedge for a convertible security is a triplet (V0 , ζ, τp ) such that: • (V0 , ζ) is a self-financing strategy with the wealth process V , • the put time τp belongs to GT0 , • the following inequality is valid, for every call time τc ∈ G¯T0 , βτ Vτ ≥ −β0 π(0; τp , τc ),

a.s.

(32)

Remarks 3.4. Definition 3.6 can be easily extended to hedge starting at any initial date t ∈ [0, T ], as well as specified to the particular cases of puttable and elementary securities (see [5, 6]). By applying the general results of [5, 6], we obtain the following (super-) hedging result. Obviously, the conclusion of Proposition 3.2 hinges on the temporary assumption that the related BSDE (E) has a solution. The issues of existence and uniqueness of a solution to (E) will be addressed in the foregoing subsection. See also Remarks 3.9 for a more explicit representation of a hedging strategy. ? Z, K) to the doubly reflected BSDE (E) Proposition 3.2. Assume that a solution (Π, + + ? + A. Then Π is the unique arbitrage exists. Let Πt denote 1{t<τd } Πt with Π := Π price process of a convertible security. (i) For any t ∈ [0, T ], an issuer’s hedge with the initial wealth Πt is furnished by , ? u = hu ∧ T τc∗ = inf u ∈ [¯ τ ∨ t, T ]; Π and

' ( + u− Λu , u ∈ [t, T ], ζu∗ := 1{u≤τd } Zu , Ru − Π (33) + + where [Zu , Ru − Πu− ] denotes the concatenation of Zu and Ru − Πu− and where Λ denotes of the inverse of the matrix-valued process Σ over [0, τd ∧ T ] (cf. Assumption 2.2). Moreover, Πt is the smallest initial wealth of an issuer’s hedge.

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(ii) For any t ∈ [0, T ], a holder’s hedge with the initial wealth −Πt is furnished by , ? u = u ∧ T τp∗ = inf u ∈ [t, T ] ; Π and ζ = −ζ ∗ with ζ ∗ given by (33). Moreover, in case of a CS with bounded cash cash flows, −Πt is the smallest initial wealth of a holder’s hedge. Proof. In view of the general results of [5, 6], we see that the process Π defined in the statement of the proposition satisfies all the assumptions for the process Π introduced in Proposition 3.1. Hence it is the unique arbitrage price process of a CS. As for statements (i) and (ii), they are rather straightforward consequences of the general results of [5, 6].  Proposition 3.2 shows that in the present set-up a CS has a bilateral hedging price, in the sense that the price Πt ensures super-hedging to both its issuer and holder, starting from the initial wealth Πt for the former and −Πt for the latter, where process Π is also the unique arbitrage price. Note also that in the case of an elementary security, there are no stopping times involved and process K is equal to 0, so that (Πt , ζ ∗ ) in fact defines a replicating strategy. ? is aimed to represent either a fixed-maturity Remarks 3.5. Let us recall that B CDS or a rolling CDS. Since Assumption 2.2 was postulated for both cases then the underlying probability Q is the unique risk-neutral probability on [0, τd ∧ Θ] no matter whether a fixed-maturity CDS or a rolling CDS is chosen to be a traded primary asset. Consequently, the hedging price of a CS does not depend on the choice of primary traded CDSs. By contrast, the super-hedging strategies of Proposition 3.2 are clearly dependent on the choice of traded CDSs through the matrix-valued process Λ = Σ−1 , where Σ is given either by (10) or by (13). 3.2.2. Solutions of the doubly reflected BSDE. As mentioned above, the existence of hedging strategies for a convertible security will be derived from the existence of a solution to the related doubly reflected BSDE. To establish the latter, we need to impose further technical assumptions on a convertible security under study. Let then P stand for the class of functions Π of the real variable S bounded by C(1 + |S|p ) for some real C and integer p that may depend on Π. By a slight abuse of terminology, we shall say that a function Π(S, . . . ) is of class P if it has polynomial growth in S, uniformly in other arguments. We postulate henceforth the following additional assumptions regarding the specification of a convertible security. Assumption 3.3. The functions R, g, h,  associated to a CS are of class P (or h = +∞, in the case of a puttable security, and  = −∞, in the case of an elementary security), and τ¯ is given as ¯ ∧ T¯ τ¯ = inf{ t > 0 ; S+t ≥ S} (34) ¯ ¯ for some constants T ∈ [0, T ] and S ∈ R+ ∪ {+∞} (so, in particular, τ¯ = 0 in case S¯ = 0, and τ¯ = T¯ in case S¯ = +∞). As for , it satisfies, more specifically, the

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following structure condition: (t, S) = λ(t, S)∨c for some constant c ∈ R∪{−∞}, and a function λ of class C 1,2 with λ, ∂t λ, S∂S λ, S 2 ∂S2 2 λ ∈ P

(35)

(or  = −∞, in the case of an elementary security). Example 3.1. The standard example of the function λ(t, S) satisfying (35) is λ(t, S) = S. In that case,  corresponds to the payoff function of a call option (or, more precisely, to the lower payoff function of a convertible bond, see Section 4). By an application of the general results of [6, 18], we then have the following proposition, which complements Proposition 3.2. ? Proposition 3.3. The doubly reflected BSDE (E) admits a unique solution (Π,Z,K).  In the next section, we will study the variational inequalities approach to convertible securities in the present Markovian set-up, as well as the link between the variational inequalities and the doubly reflected BSDEs. 3.3. Variational Inequalities Approach In Section 3.3, we will give analytical characterizations of the so-called pre-default clean prices (that is, the pre-default prices less accrued interest) in terms of viscosity solutions to the associated variational inequalities. In the context of convertible bonds, the variational inequalities approach was examined, though without formal proofs, in Ayache et al. [2]. Convention. Unless explicitly stated otherwise, by a ‘price’ of a convertible security we mean henceforth its ‘pre-default clean price.’ ? of a solution Note that the clean prices correspond to the state-process Π to (E); see Proposition 3.2 and [6]. To obtain the corresponding pre-default price, it suffices to add to the clean price process the related accrued interest given by (23), provided, of course, that there are any discrete coupons present in the product under consideration. For any τ¯ ∈ FT0 , the associated price coincides on [¯ τ , T ] with the price corresponding to a lifting time of call protection given by τ¯0 := 0. This observation follows from the general results in [5], using also the fact that, under the standing assumptions, the BSDEs related to the problems with lifting times of call protection τ¯ and τ¯0 both have solutions. The no-protection prices (i.e., prices obtained for the lifting time of call protection τ¯0 = 0) can thus also be interpreted as post-protection prices for an arbitrary stopping time τ¯ ∈ FT0 , where by the post-protection price we mean the price restricted to the random time interval [¯ τ , T ]. Likewise, we define the protection prices as prices restricted to the random time interval [0, τ¯]. For a closed domain D ⊆ [0, T ] × R, let Intp D and ∂p D stand for the parabolic interior and the parabolic boundary of D, respectively. For instance, if D = [0, T¯ ]×

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¯ =: D(T¯ , S) ¯ for some T¯ ∈ [0, T ] and S¯ ∈ R (cf. formula (34) in Assumption (−∞, S] 3.3) then     ¯ , ∂p D = [0, T¯ ] × {S} ¯ ∪ {T¯ } × (−∞, S) ¯ . Intp D = [0, T¯ ) × (−∞, S) If D = [0, T¯ ] × R =: D(T¯ , +∞) for some T¯ ∈ [0, T ] then Intp D = [0, T¯) × R , ∂p D = {T¯} × R. Definition 3.7. Assume that we are given a closed domain D ⊆ [0, T ] × R and a continuous boundary condition b of class P on ∂p D. We then introduce the following obstacles problem (or variational inequality, (VI) for short) on Intp D )  max min − LΠ(t, S) − f (t, S, Π(t, S)), * (VI)  Π(t, S) − (t, S) , Π(t, S) − h(t, S) = 0 with the boundary condition Π = b on ∂p D, where L, , h, f are defined in (2), (24) and (25). Remarks 3.6. Note that the problem (VI) is defined over a domain in space variable S ranging to −∞, although only the positive part of this domain is meaningful for the financial purposes. Had we decided instead to pose the problem (VI) over bounded spatial domains then, in order to get a well-posed problem, we would need to impose some appropriate non-trivial boundary condition at the lower space boundary. The foregoing remarks, in which we deal with special cases of convertible securities, corresponds to Remarks 3.2. Remarks 3.7. (i) For a puttable security, we have that h = +∞ and thus the associated problem (VI) simplifies to   min − LΠ(t, S) − f (t, S, Π(t, S)), Π(t, S) − (t, S) = 0 (VI.1) with the boundary condition Π = b on ∂p D. (ii) For an elementary security, we also have that  = −∞ and thus the corresponding problem (VI) reduces to the linear parabolic PDE −LΠ(t, S) − f (t, S, Π(t, S)) = 0

(VI.2)

with the boundary condition Π = b on ∂p D. Let us state the definition of a viscosity solution to the problem (VI), which is required to handle potential discontinuities in time of f at the Ti s in case there are discrete coupons (cf. (25)). Given a closed domain D ⊆ [0, T ] × R, we denote, for i = 1, 2, . . . , I, Di = D ∩ {Ti−1 ≤ t ≤ Ti } , Intp D i = Intp D ∩ {Ti−1 ≤ t < Ti }. Note that the sets Intp Di provide a partition of Intp D.

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Definition 3.8. (i) A locally bounded upper semicontinuous function Π on D is called a viscosity subsolution of (VI) on Intp D if and only if Π ≤ h, and Π(t, S) > (t, S) implies −Lϕ(t, S) − f (t, S, Π(t, S)) ≤ 0 for any (t, S) ∈ Intp Di and ϕ ∈ C 1,2 (Di ) such that Π − ϕ is maximal on Di at (t, S), for some i ∈ 1, 2, . . . , I. (ii) A locally bounded lower semicontinuous function Π on D is called a viscosity supersolution of (VI) on Intp D if and only if Π ≥ , and Π(t, S) < h(t, S) implies −Lϕ(t, S) − f (t, S, Π(t, S)) ≥ 0 for any (t, S) ∈ Intp D i and ϕ ∈ C 1,2 (D i ) such that Π − ϕ is minimal on Di at (t, S), for some i ∈ 1, 2, . . . , I. (iii) A function Π is called a viscosity solution of (VI) on Intp D if and only if it is both a viscosity subsolution and a viscosity supersolution of (VI) on Intp D (in which case Π is a continuous function). Remarks 3.8. (i) In the case of a CS with no discrete coupons, the previous definitions reduce to the standard definitions of viscosity (semi-)solutions for obstacles problems (see, for instance, [17, 28]). (ii) A classical solution of (VI) on Intp D is necessarily a viscosity solution of (VI) on Intp D. (iii) A viscosity subsolution (resp. supersolution) Π of (VI) on Intp D does not need to verify Π ≥  (resp. Π ≤ h) on Intp D. A viscosity solution Π of (VI) on Intp D necessarily satisfies  ≤ Π ≤ h on Intp D. Building upon Definition 3.8, we introduce the following definition of P-(semi-)solutions to (VI) on D. Definition 3.9. By a P-subsolution (resp. P-supersolution, resp. P-solution) Π of (VI) on D for the boundary condition b, we mean a function of class P on Intp D, which is a viscosity subsolution (resp. supersolution, resp. solution) of (VI) on Intp D, and such that Π ≤ b (resp. Π ≥ b, resp. Π = b) pointwise on ∂p D. 3.3.1. Pricing and hedging through variational inequalities. In the following re? represents the state-process of the solution to the doubly sults, the process Π reflected BSDE (E) in Proposition 3.3. It thus depends, in particular, on the stopping time τ¯ representing the end of call protection period. Lemma 3.1 (No-protection price). Assume that τ¯ := τ¯0 = 0. Then the solution ? 0t = Π ? 0 (t, S+t ), where the to the doubly reflected BSDE (E) can be represented as Π ? 0 is a P-solution of (VI) on [0, T ] × R, with the terminal condition function Π 0 ? Π (T, S) = g(S), where g is given by (24). Proof. This follows by the application of the results from [18].



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Proposition 3.4. Let τ¯ be given by (34) for some constants T¯ ∈ [0, T ] and S¯ ∈ R+ ∪ {+∞}. (i) Post-protection price. On [¯ τ , T ], the solution to the doubly reflected BSDE 0 ? ? 0 (t, S+t ), where Π ? 0 is the function defined in (E) can be represented as Πt = Π Lemma 3.1; (ii) Protection price. On [0, τ¯], the solution to the reflected BSDE (E.1) can be ? 1t = Π ? 1 (t, S+t ), where the function Π ? 1 is a P-solution of the represented as Π ¯ and the boundary condition Π ?1 = Π ? 0 on ∂p D. problem (VI.1) on D = D(T¯ , S) Proof. In view of the observations made above, Lemma 3.1 immediately implies ?0 = Π ? 0 (¯ ?0 (i). In particular, we then have that Π τ , S+τ¯ ), where the restriction of Π τ¯ to ∂p D defines a continuous function of class P over ∂p D. Part (ii) then follows by the application of the results from [18].  We are in a position to state the following corollary to Propositions 3.2 and 3.4. Corollary 3.1. (i) Post-protection optimal exercise policies. The post-protection optimal put and call times (τp∗ , τc∗ ) after time t ∈ [0, T ] for the CS are given by , τp∗ = inf u ∈ [t, T ] ; (u, S+u ) ∈ Ep ∧ T, , τc∗ = inf u ∈ [t, T ] ; (u, S+u ) ∈ Ec ∧ T, where

, ? 0 (u, S) = (u, S) , Ep = (u, S) ∈ [0, T ] × R ; Π , ? 0 (u, S) = h(u, S) , Ec = (u, S) ∈ [0, T ] × R ; Π

are the post-protection put region and the post-protection call region, respectively. (ii) Protection optimal exercise policy. The protection optimal put time τp∗ after time t ∈ [0, T ] for the CS is given by , τp∗ = inf u ∈ [t, τ¯] ; (u, S+u ) ∈ E¯p , where

, ? 1 (u, S) = (u, S) E¯p = (u, S) ∈ [0, T ] × R ; Π

is the protection put region.



Assume that the call protection has not been lifted yet (t < τ¯) and that the CS is still alive at time t. Then an optimal strategy for the holder of the CS is to put the CS as soon as (u, S+u ) hits E¯p for the first time after t, if this event actually happens before τd ∧ τ¯. If we assume instead that the call protection has already been lifted (t ≥ τ¯) and that the CS is still alive at time t then:

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• an optimal call time for the issuer of the CS is given by the first hitting time of Ec by (u, S+u ) after t, provided this hitting time is realized before T ∧ τd ; • an optimal put policy for the holder of the CS consists in putting when (u, S+u ) hits Ep for the first time after t, if this event occurs before T ∧ τd . Remarks 3.9. Let us set (see Proposition 3.4) ? S+t ) = 1{t≤¯τ } Π ? 1 (t, S+t ) + 1{t>¯τ } Π ? 0 (t, S+t ) Π(t, + t with Π + =Π ? + A. It then follows from Proposition 3.2 that and let Πt = 1{t<τd } Π (Πt )t∈[0,T ] is the arbitrage price process of the CS and the issuer’s hedge with the ? 0 is furnished by initial wealth Π0 = Π , ? t = ht ∧ T τc∗ = inf t ∈ [¯ τ , T ]; Π and

' ( + t− Λt , ζt∗ = [ζt∗1 , ζt∗2 ] = 1{t≤τd } Zt , R(t, S+t ) − Π (36) where, as usual, Λ denotes the inverse of the matrix-valued process Σ and the process Z is given by the expression   ? 1 (t, S+t ) + 1{t>¯τ } ∂S Π ? 0 (t, S+t ) , Zt = σ(t, S+t )S+t 1{t≤¯τ } ∂S Π (37) ? 0 and Π ? 1 are where the last equality holds provided that the pricing functions Π sufficiently regular for the Itˆ o formula to be applicable. Recall that the process Σ is given either by (10) or by (13), depending on whether we choose a fixed-maturity ? CDS of Section 2.2 or a rolling CDS of Section 2.3 as a traded asset B. 3.3.2. Approximation Schemes for Variational Inequalities. We now come to the issues of uniqueness and approximation of solutions for (VI). For this, we make the following additional standing Assumption 3.4. The functions r, q, γ, σ are locally Lipschitz continuous. We refer the reader to Barles and Souganidis [3] (see also Cr´epey [18]) for the definition of stable, monotone and consistent approximation schemes to (VI) and for the related notion of convergence of the scheme, involved in the following Proposition 3.5. ? 0 introduced in Proposition 3.4(i) is the (i) Post-protection price. The function Π unique P-solution, the maximal P-subsolution, and the minimal P-super? 0 )h>0 denote a solution of the related problem (VI) on D = [0, T ] × R. Let (Π h ? 0. stable, monotone and consistent approximation scheme for the function Π 0 0 + ? ? Then Πh → Π locally uniformly on D as h → 0 . ? 1 introduced in Proposition 3.4(ii) is the (ii) Protection price. The function Π unique P-solution, the maximal P-subsolution, and the minimal P-super¯ Let (Π ? 1 )h>0 denote a solution of the related problem (VI.1) on D = D(T¯, S). h ? 1. stable, monotone and consistent approximation scheme for the function Π 1 1 + ? ? ¯ Then Πh → Π locally uniformly on D as h → 0 , provided (in case S < +∞) ?1 → Π ?1 = Π ? 0 on [0, T¯ ] × {S}. ¯ Π h

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Proof. Note, in particular, that under our assumptions: • the functions (r(t) − q(t) + ηγ(t, S))S and σ(t, S)S are locally Lipschitz continuous; • the function f admits a modulus of continuity in S, in the sense that for every constant c > 0 there exists a continuous function ηc : R+ → R+ with ηc (0) = 0 and such that, for any t ∈ [0, T ] and S, S  , x ∈ R with |S| ∨ |S  | ∨ |x| ≤ c, |f (t, S, x) − f (t, S  , x)| ≤ ηc (|S − S  |). The assertions are then consequences of the results in [18].



Remarks 3.10. In particular, we refer the reader to [18] in regard to the fact that the potential discontinuities of f at the Ti s (which represent a non-standard feature from the point of view of the classic theory of viscosity solutions as presented, for instance, in Crandall et al. [17]) are not a real issue in the previous results, provided one works with Definition 3.8 of viscosity solutions to our problems.

4. Convertible bonds As was already pointed out, a convertible bond is a special case of a convertible security. To describe the covenants of a typical convertible bond (CB), we introduce the following additional notation (for a detailed description and discussion of typical covenants of a CB, see, e.g., [2, 4, 35]): ¯ the par (nominal) value, N η the fractional loss on the underlying equity upon default, C the deterministic coupon process given by (22), ¯ R the recovery process on the CB upon default of the issuer at time t, given by ¯ t = R(t, ¯ St− ) for a continuous bounded function R, ¯ R κ the conversion factor, cb ¯ t : the effective recovery process, Rt = Rcb (t, St− ) = (1 − η)κSt− ∨ R cb ¯ ∨ κST + A(T ): the effective payoff at maturity; with A given by (23), ξ =N ¯ ¯ ¯ ≤ C, ¯ P ≤ C: the put and call nominal payments, respectively, such that P¯ ≤ N δ ≥ 0: the length of the call notice period (see below), tδ = (t + δ) ∧ T : the end date of the call notice period started at t. Note that putting a convertible bond at τp effectively means either putting or converting the bond at τp , whichever is best for the bondholder. This implies that, accounting for the accrued interest, the effective payment to the bondholder who decides to put at time t is Ptef := P¯ ∨ κSt + A(t). (38) As for calling, convertible bonds typically stipulate a positive call notice period δ clause, so that if the bond issuer makes a call at time τc , then the bondholder

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has the right to either redeem the bond for C¯ or convert it into κ shares of stock at any time t ∈ [τc , τcδ ], where τcδ = (τc + δ) ∧ T . If the bond has been called at time t then, accounting for the accrued interest, the effective payment to the bondholder in case of exercise at time u ∈ [t, (t+δ)∧T ] equals Ctef := C¯ ∨ κSt + A(t). (39) 4.1. Reduced convertible bonds A CB with a positive call notice period is rather hard to price directly. To overcome this difficulty, it is natural to use a two-step valuation method for a CB with a positive call notice period. In the first step, one searches for the value of a CB upon call, by considering a suitable family of puttable bonds indexed by the time variable t (see Propositions 4.7 and 4.8). In the second step, the price process obtained in the first step is used as the payoff at a call time of a CB with no call notice period, that is, with δ = 0. To formalize this procedure, we find it convenient to introduce the concept of a reduced convertible bond, i.e., a particular convertible bond with no call notice period. Essentially, a reduced convertible bond associated with a given convertible bond with a positive call notice period is an ‘equivalent’ convertible bond with no call notice period, but with the payoff process at call adjusted upwards in order to account for the additional value due to the option-like feature of the positive call period for the bondholder. Definition 4.1. A reduced convertible bond (RB) is a convertible security with coupon process C, recovery process Rcb and terminal payoffs Lcb , U cb , ξ cb such that (cf. (38)–(39)) ¯t , Rtcb = (1 − η)κSt− ∨ R

ef ¯ Lcb t = P ∨ κSt + A(t) = Pt ,

¯ ∨ κST + A(T ), ξ cb = N

and, for every t ∈ [0, T ], + cb (t, St ) + 1{t≥τ } Ctef , Utcb = 1{t<τd } U d

(40)

+ cb (t, S) jointly continuous in time and space variables, except for for a function U + cb (t, St ) ≥ C ef on the event negative left jumps of −ci at the Ti s, and such that U t {t < τd } (so Utcb ≥ Ctef for every t ∈ [0, T ]). 

The discounted dividend process of an RB is thus given by, for every t ∈ [0, T ],  t∧τd  cb βu dDu = βu c(u) du + βTi ci + 1{0≤τd ≤t} βτd Rτcbd . (41)

[0,t]

0

0≤Ti ≤t, Ti <τd

Clearly, a CB with no call notice period (i.e., with δ = 0) is an RB with the + cb (t, S) given by the formula U + cb (t, S) = C¯ ∨ κS + A(t). More generally, function U + cb in an RB is that U + cb represents the the financial interpretation of the process U value of the RB upon a call at time t. In Section 4.2, we shall formally prove that, under mild regularity assumptions in our model, any CB (no matter whether the call period is positive or not) can be interpreted and priced as an RB prior to call.

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In order to perform a deeper analysis of the bond and option features of a reduced convertible bond, it is useful to decompose an RB into the straight bond component, referred to as the embedded bond, and the option component, called the embedded game exchange option. 4.1.1. Embedded bond. For an RB with the dividend process Dcb given by (41), we consider an elementary security with the same coupon process as the RB and with the quantities Rb and ξ b given as follows: ¯t , Rtb = R

¯ + A(T ), ξb = N

(42)

so that

  ¯ t + ≥ 0, Rtcb − −Rtb = (1 − η)κSt − R

¯ )+ ≥ 0. ξ cb − ξ b = (κST − N

This elementary security corresponds to the defaultable bond with discounted cash flows given by the expression  T βt φ(t) = βu dDub + 1{τd >T } βT ξ b (43) t



T ∧τd

:=

βu c(u) du + t



βTi ci + 1{t<τd ≤T } βτd Rτbd + 1{τd >T } βT ξ b

t<Ti ≤T,Ti <τd

and the associated functions (cf. (24)–(25)) ¯ S) + Γ(t, S) − µ(t, S)x, f (t, S, x) = γ(t, S)R(t,

¯. g(S) = N

Definition 4.2. The RB with discounted cash flows given by (42)–(43) is called the bond embedded into the RB, or simply the embedded bond. It can be seen as the ‘straight bond’ component of the RB, that is, the RB stripped of its optional clauses. In the sequel, in addition to the assumptions made so far, we work under the following reinforcement of Assumption 3.4. ¯ S) and c(t) Assumption 4.1. The functions r(t), q(t), γ(t, S)S, σ(t, S)S, γ(t, S)R(t, are continuously differentiable in time variable, and thrice continuously differentiable in space variable, with bounded related spatial partial derivatives. Note that these assumptions cover typical financial applications. In particu¯ is constant and for well-chosen parameterizations of lar, they are satisfied when R σ and γ, which can be enforced at the time of the calibration of the model. Proposition 4.1. (i) In the case of an RB, the BSDE (E) (see part (ii) in Remarks 3.2) as? Z, K = 0). sociated with the embedded bond admits a unique solution (Φ, +=Φ ? + A, the embedded bond admits the unique arbitrage price Denoting Φ + t, Φt = 1{t<τd } Φ

t ∈ [0, T ].

(44)

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? t = Φ(t, ? S+t ) where the function Φ(t, ? S) is bounded, (ii) Moreover, we have that Φ jointly continuous in time and space variables, twice continuously differentiable in space variable, and of class C 1,2 on every time interval [Ti−1 , Ti ) ? S+t ) is an Itˆ (or [TI−1 , T ), in case i = I). The process Φ(t, o process with true martingale component; specifically, we have   ? t = µt Φ ? t − (γt Rb + Γt ) dt + σ(t, S+t )S+t ∂S Φ ? t dWt Φ t (45) = ut dt + vt dWt , where the process v belongs to H2 . ¯+ Proof. (i) By standard results (see, e.g., [25, 27]), the BSDE (E) with data (γ R ¯ ? Γ − µx, N ) admits a unique solution (Φ, Z, K = 0). Therefore, from Proposition 3.2 specified to the particular case of an elementary security, we deduce that the embedded bond admits a unique arbitrage price given by (44). (ii) The BSDE yields, for every t ∈ [0, T ], ) T     * ? ? u du + ξ b − A(T )  Ft Φt = EQ γu Rub + Γu − µu Φ t

or, equivalently, ? t = EQ αt Φ

)

T

     * αu γu Rub + Γu du + αT ξ b − A(T )  Ft .

(46)

t

Note that we have (cf. (23) and (28) with, by convention A(0−) = 0)   T    αT A(T ) = d(αu A(u)) = αu a(u) − µu A(u) du − αTi ci . [0,T ]

0

0≤Ti ≤T

+=Φ ? + A and (26), we obtain By plugging this into (46) and using the equalities Φ  * ) T     + αt Φt = EQ αu γu Rub + c(u) du + αTi ci + αT ξ b  Ft . t

Let us set

)

t<Ti ≤T

 *   ¯ + A(T ))  Ft , t ≤ T, αu γu Rub + c(u) du + αT (N (47) t    ? i = EQ αT ci  Ft , t ≤ Ti . αt Φ (48) t i +T = Φ ? 0 and Φ +t = Φ ?0 + ?j We have Φ t T j;Ti ≤Tj ≤T Φt on [Ti−1 , Ti ) (or on [TI−1 , T ) ? 0 or Φ ? i by Θ, ? in case i = I). Let us denote generically T or T i by T , and Φ ? is bounded. In as appropriate according to the problem at hand. Note that Θ ? t = Θ(t, ? S+t ), where Θ ? belongs addition, given our regularity assumptions, we have Θ 1,2 0 ? + to C ([0, T ) × R) ∩ C ([0, T ] × R) (see [27, 40]). Therefore, Φt = Φt − A(t) is given ? S+t ) for a function Φ(t, ? S), which is jointly continuous in (t, S) on [0, T ] × R by Φ(t, ? 0 = EQ αt Φ t

T

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and twice continuously differentiable in S on [0, T ) × R. Moreover, given (47)–(48) and the above C 1,2 regularity results, we have )  * ? 0t = µt Φ ? 0t − γt Rtb + c(t) dt + σ(t, S+t )S+t ∂S Φ ? 0 (t, S+t ) dWt , t < T, dΦ ? i dt + σ(t, S+t )S+t ∂S Φ ? i (t, S+t ) dWt , ? i = µt Φ dΦ t t t < Ti ∧ T, for i = 1, 2, . . . , I, dA(t) = ρ(t) dt,

t∈ / {Ti }i=0,1,...,I .

This yields

)  * ? S+t ) = µt Φ + t − γt Rb + c(t) + ρ(t) dt + σ(t, S+t )S+t ∂S Φ(t, + S+t ) dWt dΦ(t, t = ut dt + vt dWt .

? and u in (45) are bounded, we conclude that v ∈ H2 . Moreover, since Φ



4.1.2. Embedded game exchange option. The option component of an RB is formally defined as an RB with the dividend process Dcb − D b , payment at maturity cb ξ cb − ξ b , put payment Lcb t − Φt , call payment Ut − Φt and call protection lifting time τ¯, where Φ is the embedded bond price in (44). This can be formalized by means of the following definition. Definition 4.3. The embedded game exchange option is a zero-coupon convertible security with discounted cash flows, for any t ∈ [0, T ] and (τp , τc ) ∈ GTt × G¯Tt : βt ψ(t; τp , τc ) = 1{t<τd ≤τ } βτd (Rτcbd − Rτbd ) )   + 1{τd >τ } βτ 1{τ =τp
(49)

Note that from the point of view of the financial interpretation (see [4] for more comments), the game exchange option corresponds to an option to exchange the embedded bond for either Lcb , U cb or ξ cb (as seen from the perspective of the holder), according to which player decides first to stop this game prior to or at T. Also note that in the case of the game exchange option, there are clearly no coupons involved and thus the clean price and the price coincide. 4.1.3. Solutions of the doubly reflected BSDEs. The following auxiliary result can be easily proved by inspection. Lemma 4.1. Given an RB, the associated functions f (t, S, x), g = g(S),  = (t, S) and h = h(t, S) are: ¯ ∨ κS,  = P¯ ∨ κS and h = U + cb − A for the RB; • f = γRcb + Γ − µx, g = N cb b + ¯ ¯ ? + cb − A − Φ ? • f = γ(R − R ) − µx, g = (κS − N ) ,  = P ∨ κS − Φ and h = U for the embedded game exchange option.  We will now show how our results can be applied to both an RB and an embedded game exchange option.

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Proposition 4.2. (i) The data f, g, , h (and τ¯ given, as usual, by (34)) associated to an RB satisfy all the assumptions of Propositions 3.4–3.5 (ii) The BSDEs (E) related to an RB or to the embedded game exchange option have unique solutions. Proof. (i) This can be verified directly by inspection of the related data in Lemma 4.1 (we are in fact in the situation of Example 3.1). (ii) Given part (i), the BSDE (E) related to an RB has a unique solution ? V, K), by a direct application of Proposition 3.3. Now, (Φ, ? Z, 0) denoting the (Π, solution to the BSDE (E) exhibited in Proposition 4.1(i), it is immediate to check ? Y, K) solves the game exchange option-related problem (E) iff (Φ ? + Ψ, ? Z+ that (Ψ, Y, K) solves the RB-related problem (E). Hence the result for the game exchange option follows from that for the RB.  ? and Given an RB and the embedded game exchange option, we denote by Π ? the state-processes (i.e., the first components) of solutions to the related BSDEs. Ψ The following result summarizes the valuation of an RB and the embedded game exchange option. Proposition 4.3. ? t is the unique arbitrage price of the (i) The process Ψt defined as 1{t<τd } Ψ embedded game exchange option and (Ψt , ζ ∗ , τc∗ ) (resp. (−Ψt , −ζ ∗ , τp∗ )) as defined in Proposition 3.2 is an issuer’s hedge with initial value Ψt (resp. holder’s hedge with initial value −Ψt) starting from time t for the embedded game exchange option. + t , with Π + := Π+ ? A, is the unique arbitrage (ii) The process Πt defined as 1{t<τd } Π ∗ ∗ price of the RB, and (Πt , ζ , τc ) (resp. (−Πt , −ζ ∗ , τp∗ )) as defined in Proposition 3.2 is an issuer’s hedge with initial value Πt (resp. holder’s hedge with initial value −Πt ) starting from time t for the RB. ? and Φ defined as in Proposition 4.1, we have that Π = Φ + Ψ and (iii) With Φ ? ? ? Π = Φ + Ψ. Proof. Given Proposition 4.2, statements (i) and (ii) follow by an application of Proposition 3.2., Part (iii) is then a consequence of the general results of [4].  4.1.4. Variational inequalities for post-protection prices. We consider the following problems (VI) (for a game exchange option or an RB) or (VI.2) (for the defaultable bond) on D = [0, T ] × R : • for a defaultable bond ? + µΦ ? − (γRb + Γ) = 0, −LΦ ? ¯ Φ(T, S) = N,

t < T, (50)

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283

• for a game exchange option ) )    * ? + µΨ ? − γ Rcb − Rb , Ψ ? − P¯ ∨ κS − Φ ? , max min −LΨ  cb * ?− U + −A−Φ ? = 0, Ψ

t < T,

? ¯ )+ , Ψ(T, S) = (κS − N • for an RB ) ) *   ? + µΠ ? − γRcb + Γ , Π ? − P¯ ∨ κS , max min −LΠ  cb * ?− U + − A = 0, Π ? ¯ ∨ κS. Π(T, S) = N

(51)

t < T, (52)

? the state-process (i.e., Convention. In what follows, we denote generically by Θ the first component) of the solution to the BSDE related to an RB, the embedded game exchange option or the embedded bond, as is appropriate for the problem at hand. Proposition 4.4 (Post-protection prices). For any of problems (50)–(52) there ex? S), which determines the corists a P-solution on D, denoted generically as Θ(t, responding post-protection price, in the sense that ? t = Θ(t, ? S+t ), Θ

t ∈ [¯ τ , T ].

(53)

Moreover, we have uniqueness of the P-solution and any stable, monotone and ? converges locally uniformly to Θ ? on D as consistent approximation scheme for Θ h → 0+ . In the case of the RB and the embedded game exchange option, the postprotection put/conversion region and the post-protection call/conversion region are given as , ? S) = P¯ ∨ κS , Ep = (u, S) ∈ [0, T ] × R ; Π(u, , ? + cb (u, S) − A(u) . Ec = (u, S) ∈ [0, T ] × R ; Π(u, S) = U Proof. In the case of the RB or of the embedded bond, the results follow by direct application of Propositions 4.2, 3.4(i), 3.5(i) and Corollary 3.1(i). Now, given that ? and Φ ? are P-solutions to (52) and (50), respectively, and in view of the regularity Π ? stated in Proposition 4.1(ii), therefore Ψ ? := Π ? −Φ ? is a P-solution properties of Φ ? and Φ ? satisfy the related identities (53), then so does Ψ, ? in view of to (51). Since Π Proposition 4.3(iii). Finally, given the last statement in Proposition 3.5, the game exchange option also satisfies the claimed uniqueness and convergence results.  4.1.5. Variational inequalities for protection prices. We now deal with the follow¯ where the functions Φ, ? Ψ, ? Π ? are solutions to ing problems (VI.1) on D = D(T¯ , S), (54)–(55):

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• for a game exchange option )    * ¯ + µΨ ¯ − γ Rcb − Rb , Ψ ¯ − P¯ ∨ κS − Φ ? = 0 on Intp D, min −LΨ ¯ =Ψ ? on ∂p D, Ψ

(54)

• for an RB ) *   ¯ + µΠ ¯ − γRcb + Γ , Π ¯ − P¯ ∨ κS = 0 on Intp D, min − LΠ ¯ =Π ? on ∂p D. Π

(55)

Proposition 4.5 (Protection prices). For any of the problems (54)–(55) there exists ¯ that determines the corresponding a P-solution on D, denoted generically as Θ, protection price, in the sense that ? t = Θ(t, ¯ S+t ), t ∈ [0, τ¯]. Θ Moreover, we have uniqueness of the P-solution and any stable, monotone and ¯ converges locally uniformly to Θ ¯ on D as consistent approximation scheme for Θ + ¯ ¯ ? ¯ h → 0 , provided (in case S < +∞) it converges to Θ(=Θ) at S. In the case of the RB and the embedded game exchange option, the protection put/conversion region is given as , ¯ E¯p = (u, S) ∈ [0, T ] × R ; Π(u, S) = P¯ ∨ κS . Proof. In the case of the RB, the results follow by direct application of Propositions 4.2, 3.4(ii), 3.5(ii) and Corollary 3.1(ii). In the case of the game exchange option, ? denoting we proceed by taking the difference, as in the proof of Proposition 4.4 (Φ the same function as before).  4.2. Convertible bonds with a positive call notice period We now consider the case of a convertible bond with a positive call notice period. Note that between the call time t and the end of the notice period tδ = (t + δ) ∧ T , a CB actually becomes a CB with no call clause (or puttable bond ) over the time interval [t, tδ ], which is a special case of a puttable security (cf. Definition 3.2; formally, we set τ¯ = tδ in the related BSDE). For a fixed t, this puttable bond, denoted henceforth as the t-PB, has the effective payment given by the process Cuef , u ∈ [t, tδ ] for the original CB (see (39)). Lemma 4.2. In the case of the t-PB, the associated functions f (u, S, x), g = g(S) and  = (u, S) are (h = +∞ in all three cases below): • embedded bond (called the t-bond from now on): ¯ and (u, S) = −∞; f (u,S,x) = γ(u,S)Rb (u,S) + Γ(u,S) − µ(u,S)x, g(S) = C • embedded game exchange option (called the t-game exchange option from now on): f (u, S, x) = γ(u, S)(Rcb − Rb )(u, S) − µ(u, S)x, g(S) = C¯ ∨ κS − ? t (tδ , S) and (u, S) = C¯ ∨ κS − Φ ? t (u, S), where Φ ? t is the pricing function Φ of the t-bond (obtained by an application of Proposition 4.4, see also (56) below);

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• t-PB: f (u, S, x) = γ(u, S)Rcb(u, S) + Γ(u, S) − µ(u, S)x, g(S) = C¯ ∨ κS and (u, S) = C¯ ∨ κS. Note that in view of the proof of Proposition 4.7 below, it is convenient to define the related pricing problems on [0, tδ ] × R, rather than merely on [t, tδ ] × R. Specifically, given t ∈ [0, T ], we define the following problems (VI) on [0, tδ ] × R : • for the t-bond   ? t + µΦ ? t − γRb + Γ = 0, u < tδ , −LΦ ? t (tδ , S) = C, ¯ Φ

(56)

• for the t-game exchange option )   t  * ? t + µΨ ? t − γ Rcb − Rb , Ψ ? − C¯ ∨ κS − Φ ? t = 0, min −LΨ

u < tδ ,

? t (tδ , S) = C¯ ∨ κS − Φ ? t (tδ , S), Ψ • for the t-PB ) *   t ? t + µΠ ? t − γRcb + Γ , Π ? − C¯ ∨ κS = 0, min −LΠ

(57) u < tδ ,

? t (tδ , S) = C¯ ∨ κS. Π

(58)

Proposition 4.6. For any of problems (56)–(58), the corresponding BSDE (E) has a ? t can be represented as Θ ? t (u, S+u ), where solution, and the related t-price process Θ u t δ ? the function Θ is a P-solution of the related problem (VI) on [0, t ]×R. Moreover, the uniqueness of the P-solution holds and any stable, monotone and consistent ? t converges locally uniformly to Θ ? t on [0, tδ ] × R as approximation scheme for Θ + h → 0 . In the case of the t-PB and the t-game exchange option, the protection put/conversion region is given as , ? t (u, S) = C¯ ∨ κS . Ept = (u, S) ∈ [t, tδ ] × R ; Π Proof. In view of Lemma 4.2, the assertion follows by an application of Proposition 4.4.  ? (t, S) := Proposition 4.7 (Continuous aggregation property). The function U t ? Π (t, S) is jointly continuous in time and space variables. Hence the function + (t, S) = U ? (t, S) + A(t) is also continuous with respect to (t, S), except for left U jumps of size −ci at the Ti s. Proof. Let (tn , Sn ) → (t, S) as n → ∞. We decompose ? tn (tn , Sn ) = Π ? t (tn , Sn ) + (Π ? tn (tn , Sn ) − Π ? t (tn , Sn )). Π ? t (tn , Sn ) → Π ? t (t, S) as n → ∞. Moreover, denoting C ?t = By Proposition 4.6, Π cb C¯ ∨ κS+t and F = γR + Γ, we have that  * )  τp t ? ?τp  Fu , u ≤ tδ . αu Πu = esssupτp ∈F uδ EQ αv Fv dv + ατp C t

u

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So, assuming tn sufficiently close to the left of t, and in view of the Markov property + we obtain, on the event {S+tn = Sn }, of the process S,  )  τp * ? tn (tn , Sn ) = esssup ?τ  Ft tn EQ αtn Π C α F dv + α v v τ p p n τp ∈F δ tn tn  )  τp * ?τ  Ft = αt Π ? t (tn , Sn ). ≤ esssupτp ∈F tn EQ αv Fv dv + ατp C p n n tδ

tn

Fttδn ,

Conversely, for any τp ∈ we have τpδ := τp ∧ tδn ∈ Fttδn , 0 ≤ τp − τpδ ≤ t − tn n and  τpδ   τp   ?τp − ?τ δ  αv Fv dv + ατp C αv Fv dv − ατpδ C  p tn tn  τp   ?τp − ατ δ C ?τ δ . ≤ αv |Fv | dv + ατp C p p τpδ

Therefore,  )  EQ

  * )  τpδ * ?τp  Ftn − EQ ?τ δ  Ftn  αv Fv dv + ατp C αv Fv dv + ατpδ C p tn tn  )  τp * ) *   ?τp − ατ δ C ?τ δ   Ftn ≤ EQ αv |Fv | dv  Ftn + EQ ατp C p p τp

τpδ

) *  √ ?τp − ατ δ C ?τ δ   Ftn ≤ c t − tn F H2 + EQ ατp C p p ? tn (tn , Sn )−Π ? t (tn , Sn ) → 0 as tn → t− . But for some constant c. We conclude that Π + ? tn (tn , Sn )− Π ? t (tn , Sn ) → this is also true, with the same proof, as tn → t . Hence Π t t ? n (tn , Sn ) → Π ? (t, S) as tn → t, as desired.  0 as tn → t. Finally, Π The next result shows that a CB can be formally reduced to the corresponding RB. Proposition 4.8. A CB with a positive notice period δ > 0 can be interpreted as an + cb(t, S) = U + (t, S), where U + (t, S) is the function defined in Proposition RB with U 4.7, so that (cf. (40)) + (t, St ) + 1{τ ≤t} (C¯ ∨ κSt + A(t)). Utcb = 1{τ >t} U (59) d

d

Proof. The t-PB related reflected BSDE (E) has a solution, and thus, by Propo+ tu with sition 3.2, the t-PB has a unique arbitrage price process Πtu = 1{u<τd } Π +t = Π ? t + A(u). Hence the arbitrage price of the CB upon call time t (assuming Π u u + t = U(t, + S+t ) (cf. Proposithe CB still alive at time t) is well defined, as Πtt = Π t tion 4.7). + S) is jointly continuous in Moreover, by Proposition 4.7, the function U(t, time and space, except for negative left jumps of −ci at the Ti s, and we also have that Πtt ≥ C¯ ∨ κSt + A(t) on the event {τd > t}. Hence U cb defined as (59) satisfies all the requirements in (40). 

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Conclusion. An important conclusion from Proposition 4.8 is that the results of Section 4.1 are applicable to a CB also in the case of a positive call notice period, since, in the Markovian model of the present paper, a CB may always be interpreted as an RB. 4.3. Numerical analysis of a convertible bond The remaining part of the paper is devoted to a numerical study of a convertible bond within the present Markovian framework. In particular, we examine the price decomposition into embedded bond and game exchange option, the behavior of hedge ratios, the impact of the call protection period, as well as the values of implied spread and implied volatility of a convertible bond. 4.3.1. Numerical issues. Let us first comment briefly on numerical issues related to the valuation of a convertible bond and of its bond and option components. Assume that τ¯ = τ¯0 := 0 (no call protection) and that we have already specified all the parameters for one of the problems (50), (51) or (52), including, + cb . in the case of (51) or (52), the function U By Proposition 3.5 and the results of Sections 4.1, standard deterministic approximation schemes (see, e.g., [2, 35]) converge towards the P-solution of the problem as the discretization step goes to 0. Solving the PDEs related to the embedded bond is complete routine and thus we shall not comment on this issue. To have a fully endogenous specification of the problem in the case of the + cb(t, S) = U + (t, S) as convertible bond and/or its option component, one can take U + defined in Proposition 4.7 in (51) or (52), where U (t, S) is numerically computed by solving the related obstacles problems, using Proposition 4.6. We provide be+ cb (t, S) = U + (t, S), using, low a practical algorithm for solving, say (52), with U for example, a fully implicit finite difference scheme (see, for instance, [39]) to discretize L : 1. Localize problems (58) for the embedded t-PBs and problem (52) for the CB. A natural choice, for the t-PBs and the CB, is to localize the problems on ¯ the spatial domain (−∞, Cκ ], with a Dirichlet boundary condition equal to ¯ κS (or a Neumann boundary condition equal to κ) at level Cκ ; ¯ 2. Discretize the localized domain D lo = [0, T ] × (−∞, C κ ], using, say, one time step per day between 0 and T ; 3. Discretize problems (58) for the embedded t-PBs on the subdomain [t, tδ ] of Dlo for t in the time grid (one problem per value of t in the time grid); ? t the discretized problems (58) corresponding to the embedded 4. Solve for Π t-PBs for t in the time grid (one problem per value of t in the time grid); 5. Discretize problem (52) for the CB on Dlo and solve the discretized problem, + (t, S) := Π ? t (t, S) + A(t) as an input using the numerical approximation of U + cb (t, S) in (52). for U Note that the problem for the t-PB only has to be solved on the time-strip [t, tδ ] of Dlo . Hence the overall computational cost for solving a CB problem (52)

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with a positive call notice period is roughly the same as that required for solving one CB problem without call notice period, plus the cost of solving n PB problems that would be defined on the whole grid, where n is the number of time mesh points in the call notice period. For instance, for a call notice period δ equal to one month and a time step of one day, we have n = 30. Finally, if a call protection is in force then we proceed along essentially the same lines, using the results of Section 4.1.5. 4.3.2. Embedded bond and game exchange option. On Figure 1,1 we plot the ? 0 (0, S) of a convertible bond with no call protection, its bond component price Π and its game exchange option component, obtained through the method described in Section 4.3.1, as a function of the stock level S at time t = 0, in the simple case where there are no dividends on the convertible bond (neither coupons nor recovery), and for the remaining parameters as given in Table 1. Note that we take here η = 0; this corresponds to no jump of the stock price at default. For each of the three products, we plot the curves corresponding to default intensities of the form γ(t, S) = γ0 ( SS0 )γ1 where γ0 = 0.02, and γ1 equals either 1.2 (strong interaction between equity and credit), or 0 (no interaction between equity and credit). The corresponding curves are labeled local (red/solid curves) and implied (blue/dashed curves), respectively. Note that in case γ1 = 1.2, consistently with typical market data, the price of the CB as a function of S exhibits the so-called ski-jump behavior, namely, it is convex for high values of S and collapsing at the low values. This behavior at low levels of S comes from the collapse of the bond component of a CB (‘collapse of the bond floor’ at low S’s in case γ1 > 0 cf. the red/solid local embedded bond curve on Figure 1). r

q

η

5% 0

0

T

P¯

20% 5y

0

σ

¯ N

C¯

100 103

κ 1

Table 1. Parameter values (all dividends set to 0).

? 0 (0, S) as a func4.3.3. Hedge ratios. Figure 2 shows the no-protection delta ∂S Π tion of S for the data of the previous subsection. Note that in the present context ? 0 (t, S+t ). As we have that τ¯ = 0 and thus formula (37) reduces to Zt = σ S+t ∂S Π explained in Remarks 3.9, the process Z is the essential ingredient in the computation of the hedge ratios [ζt∗1 , ζt∗2 ] with respect to traded primary assets (see, in particular, formula (36)). Under the present assumptions that η = 0 and A ≡ 0 1 We

thank Abdallah Rahal from the Mathematics Departments at University of Evry, France, and Lebanese University, Lebanon, for numerical implementation of the model.

Convertible Bond

160 140 120 100 80 60 40 20 0

embedded Bond

80 70 60 50 40 30 20 10 0

embedded Option

Convertible Bonds in a Defaultable Diffusion Model

90 80 70 60 50 40 30 20 10 0

289

local implied 0

20

40

60

80

100

120

140

160

140

160

140

160

local implied 0

20

40

60

80

100

120

local implied 0

20

40

60

80 Stock

100

120

Figure 1. The ski-jump diagram and its decomposition. +0 = Π ? 0 is continuous), after straightforward computations (so that the process Π we obtain ) * +0 + + + 0 (t, S+t ) + Π (t, St ) − R(t, St ) ∂S B(t, + S+t ) ζt∗1 = 1{t≤τd } ∂S Π (60) + S+t ) ν(t) − B(t, and ζt∗2 = 1{t≤τd }

+ 0 (t, S+t ) R(t, S+t ) − Π . + S+t ) ν(t) − B(t,

(61)

Equality (61) is rather clear since for η = 0 the stock price does not jump at default and thus jump risk should be hedged using a fixed-maturity CDS only. Formula (60) shows that in order to find the hedge ratio with respect to the stock + 0 (t, S+t ) using the specific features of price, one needs to adjust the usual delta ∂S Π the CDS used for hedging. + 0 (0, S) by the finite difSince the computation of the partial derivative ∂S Π ference approximation on the grid requires more accuracy than that of the prices, we only represented the values of deltas for S ‘not too close’ from the boundaries of the computational domain (specifically, on the subset 10 ≤ S ≤ 140 of the computational domain 5 ≤ S ≤ 160), where numerical boundary effects are not too important. Observe that in the case of the convertible bond and of the embedded op+ 0 (0, S) and ∂S Ψ(0, S) exhibit a rather irregular tion, the corresponding deltas ∂S Π behavior (vary faster) in a small zone around S = Sc ≈ 99 (in the case γ1 = 1.2) or S = Sc ≈ 103 (in the case γ1 = 0). For greater values of S both deltas are equal

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embedded Option

embedded Bond

Convertible Bond

to one; this behavior is consistent, in particular, with the fact (already present, but less easy to see, on Figure 1) that, in view of the values of the parameters in Table 1 and since there are no coupons involved, the price of the convertible bond is equal to P¯ ∨ κS = C¯ ∨ κS = S ¯ for any S ≥ C = 103. 3

local implied

2.5 2 1.5 1 0.5 0 −0.5

0

20

40

60

80

100

120

140

3

local implied

2.5 2 1.5 1 0.5 0 −0.5 1.2 1 0.8 0.6 0.4 0.2 0 −0.2

0

20

40

60

80

100

120

140

local implied

0

20

40

60

80

100

120

140

Stock

Figure 2. Deltas corresponding to the prices of Figure 1. 4.3.4. Separation of credit and volatility risks. At the intuitive level, it seems clear that the embedded bond should concentrate most of the interest rate and credit risks of a convertible bond, whereas the value of the embedded game exchange option should explain most of the volatility risk (note in this respect that the embedded game exchange option always has a null coupon process). In the specific context of the model of this paper, this intuition can be assessed numerically. In what follows, we consider the case when η = 1, which corresponds to the total default of the stock. Figure 3 shows the prices of a convertible bond and of its bond and option components as functions of the volatility parameter σ for a local default intensity γ(t, S) = γ0 ( SS0 )γ1 with γ0 = 0.02 and γ1 = 1.2. The embedded bond price varies between 69.19 and 72.98, whereas the embedded option price grows from 9.52E-05 to 8.20, as σ increases from 3% to 61%. Moreover, the embedded option price increases with σ, whereas the prices of the convertible bond and of the embedded bond exhibit a more complex behavior. Figure 4 shows the same prices as functions of the parameter γ = γ0 (constant default intensity) for the volatility σ = 20%. The embedded bond price decreases

Convertible Bonds in a Defaultable Diffusion Model 80

291

9 embedded Option

Convertible Bond embedded Bond

8

78 7 76

6 5

74 4 3

72

2 70 1 68

0 0

0.1

0.2

0.3 0.4 sigma

0.5

0.6

0.7

0

0.1

0.2

0.3 0.4 sigma

0.5

0.6

0.7

Figure 3. Price of a convertible bond and of its option and bond components as a function of a constant volatility parameter σ (S = 100, η = 1, r = 5%, q = 0, γ0 = 0.02, γ1 = 1.2). from 97.72 to 60.89, whereas the embedded option price only rises from 0.0031 to 2.59, as γ0 increases from 0.01 to 0.29. 4.3.5. Call protection period. The diversity and combination of all the possible clauses can give rise to a virtually infinite variation of situations. We shall thus not push much further the numerical analysis of convertible bonds in this paper, 100

3 Convertible Bond embedded Bond

95

embedded Option 2.5

90 2

85 80

1.5

75

1

70 0.5

65 60

0 0

0.05

0.1

0.15 0.2 gamma

0.25

0.3

0

0.05

01

0.15 0.2 gamma

0.25

0.3

Figure 4. Price of a convertible bond and of its option and bond components as a function of a constant default intensity parameter γ = γ0 (S = 100, η = 1, r = 5%, q = 0, γ1 = 0, σ0 = 20%).

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referring the interested reader to, e.g., the studies by Ayache et al. [2] and Kwok and Lau [35]. As for the impact of the call protection, in particular, we shall only present ? 1 (0, S) here few preliminary results in Table 3, which shows protection CB prices Π ¯ ¯ for τ¯ of the form (34), with T = T therein, for various pairs (S, S). These prices were computed by the deterministic numerical scheme of Section 4.1.5, using the general data of Table 2 (in particular, a discrete coupon tenor is treated numerically by the transformation into clean prices). The protection prices are increasing with ¯ which means that a longer call protection period makes a convertible respect to S, bond more expensive to the investor, as expected. Table 2. General data for the numerical results of Table 3. T

P

N

C

R

6m

0

100 103

0

ci

c

r

q

η

1.2/month 0 5% 0

1

γ0

γ1

0.02 1.2

σ 20%

Note that real-life lifting times of call protections are typically highly path¯ dependent, e.g. (given a trigger level S), τ¯ = inf { t > 0 , Sti ≥ S¯ at d among l consecutive monitoring times ti s }. For such call protections, or, more generally, for contracts with highly pathdependent features, deterministic numerical schemes as above are ruled out by the curse of dimensionality, and simulation methods appear to be the only viable alternative. The related simulation methods, which can be viewed as extensions to game problems of simulation methods for American options [36, 37, 44], correspond more generally to simulation numerical schemes for reflected BSDEs [10, 15, 38]. The possibility to effectively deal with various kinds of call protection through simulation pricing schemes is currently under study in [16]. Table 3. Time-0 protection price of the convertible bond of Table 2 ¯ for various pairs (S, S). S¯ = 80

S

78.55

79.55

? 1 (0, S) 103.18 103.11 Π S¯ = 103

S

80.55

81.55

103

103

100.55 101.55 102.55 103.55

? 1 (0, S) 103.87 103.78 103.69 103.55 Π S¯ = 120

S

100.55 101.55 102.55 103.55

? 1 (0, S) 110.32 110.89 111.48 112.09 Π

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It is interesting to note that besides computational tractability in highdimension, simulation methods also present the theoretical interest of being amenable to a more complete convergence analysis than deterministic schemes. For instance, one can deal reasonably well with the issue of the numerical approximation by time discretization and simulation of the component Z of the solution to a BSDE, whereas in a PDE viscosity solution set-up underlying the convergence analysis of a deterministic scheme, this is more difficult. This is because the representation (cf. (37))   ? 1 (t, S+t ) + 1{t>¯τ } ∂S Π ? 0 (t, S+t ) Zt = σ(t, S+t )S+t 1{t≤¯τ } ∂S Π ? 0 , unless further regularity is only tentative for a continuous pricing function Π 0 ? estimates for the solution Π may be established, or other notions of weak solutions to the pricing PDEs may be considered. 4.3.6. Implied credit spread and implied volatility. As argued in [4], in the context of hybrid equity-and-credit derivatives like convertible bonds, the standard notion of a (default-free) Black–Scholes implied volatility is not adequate. One should use instead a suitable notion of implied credit spread and volatility (two implied numbers rather than a single one), defined relatively to the prices of two financial contracts (rather than one, as for the Black–Scholes implied volatility), and in reference to a suitable benchmark model of equity-to-credit derivatives (rather than the Black–Scholes model). In this regard, the decomposition of a convertible bond into the embedded bond and the game exchange option, which was established in [4] and is further studied in this paper in the context of model (1), allows one to give a rigorous definition of implied spread and volatility of a convertible bond, in terms of a special case of model (1) with a constant default intensity γ(t, S) ≡ γ0 and a constant equity volatility σ(t, S) ≡ σ0 , that is, ) *  dS+t = S+t r(t) − q(t) + ηγ0 dt + σ0 dWt . (62) Let us thus recall from [4] the following definition of the implied spread and the implied volatility of a convertible bond at time 0, say, and for already known deterministic short-term risk-free interest-rate curve r(t), the dividend yield curve q(t), and the fractional loss parameter η upon default. Definition 4.4. Given time-0 arbitrage prices Φ0 of the bond component and Ψ0 of the game exchange option component of a convertible bond, the implied credit spread and implied volatility at time 0 of the convertible bond are defined as any pair of constant positive parameters (γ0 , σ0 ) that are consistent with the prices Φ0 and Ψ0 , in the sense that the price at time 0 of the bond component (resp. the game exchange option component) in model (62) is equal to Φ0 (resp. Ψ0 ). Figure 5 shows the implied credit spread and the implied volatility of the convertible bond as a function of S0 for the data of Table 1 (except that we now set η = 1). This means that the ‘observed’ prices Φ0 and Ψ0 are in fact computed

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0.6

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0.5

0.4

0.3

0.2

0.1

0 20

30

40

50 Stock

60

70

80

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Figure 5. Implied credit spread and implied volatility of a convertible bond at time 0 as a function of S0 (γ0 = 0.02, γ1 = 1.2). using model (1) with the local default intensity γ(t, S) = γ0 ( SS0 )γ1 where γ0 = 0.02 and γ1 = 1.2 and a constant volatility σ = 20%. Remarks 4.1. In view of the no-arbitrage relationship Φ0 + Ψ0 = Π0 , a formally equivalent definition would be that of a pair (γ0 , σ0 ) consistent with the prices Π0 and Φ0 of the convertible bond and its embedded bond in a model (1) with a constant default intensity γ0 and a constant equity volatility σ0 . We have argued in Section 4.3.2, however, that the embedded bond concentrates most of the interest rate and credit risks (cf. Figure 4), whereas the value of the embedded game exchange option explains most of the volatility risk (cf. Figure 3). It is thus more natural to use the prices of the embedded bond and game exchange option to infer the implied credit spread and the implied volatility. 4.3.7. Calibration issues. A further numerical issue is the calibration of the model, which consists in fitting some specific parameters of the model, such as the local volatility σ and the local intensity γ in our case, to market quotes of related liquidly traded assets. Various sets of input instruments can be used in this calibration process, including traded options on the underlying equity and/or CDSs related to bond issues of the reference name. The specification of a pertaining calibration procedure typically depends on the specification of the model at hand (non-parametric model with arbitrary local volatility and/or default intensity functions σ and γ or versus parameterized subclasses of models with specific parametric forms for σ and γ), as well as on the nature of the calibration input instruments.

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We shall not dwell more on calibration issues in the present paper, referring instead the reader to Andersen and Buffum [1] for interesting developments and numerical algorithms in this regard. Let us only observe that, as it can be seen in Figure 1, the price of the embedded game exchange option enjoys much better properties than the price of the CB in terms of convexity with respect to the stock price, and thus in turn (see Figure 3) in terms of monotonicity with respect to the volatility. Moreover, as we have argued above, the embedded bond concentrates most of the interest rate and credit risks of a convertible bond, whereas the value of the embedded game exchange option explains most of the volatility risk. These features suggest that for the calibration purposes it could be advantageous to use the (synthetic) prices of the embedded option and of the embedded bond, rather than those of the CB and of the embedded bond (in a situation where the related market prices would be available or could be reconstructed ‘synthetically’ in terms of prices of liquidly traded bonds and/or other traded instruments). Acknowledgment • The research of T.R. Bielecki was supported by NSF Grant 0202851 and Moody’s Corporation grant 5-55411. • The research of S. Cr´epey benefited from the support of Ito33, the ‘Chaire Risque de cr´edit’ and the Europlace Institute of Finance. • The research of M. Jeanblanc was supported by Ito33, the ‘Chaire Risque de cr´edit’ and Moody’s Corporation grant 5-55411. • The research of M. Rutkowski was supported by the ARC Discovery Project DP0881460.

References [1] Andersen, L. and Buffum, D.: Calibration and implementation of convertible bond models. Journal of Computational Finance 7 (2004), 1–34. [2] Ayache, E., Forsyth, P. and Vetzal, K.: Valuation of convertible bonds with credit risk. Journal of Derivatives 11 (2003), 9–29. [3] Barles, G. and Souganidis, P.E.: Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Analysis 4 (1991), 271–283. [4] Bielecki, T.R., Cr´ epey, S., Jeanblanc, M. and Rutkowski, M.: Arbitrage pricing of defaultable game options with applications to convertible securities. Quantitative Finance 8 (2008), 795–810. [5] Bielecki, T.R., Cr´ epey, S., Jeanblanc, M. and Rutkowski, M.: Valuation and hedging of defaultable game options in a hazard process model. Journal of Applied Mathematics and Stochastic Analysis, Article ID 695798, 2009. [6] Bielecki, T.R., Cr´ epey, S., Jeanblanc, M. and Rutkowski, M.: Defaultable options in a Markovian intensity model of credit risk. Mathematical Finance 18 (2008), 493–518.

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[7] Bielecki, T.R., Jeanblanc, M. and Rutkowski, M.: Pricing and trading credit default swaps in a hazard process model. Annals of Applied Probability 18 (2008), 2495–2529. [8] Bielecki, T.R. and Rutkowski, M.: Credit Risk: Modeling, Valuation and Hedging. Springer-Verlag, Berlin, 2002. ¨ rk, T.: Arbitrage Theory in Continuous Time. Oxford University Press, Oxford, [9] Bjo 1998. [10] Bouchard, B. and Chassagneux, J.-F.: Discrete-time approximation for continuously and discretely reflected BSDEs (2008). Stochastic Processes and their Applications, 118(12), 2269–2293. [11] Bouchard, B. and Touzi, N.: Discrete-time approximation and Monte Carlo simulation of backward stochastic differential equations. Stochastic Processes and their Applications 111(2) (2004), 175–206. [12] Brennan, M.J. and Schwartz, E.S.: Convertible bonds: valuation and optimal strategies for call and conversion. Journal of Finance 32 (1977), 1699–1715. [13] Brennan, M.J. and Schwartz, E.S.: Analyzing convertible bonds. Journal of Financial and Quantitative Analysis 15 (1980), 907–929. [14] Carr, P. and Linetsky, V.: A jump to default extended CEV model: an application of Bessel processes. Finance and Stochastics 10 (2006), 303–330. [15] Chassagneux, J.-F.: A discrete-time approximation for doubly reflected BSDE (2009). Advances in Applied Probability, 41. ´pey, S., and Rahal, A.: Pricing convertible bonds with [16] Chassagneux, J.F., Cre call protection. In preparation. [17] Crandall, M., Ishii, H., and Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bulletin of the American Mathematical Society 27(1) (1992), 1–67. ´pey, S.: About the pricing equations in finance (2011). Paris-Princeton Lectures [18] Cre in Mathematical Finance 2010, Lecture Notes in Mathematics, Springer, 63–203, 2011. [19] Cr´ epey, S. and Matoussi, A.: Reflected and doubly reflected BSDEs with jumps: A priori estimates and comparison principle. Annals of Applied Probability 18 (2008), 2041–2069. ´, J. and Karatzas, I.: Backward stochastic differential equations with [20] Cvitanic reflection and Dynkin games. Annals of Probability 24 (1996), 2024–2056. [21] Darling, R. and Pardoux, E.: Backward SDE with random terminal time and applications to semilinear elliptic PDE. Annals of Probability 25 (1997), 1135–1159. [22] Davis, M. and Lischka, F.R.: Convertible bonds with market risk and credit risk. In: Applied Probability, Studies in Advanced Mathematics 26, edited by Chan R., Kwok Y.K., Yao D. and Zhang Q., American Mathematical Society and International Press, 2002, pp. 45–58. [23] Delbaen, F. and Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Mathematische Annalen 312 (1997), 215–250. [24] Duffie, D. and Singleton, K.J.: Modeling term structures of defaultable bonds. Review of Financial Studies 12 (1999), 687–720.

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[25] El Karoui, N., Hamad` ene, S. and Matoussi, A.: Backward stochastic differential equations and applications. Chapter 8 of “Indifference Pricing: Theory and Applications”, Ren´e Carmona, ed., Springer-Verlag, 267–320 (2008). [26] El Karoui, N., Kapoudjian, E., Pardoux, C., Peng, S., and Quenez, M.C.: Reflected solutions of backward SDEs, and related obstacle problems for PDEs. Annals of Probability 25 (1997), 702–737. [27] El Karoui, N., Peng, S., and Quenez, M.-C.: Backward stochastic differential equations in finance. Mathematical Finance 7 (1997), 1–71. [28] Fleming, W.H. and Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, Second Edition. Springer-Verlag, Berlin, 2006. [29] Jacod, J. and Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin, 2003. [30] Jeanblanc, M. and Rutkowski, M.: Default risk and hazard processes. In: Mathematical Finance – Bachelier Congress 2000, edited by Geman, H., Madan, D., Pliska, S.R. and Vorst, T., Springer-Verlag, Berlin, 2002, pp. 281–312. ¨ hn, C.: Convertible bonds: financial derivatives of game type. [31] Kallsen, J. and Ku In: Exotic Option Pricing and Advanced L´evy Models, edited by Kyprianou, A., Schoutens, W. and Wilmott, P., Wiley, 2005, pp. 277–288. [32] Kifer, Y.: Game options. Finance and Stochastics 4 (2000), 443–463. [33] Kusuoka, S.: A remark on default risk models. Advances in Mathematical Economics 1 (1999), 69–82. [34] Linetsky, V.: Pricing equity derivatives subject to bankruptcy. Mathematical Finance 16(2) (2006), 255–282. [35] Kwok, Y. and Lau, K.: Anatomy of option features in convertible bonds. Journal of Futures Markets 24 (2004), 513–532. [36] Longstaff, F.A. and Schwartz, E.S.: Valuing American options by simulations: a simple least-squares approach. Review of Financial Studies 14(1) (2001), 113–147. [37] Lvov, D., Yigitbasioglu, A.B. and El Bachir, N.:. Pricing convertible bonds by simulation. Working paper, 2004. [38] Ma, J. and Zhang, J.: Representations and regularities for solutions to BSDEs with reflections. Stochastic Processes and their Applications 115(4) (2005), 539–569. [39] Morton, K. and Mayers, D.: Numerical Solution of Partial Differential Equations. Cambridge University Press, Cambridge, 1994. [40] Pardoux, E. and Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations, In: Stochastic Partial Differential Equations and Their Applications. Lecture Notes in Control Inf. Sci. 176, Springer-Verlag, Berlin, 1992, pp. 200–217. [41] Romano, M. and Touzi, N.: Contingent claims and market completeness in a stochastic volatility model. Mathematical Finance 7 (1997), 399–412. [42] Sˆırbu, M., Pikovsky, I. and Shreve, S.: Perpetual convertible bonds. SIAM J. Control Optim. 43 (2004), 58–85. [43] Takahashi, A., Kobayashi, T. and Nakagawa, N.: Pricing convertible bonds with default risk. Journal of Fixed Income 11 (2001), 20–29.

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[44] Tsitsiklis, J.N. and Van Roy, B.: Regression methods for pricing complex American-style options. IEEE Transactions on Neural Networks 12(4) (2001), 694– 703. [45] Tsiveriotis, K. and Fernandes, C.: Valuing convertible bonds with credit risk. Journal of Fixed Income 8 (1998), 95–102. Tomasz R. Bielecki Department of Applied Mathematics Illinois Institute of Technology Chicago, IL 60616, USA e-mail: [email protected] St´ephane Cr´epey Laboratoire Analyse et probabilit´es ´ Universit´e d’Evry Val d’Essonne ´ F-91025 Evry Cedex, France e-mail: [email protected] Monique Jeanblanc Laboratoire Analyse et probabilit´es ´ Universit´e d’Evry Val d’Essonne ´ F-91025 Evry Cedex, France and Europlace Institute of Finance e-mail: [email protected] Marek Rutkowski School of Mathematics and Statistics University of Sydney NSW 2006, Australia e-mail: [email protected]

Progress in Probability, Vol. 65, 299–315 c 2011 Springer Basel AG


A Convexity Approach to Option Pricing with Transaction Costs in Discrete Models Tzuu-Shuh Chiang and Shuenn-Jyi Sheu Abstract. We study the option pricing problems under transaction costs in a discrete financial model with a riskless bond and one risky asset. For general models with a transaction fee, a perfectly replicating portfolio (if it exists) may not be optimal, i.e., a perfectly replicating portfolio may cost more than certain portfolios which super replicate the contingent claim. In this paper, we shall define two adapted convex set-valued processes H t (2.1) and Rt (2.2) associated with a contingent claim and demonstrate how its (seller’s) price and optimal portfolio processes may depend on them (Theorem 2.6). A sufficient condition (Theorem 2.12) is given to ensure the uniqueness of optimal portfolio process. As an application, we investigate when a perfect replication (if it exists) is optimal for a contingent claim and prove in a unified way some known results regarding the optimality of perfect replication using H t and Rt . Finally, we shall demonstrate our methods in trinomial models and define two classes of contingent claims called AS and DS which include long and some short European call options respectively and under which we are able to determine their prices and the exact shapes of the H t and Rt . Mathematics Subject Classification (2000). Primary 90A99; Secondary 60F99. Keywords. Super(perfect) replication, (unique)optimal replicating portfolio, transaction costs, trinomial model.

1. Introduction Numerous papers have been published addressing the problems of option pricing in discrete models when there is a transaction fee. Boyle and Vorst [2] works on multiplicative binomial models (CRR model, [4]) and finds self-financing portfolios replicating the long European call options assuming proportional transaction cost in the stock and no cost on transaction of the riskless bond. Kusuoka [9] and Bensaid et al. [1] showed that the replicating portfolio process found in [2] is optimal among all super replicating ones and the price derived there is the seller’s

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price. Under the assumption that transaction fee is small, both [1] and [2] showed a replicating portfolio of short European call options exists and [1] also shows it is optimal among all super replicating strategies. Various generalizations of [1] and [2] can be found in [6], [7], [8], [11] and [12]. It is worth noting that in continuous time models with transaction costs, the only optimal hedging is trivial (buy and hold) ([3]). One can find various information on pricing in continuous time models in [5] and [10]. An algorithm was proposed in [1] to search for optimal portfolio processes and determine the seller’s price of a contingent claim. Almost all the results obtained in above-mentioned papers deal with the multiplicative binomial models and emphasize on the question whether a (perfectly) replicating portfolio process is optimal. In this paper, we shall consider option pricing problems in general discrete models where a replicating portfolio may or may not exist. Convexity arguments are used to simplify and sometimes generalize the results in previous papers. In Section 2, we study the general phenomena of super replication of contingent claims in discrete models with a transaction fee and introduce the hedging process H t and super replicaing process Rt associated with a contingent claim. Theorem 2.6 explains the relationship between H t and optimal super replicating portfolio processes and Theorem 2.12 establishes a uniqueness criterion for the optimal super replicating portfolio process. In section 3, we consider the question when a perfect replication is optimal and use Theorems 2.6 and 2.12 to re-derive some results in [1], [2] and [9]. While most of the proofs ([1], [2], [9] and [12]) of the optimality of perfect replication are analytical, our proofs exploit the convexity of the sets H t and Rt and are simpler and more intuitive. Actually, some seemingly unrelated theorems in [1], [2] and [9] hold true because of the same reason. A more detailed explaination can be found in Section 3. Also, the question of uniqueness of the optimal portfolio process can be treated simultaneously. To illustrate how to determine H t and Rt explicitly beyond CRR models, Section 4 is devoted to trinomial models. We define two classes of contingent claims called AS and DS which include long and some short European call options (Corollary 4.13) respectively and under which we are able to find their prices and determine the exact shapes of H t and Rt . However, the optimal super replicating portfolio process is not unique. In this regard, some similar results using analytic methods can be found in [7].

2. Model description and general results Let S0 , S1 , . . . , ST be the adapted stock price process on a probability space Ω taking finitely many values with the filtration {Ft }, t = 0, 1, . . . , T . (The space Ω can be taken as the set {ω : ω = (ω0 , ω1 , . . . ωT )}, where ωi is a possible stock price at time i). Let B0 , B1 , . . . , BT be the riskless bond process. We shall assume that the interest rate is 0 and thus Bt = 1 for every t. For ω ∈ Ω, we use ω(t) to denote the first t + 1 components (ω0 , ω1 , . . . , ωt ) of ω. If, conditioned on Ft , St+1 takes only two values St u and St d where u > 1 > d for every t = 0, . . . , T − 1, it is then the usual binomial CRR model. It is called multiplicative trinomial if conditioned

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on Ft , St+1 take values of St u, St or St u−1 , for some u > 1. In this paper, R2 is considered the portfolio space and any z = (x, y) ∈ R2 is a portfolio with x the shares of stock and y the shares of bond. A portfolio process {Zt = (xt , yt )}, t = 1, . . . , T + 1 is a predictable R2 -valued process representing the post-trade portfolio at time t−1 and carried over to time t. It is always assumed self-financing in the sense that for Zt (ω) = (xt (ω), yt (ω)) and Zt+1 (ω) = (xt+1 (ω), yt+1 (ω)), we have for every ω ∈ Ω and t = 1, . . . , T , xt (ω)St (ω) + yt (ω) = xt+1 (ω)St (ω) + yt+1 (ω)+ | xt+1 (ω) − xt (ω) | kSt (ω) where 0 ≤ k < 1 is the proportional constant of transaction cost. Here we assume no transaction cost for the bond. A contingent claim X is a random portfolio on Ω, X(ω) = (x(ω), y(ω)) for some real functions x and y. A portfolio process {Zt} is said to super replicate (replicate) X if xT +1 (ω) ≥ (=) x(ω) and yT +1 (ω) ≥ (= ) y(ω) for every ω ∈ Ω. For two portfolios z1 = (x1 , y1 ) and z2 = (x2 , y2 ) in R2 , we say that z1 super replicates (replicates) z2 at price s if x1 s + y1 ≥ (=)x2 s + y2 + | x1 − x2 | ks. This means that including transaction cost, z2 can be the post trade portfolio of z1 with some (no) cash left if the present stock price is s. For a portfolio z0 , we use R(z0 , s) to denote the set of all portfolios that can super replicate z0 when the price is s. Thus if z0 = (x0 , y0 ), then R(z0 , s) = {z = (x, y), xs + y ≥ x0 s + y0 + |x − x0 |sk}. Definition 2.1. For r > 0 and z0 = (x0 , y0 ) ∈ R2 , we define two half-lines (r, z0 ) = {(x, y), (y − y0 ) + r(x − x0 ) = 0, x ≤ x0 , y ≥ y0 } and (z0 , r) = {(x, y), (y − y0 ) + r(x − x0 ) = 0, x ≥ x0 , y ≤ y0 }. Remark 2.2. Since we shall be dealing with lines with negative slopes throughout the paper, we simply use the absolute value of their slopes to denote the steepness. Note that the roles of z0 in (l, z0 ) and (z0 , r) in Definition 2.1 are different. For two points z1 = (x1 , y1 ) and z2 = (x2 , y2 ) where x1 ≤ x2 and y1 ≥ y2 , (i.e., z1 is to the northwest of z2 ) we shall use |z1 z2 | to denote the absolute value of the 2 slope of the line through z1 and z2 . (i.e., |z1 z2 | = xy12 −y ) and use (z1 , |z1 z2 |, z2 ) −x1 for the line segment connecting z1 and z2 . Similarly, for z1 , . . . , zn in R2 with zi to the northwest of zi+1 for each i, (z1 , |z1 z2 |, z2 , . . . , zn ) is the polygon connecting z1 , z2 , . . . to zn . Definition 2.3. Let z0 = (x0 , y0 ) ∈ R2 and r > 0. A point z = (x, y) ∈ R2 is said to the left of (r, z0 ) (or (z0 , r)) if y + rx ≤ y0 + rx0 and to the right of (r, z0 ) (or (z0 , r)) if y + rx ≥ y0 + rx0 . Definition 2.4. For two half-lines (l, z0 ) and (z0 , r) with l > r, let (l, z0 , r) = (l, z0 ) ∪ (z0 , r) and let conv(l, zo , r) be the convex set enclosed by (l, z0 , r). We call conv(l, z0 , r) a wedge and obviously, (l, z0 , r)= ∂ conv(l, z0 , r). For l1 > l2 > · · · > ln+1 and z1 , . . . , zn in R2 where zi is to the northwest of zi+1 with |zi zi+1 | = li+1 , let conv(l1 , z1 , l2 , z2 , . . . , ln , zn , ln+1 ) = ∩ni=1 conv(li , zi , li+1 ) and obviously, (l1 , z1 , . . . , zn , ln+1 ) = ∂ conv(l1 , z1 , l2 , z2 , . . . , ln , zn , ln+1 ). We also define (z1 , l2 , . . . , zn ) to be the effective boundary of conv(l1 , z1 , l2 , . . . , ln , zn , ln+1 ).

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and

, conv(l1 , z1 , . . . , ln , zn , ln+1 ) = z ∈ R2 , z is to the right of (l1 , z1 ), . . . . . . , (ln , zn ), (zn , ln+1 ) .

Lemma 2.5. For any z0 = (x0 , y0 ) ∈ R2 and s > 0, we have R(z0 , s) = conv(l, z0 , r) where l = s(1 + k) and r = s(1 − k). And ∂R(z0 , s) consists of all replicating portfolios of z0 when the stock price is s. Proof. A point z = (x, y) ∈ R(z0 , s) if xs+y ≥ x0 s+y0 +|x−x0 |sk. If x ≤ x0 , then the inequality becomes xs + y ≥ x0 s + y0 + (x0 − x)sk, i.e., xs(1 + k) + y ≥ x0 s(1 + k) + y0 . The case x ≥ x0 is similar. Hence R(z0 , S) = conv(s(1 + k), z0 , s(1 − k)). Trivially, ∂R(z0 , s) is the set of all portfolios replicating z0 when the stock price is s.  For a contingent claim X(ω) = (x(ω), y(ω)), let H T (ω) = {z = (¯ x, y¯), x ¯≥ x(ω), y¯ ≥ y(ω)} and RT (ω) = R(X(ω), ST (ω)) be the set of super replicating portfolios of X(ω) at time T . We next define successively the adapted set-valued hedging process H t and super replicating process Rt , t = 1, 2, . . . , T − 1 as follows. First, @ H T −1 (ω) = H T −1 (ω(T − 1)) = RT (¯ ω) ω ¯ (T −1)=ω(T −1)

and

A

RT −1 (ω) = RT −1 (ω(T − 1)) =

R(z, ST −1 (ω)) (Figure 1)

z∈H T −1 (ω)

In general, having defined Rt+1 (ω) = Rt+1 (ω(t + 1)), t ≥ 1, let @ H t (ω) = Rt+1 (¯ ω)

(2.1)

ω(t)=ω(t) ¯

and

A

Rt (ω) =

R(z, St (ω)).

(2.2)

z∈H t (ω)

Finally, at t = 0, let H0 =

@

R1 (ω).

(2.3)

ω

For each ω and 1 ≤ t ≤ T − 1, Rt (ω) consists of the portfolios that can super replicate a portfolio in H t (ω) when the price is St (ω) and H t (ω) is characterized as the set that if z is in H t (ω), then carried over to time t+ 1, z belongs to Rt+1 (¯ ω) for any ω ¯ with ω ¯ (t) = ω(t). The following theorem is an easy consequence of the above definitions. We omit the proof.

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Figure 1. H T −1 (ω) and RT −1 (ω) in CRR model Theorem 2.6. For a portfolio process {Zt } to super replicate a contingent claim X, it is necessary and sufficient that Zt+1 (ω) ∈ H t (ω) for each ω (or equivalently, Zt+1 (ω) ∈ Rt+1 (¯ ω ) with ω(t) = ω ¯ (t)) and 0 ≤ t ≤ T . If no transaction fee is charged at t = 0, then the (seller’s) price for X is obviously p(X) = inf z=(x,y)∈H 0 (S0 x + y). A point (x, y) ∈ H 0 is said to determine p(X) if the minimum is attained at (x, y) in p(X). The seller’s price is the minimal cost to create a portfolio process to super replicate X. A portfolio process Zt which super replicates X is optimal if Z1 = (x, y) for some (x, y) determining p(X). It is uniquely optimal if for any other optimal process Zt , we have Zt = Zt for any 1 ≤ t ≤ T + 1. Trivially, RT (ω) is the wedge conv(ST (ω)(1 + k), X(ω), ST (ω)(1 − k)). But immediately H T −1 becomes complicated because it is the intersection of several wedges. Some simple but important observations are as follows. Lemma 2.7. For wedges conv(li , zi , ri ), i = 1, . . . , n let l = max(l1 , . . . , ln ) and r = min(r1 , . . . , rn ). Then for some k and points w1 , . . . , wk , ∩ni=1 conv(li , zi , ri ) = conv(c1 , w1 , c2 , w2 , . . . , ck , wk , ck+1 ) where c1 = l > c2 > · · · > ck+1 = r. Proof. Let conv(l1 , z1 , r1 ) and conv(l2 , z2 , r2 ) be two wedges and l1 > l2 , r1 > r2 . If z2 ∈ conv(l1 , z1 , r1 ), then either (l2 , z2 ) ∩ (l1 , z1 ) = φ or (l2 , z2 ) ∩ (z1 , r1 ) = φ. In

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the first case, we have conv(l1 , z1 , r1 ) ∩ conv(l2 , z2 , r2 ) = conv(l1 , w, l2 , z2 , r2 ) and in the second case, conv(l1 , z1 , r1 ) ∩ conv(l2 , z2 , r2 ) = conv(l1 , z1 , r1 , w, l2 , z2 , r2 ) where w is the intersection point. If z2 ∈ / conv(l1 , z1 , r1 ), then four cases can occur. (i) (ii) (iii) (iv)

(l2 , z2 ) ∩ (l1 , z1 ) = φ (z2 , r2 ) ∩ (l1 , z1 ) = φ (z2 , r2 ) ∩ (l1 , z1 ) = φ (l2 , z2 ) ∩ (l1 , z1 ) = φ

and (l2 , z2 ) ∩ (z1 , r1 ) = φ, but (z2 , r2 ) ∩ (z1 , r1 ) = φ, but (z2 , r2 ) ∩ (z1 , r1 ) = φ or but (l2 , z2 ) ∩ (z1 , r1 ) = φ.

In (i), conv(l1 , z1 , r1 ) ∩ conv(l2 , z2 , r2 ) = (l1 , w1 , l2 , w2 , r1 , w3 , r2 ). In (ii), conv(l1 , z1 , r1 ) ∩ conv(l2 , z2 , r2 ) = (l1 , w1 , r2 ). In (iii), conv(l1 , z1 , r1 ) ∩ conv(l2 , z2 , r2 ) = (l1 , z1 , r1 , w1 , r2 ) and in (iv), conv(l1 , z1 , r1 ) ∩ conv(l2 , z2 , r2 ) = (l1 , z1 , r1 , w1 , l2 , z2 , r2 ). Here, w1 , w2 and w3 are intersection points. The proof is similar for other cases.



After this point the convention appearing in Lemma 2.7 about the ordering of slopes is used. This should be noted. i i i Corollary 2.8. For convex sets Ki = conv(l1i , w1i , . . . , lL , wL , lL ), i = 1, . . . , n i −1 i −1 i 1 n 1 n n with l = max(l1 , . . . , l1 ) and r = min(lL1 , . . . , lLn ), ∩i=1 Ki = conv(l1 , w1 , . . . , lL−1 , wL−1 , lL ) where l1 = l and lL = r for some L and points w1 , w2 , . . . and wL−1 . i Proof. This immediately follows from Lemma 2.7 because each conv(l1i , . . . , lL ) is i i i i i i i i itself the intersection of wedges conv(l1 , w1 , l2 ), conv(l2 , w2 , l3 ), . . . and conv(lL , i −1 i i wL , l ).  Li i −1 (¯ ω) (¯ ω) Let Mt (ω) = supω¯ (t)=ω(t) SSt+1 and mt (ω) = inf ω¯ (t)=ω(t) SSt+1 . We shall t (ω) t (ω) always assume Mt > 1 and mt < 1. The following two lemmas describe the shapes of H t and Rt respectively.

Lemma 2.9. For any 0 ≤ t ≤ T − 1, H t (ω) = conv(l1 , z1 , l2 , . . . , ln , zn, ln+1 ) where l1 = St (ω)Mt (ω)(1 + k) and ln+1 = St (ω)mt (ω)(1 − k) for some points zi , i = 1, . . . , n, and Rt (ω) = conv(St (1+k), w1 , . . . , wk , St (1−k)) for some points wi , i = 1, . . . , k. Proof. For t = T − 1, H T −1 (ω) = ∩ω¯ (T −1)=ω(T −1) RT (¯ ω) = ∩ω¯ (T −1)=ω(T −1) (ST (¯ ω )(1 + k), X(ω), ST (¯ ω )(1 − k)). Hence by Lemma 2.7, for some points w1 , w2 , . . . , wk , H T −1 (ω) = conv(ST −1 (ω)MT −1 (ω)(1 + k), w1 , . . . , wk , ST −1 (ω)mT −1 (ω)(1 − k)). (2.4)

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Now for a wedge conv(l, z0 , r), the super replicating set ∪z∈conv(l,z0 ,r)R(z, s) is obviously conv(s(1 + k), z0 , s(1 − k)) if s(1 + k) ≤ l and r ≤ s(1 − k). And in general, ∪z∈conv(l1 ,z1 ,...,lL ) R(z, s) = conv(s(1 + k), zm , lm+1 , zm+1 , . . . , lM−1 , zM −1 , lM , zM , s(1 − k)) where m and M are so chosen that lm+1 < s(1+k) ≤ lm and lM +1 ≤ s(1−k) < lM . Since Mt (ω) > 1 and mt (ω) < 1, we thus have by (2.4), RT −1 (ω) = ∪z∈H T −1 (ω) R(z, ST −1 (ω)) = ∪z∈conv(ST −1 (ω)MT −1 (ω)(1+k),w1 ,...,wk ,ST −1 (ω)mT −1 (ω)(1−k)) R(z, ST −1 (ω)) = conv(ST −1 (ω)(1 + k), . . . , ST −1 (ω)(1 − k)). 

Now Corollary 2.8 and an induction yield the lemma.

Remark 2.10. For a hedging set H t = conv(l1 , z1 , . . . , ln , zn , ln+1 ) with l1 = St (1 + k)Mt > l2 > · · · > ln+1 = St (1 − k)mt as in Lemma 2.9, let m and M be so chosen that lm ≥ St (1 + k) > lm+1

and lM > St (1 − k) ≥ lM +1 , t = 1, 2, . . . , T,

(2.5)

respectively. Then R = conv(St (1 + k), zm , lm+1 , . . . , lM , zM , St (1 − k)). Note that the effective boundary of Rt (= (zm , lm+1 , . . . , lM , zM )) is contained in ∂H t . If we have strict inequalities t

lm > St (1 + k) > lm+1

and lM > St (1 − k) > lM +1 , t = 1, 2, . . . , T,

(2.6)

and if w is a portfolio in ∂Rt , then at the price St , there can be only one posttrade portfolio of w to be in H t . To be precise, if w ∈ (zM , St (1 − k)), then zM is the only possible post trade portfolio of w in H t when the stock price is St . If w ∈ (St (1 + k), zm ), zm is the only possible post trade portfolio of w in H t . If w ∈ (zm , lm+1 , . . . , lM , zM ), then w itself is the only element in H t such that w ∈ R(w, St ). Hence if Zt super replicates X and Zt (ω) ∈ ∂Rt (¯ ω ) for some ω ¯ with ω ¯ (t − 1) = ω(t − 1), then Zt+1 (¯ ω ) must be contained in the effective boundary of Rt (¯ ω )(∈ ∂H t (¯ ω )). In particular, if Rt (¯ ω ) is a wedge conv(St (1 + k), z, St (1 − k)), then Zt+1 (¯ ω ) = z. Remark 2.11. Since H 0 = conv(l1 , z1 , . . . , ln , zn , ln+1 ) with l1 = S0 M0 (1 + k) > S0 > ln+1 = S0 m0 (1 − k), there is a first i such that li ≥ S0 > li+1 and p(X) = xi S0 + yi where zi = (xi , yi ). (Actually, if li = S0 , then p(X) = xS0 + y for any z = (x, y) ∈ (li , zi ).) If li > S0 > li+1 , (2.7) then zi is the unique portfolio determining p(X) and any optimal portfolio process must have Z1 = zi . We next state the situation that the optimal portfolio process is unique.

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Theorem 2.12. Let St be a price process and X be a contingent claim. Suppose (2.6) and (2.7) are satisfied and let z be the unique portfolio that determines p(X). Then the optimal portfolio process super replicating X is unique if z ∈ ∂R1 (ω) a.s. and the effective boundary of Rt (ω) is contained in ∂Rt+1 (¯ ω ) with ω ¯ (t) = ω(t) for t = 1, . . . , T . Proof. If {Zt } is optimal, then Z1 = z by Remark 2.11. Since Z1 = z ∈ ∂R1 , by Remark 2.10, Z2 can take the unique choice in the effective boundary of R1 , which is contained in ∂R2 . The induction then implies that the process is unique. 

3. Perfect replication and optimality In this section, we re-derive three theorems dealing with the question whether a perfect replication is optimal. Theorem 3.1 was proved in (Kusuoka [9], p. 199) considering in general models when a perfectly replicating portfolio process (assuming it exists) of a contingent claim is optimal. Here, we can also establish the uniqueness of optimal process. It was proved in (Bensaid et al. [1], p. 74) that in binomial models with d1 = u > (1+k)2 (i.e., the transaction fee is small), a perfect replication is the unique optimal portfolio process. Theorem 3.2 was proved in [12, Theorem 9] generalizing the theorem to binomial models with u(1−k) > 1+k > 1−k > d(1+k). Theorem 3.3 was proved in (Bensaid et al. [1], p. 75) generalizing the fact (Boyle and Vorst, [2]) that perfect replication is optimal for long European calls in binomial models. We shall present much simpler proofs of the three theorems using the convexity method and show that all the three seemingly unrelated theorems hold true actually because of the same reason that ∂Rt is a wedge and the effective boundary of Rt is contained in ∂Rt+1 . Theorem 2.12 then establishes the uniqueness of the optimal portfolio process in all three cases. Theorem 3.1. Let S0 , S1 , . . . , ST be a stock price process on Ω with the filtration {Ft } and Z1 , Z2 , . . . , ZT +1 be a portfolio process perfectly replicating a contingent claim X (i.e., X = ZT +1 ). Let Zt = (xt , yt ). If for t = 0, 1, . . . , T − 1, P (St+1 ≥ St , xt+2 ≥ xt+1 |Ft ) > 0

(3.1)

P (St+1 ≤ St , xt+2 ≤ xt+1 |Ft ) > 0,

(3.2)

and then Zt is optimal. If we have strict inequalities in (3.1) and (3.2), then {Zt}t is unique. Proof. We first claim that Rt is the wedge conv(St (1 + k), Zt+1 , St (1 − k)), t = 1, . . . , T . By definition, RT = conv(ST (1 + k), ZT +1 , ST (1 − k)). By (3.1), for every ω ∈ Ω, there exists an ω  such that ω(T − 1) = ω  (T − 1) but ST −1 (ω  ) ≤ ST (ω  ) and xT (ω  ) ≤ xT +1 (ω  ).

(3.3)

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Since ZT (ω) replicates ZT +1 (ω  ), we thus have ZT (ω) ∈ ∂RT (ω  ). Moreover, (3.3) implies that ZT (ω) ∈ (ST (ω  )(1 + k), ZT +1 (ω  )). (3.4) Similarly, condition (3.2) implies that there exists an ω  such that ω(T − 1) = ω  (T − 1) but ST −1 (ω  ) ≥ ST (ω  ) and xT (ω  ) ≥ xT +1 (ω  ).

(3.5)

ZT (ω) ∈ (ZT +1 (ω  ), ST (ω  )(1 − k)).

(3.6)

Also, By Theorem 2.6, we have ZT (ω) ∈ H T −1 (ω). T −1

(3.7)

Since intH (ω) is contained in intR (ω ), the perfect replication of ZT +1 (ω  ) by ZT (ω) implies ZT (ω) ∈ ∂H T −1 (ω). (3.8) T



From Lemma 2.9, H T −1 (ω) takes the form conv(l1 , . . . , l, ZT (ω), r, . . . ) and (3.3) implies that l ≥ ST (ω  )(1 + k) and r ≤ ST (ω  )(1 − k). By (3.3), (3.5) and Remark 2.10, we have that RT −1 is conv(ST −1 (1 + k), ZT , ST −1 (1− k)). This completes the induction. Now at time 0, H 0 = conv(l1 , . . . , l, Z1 , r, . . . ) with l ≥ S1 (ω  )(1 + k) and r ≤ S1 (ω  )(1 − k). Since S1 (ω  ) ≥ S0 ≥ S1 (ω  ) by (3.3) and (3.5), we obviously have minz=(x,y)∈H 0 (S0 x + y) = S0 x0 + yo where Z1 = (x0 , y0 ). Hence {Zt } is optimal. The effective boundary of Rt is the point Zt and it is contained in ∂Rt because it perfectly replicates Zt+1 . The uniqueness follows from Theorem 2.12 because (2.6) and (2.7) are satisfied by the strict inequalities in (3.1) and (3.2).  It is easy to see that in a binomial model, the condition u(1 − k) > d(1 + k) is necessary and sufficient for any contingent claim to be uniquely perfectly replicated. With a slightly stronger condition, we re-derive the following theorem (Theorem 4.1 [8], Prop. 3.1 [11] and Theorem 9 [12]) and establish the uniqueness of the optimal replicating portfolio process using our method. Theorem 3.2. In a binomial model, suppose 1+k 1 1+k and > . (3.9) 1−k d 1−k Then the portfolio process which perfectly replicates a contingent claim is uniquely optimal. u>

Proof. The proof is essentially the same as that in Theorem 3.1. We first claim Rt (ω) = conv(St (ω)(1 + k), Zt+1 , St (ω)(1 − k)) t = 1, . . . , T

(3.10)

where Zt , t = 1, . . . , T + 1 is the unique portfolio process replicating X. By definition, RT = conv(ST (1 + k), X, ST (1 − k)) and we proceed to show RT −1 (ω) = conv(ST −1 (ω)(1 + k), ZT (ω), ST −1 (ω)(1 − k)).

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From (2.1), H T −1 (ω) = ∩ω¯ (T −1)=ω(T −1) RT (¯ ω) = conv(ST −1 (ω)u(1 + k), X(ω  ), ST −1 (ω)u(1 − k)) ∩ conv(ST −1 (ω)d(1 + k), X(ω  ), ST −1 (ω)d(1 − k)) where ω  , ω  ∈ Ω, ω  (T −1) = ω  (T −1) = ω(T −1) and ωT = ωT −1 u, ωT = ωT −1 d. Since ZT replicates ZT +1 , H T −1 (ω) must be one of the following four forms: conv(ST −1 (ω)u(1 + k), ZT (ω), ST −1 (ω)d(1 − k)), conv(ST −1 (ω)u(1 + k), ZT (ω), ST −1 (ω)d(1 + k), ZT +1 (ω  ), ST −1 (ω)d(1 − k)), conv(ST −1 (ω)u(1 + k), ZT +1 (ω  ), ST −1 (ω)u(1 − k), ZT (ω), ST −1 (ω)d(1 + k), ZT +1 (ω  ), ST −1 (ω)d(1 − k)) and conv(ST −1 (ω)u(1 + k), ZT +1 (ω  ), ST −1 (ω)u(1 − k), ZT (ω), ST −1 (ω)d(1 − k)). (3.11) By Remark 2.10, condition (3.9) implies (3.10) holds for t = T −1 in any of the four cases. The induction completes (3.10). Now at t = 0, H 0 takes one of the four forms in (3.11) with T = 1 and it is obvious minz=(x,y)∈H 0 (S0 x + y) = S0 x0 + y0 where Z1 = (x0 , y0 ). Thus {Zt } is optimal. Moreover,(2.6) and (2.7) hold because of (3.11) and similar to the proof in Theorem 3.1, Theorem 2.12 implies the uniqueness of optimal portfolio process.  The following theorem generalizes the long European call option ([1] and [2]). Theorem 3.3. Let X = X(ST ) = (x(ST ), y(ST )) be a contingent claim in a binomial model such that x(s) is non-decreasing and y(s) non-increasing in s. Suppose for every ω, ST −1 (ω)d(1 − k) ≤ |X(ST −1 (ω)d)X(ST −1 (ω)u)| ≤ ST −1 (ω)u(1 + k)

(3.12)

then there exists a unique portfolio process perfectly replicating X. It is also uniquely optimal. Proof. Let ZT +1 = X. Since H T −1 (ω) = conv(ST −1 (ω  )u(1 + k), X(ST (ω  )), ST −1 (ω  )u(1 − k)) ∩ conv(ST −1 (ω  )d(1 + k), X(ST (ω  )), ST −1 (ω  )d(1 − k)) where ω  , ω  ∈ Ω, ωT −1 = ωT −1 = ωT −1 and ωT = ST −1 (ω)u and ωT = ST −1 (ω)d, condition (3.12) implies there is a unique ZT (ω) = Z(ST −1 (ω)) ∈ (ST −1 (ω  )u(1 + k), X(ST (ω  ))) ∩ (X(ST (ω  )), ST −1 (ω  )d(1 − k)) such that H T −1 (ω) = conv(ST −1 (ω)u(1 + k), ZT (ω), ST −1 (ω)d(1 − k)). From Remark 2.10, it follows that RT −1 (ω) = conv(ST −1 (ω)(1 + k), ZT (ω), ST −1 (ω)(1 − k)). Now for any ω, let ω  , ω  ∈ Ω be such that ω(T − 2) = ω  (T − 2) = ω  (T − 2) and ST −1 (ω  ) = ST −2 (ω)u and ST −1 (ω  ) = ST −2 (ω)d. We claim that ST −2 (ω)d(1 − k) ≤ |ZT (ω  )ZT (ω  )| ≤ ST −2 (ω)u(1 + k).

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Let ω  be such that ωT = ωT −1 d = ωT −1 u. It is obvious that |ZT (ω  )ZT +1 (ω  )| = ST (ω  )(1 + k) and |ZT +1 (ω  )ZT (ω  )| = ST (ω  )(1 − k). Thus

ST −2 (ω  )d(1 − k) = ST (ω  )(1 − k) ≤ |ZT (ω  )ZT (ω  )| ≤ ST (ω  )(1 + k) = ST −2 (ω  )u(1 + k).

An induction implies that H 0 (ω) = conv(S0 u(1+k), Z1 (ω), S0 d(1−k)). Obviously, Z1 determines p(X) and {Zt} is optimal. It is uniquely optimal because of Theorem 2.12. 

4. Trinomial model In this section, we shall apply results in §2 to pricing contingent claims in trinomial models. In a trinomial model, perfect replication is in general not possible and therefore we need to consider super replicating portfolios. For contingent claims X(ST ) = (x(ST ), y(ST )), we shall formulate two conditions AS and DS in Definitions 4.1 and 4.2 including long and some short European calls (Corrolary 4.13) respectively and determine their optimal super replicating portfolio processes. We want to remark that opions with cash settlements do not satisfy AS and DS. Definition 4.1. Let X = X(ST ) = (x(ST ), y(ST )) be a contingent claim where x denotes the number of shares of the stock and y the number of shares of bond. We say X satisfies condition AS if (1) x(s) is increasing in s while y(s) is decreasing in s and (2) X(ST −1 ui ) is to the left of (ST −1 ui+1 (1 + k), X(ST −1 ui+1 )) and to the right of (ST −1 ui (1 − k), X(ST −1 ui+1 )) for any i = −1, 0. It is easy to see that the long position of a European call option (X(ST ) = (1, −K) if ST ≥ K and X(ST ) = (0, 0) of ST < k) satisfies Definition 4.1. (Figure 2). Definition 4.2. Let X = X(ST ) be a contingent claim as in Definition 4.1. We say X satisfies the DS condition if (1) x(s) is decreasing in s while y(s) is increasing in s and (2) X(ST −1 ui ) is to the right of (X(ST −1 ui+1 ), ST −1 ui+1 (1 − k)) and to the left of (X(ST −1 ui+1 ), ST −1 ui (1 + k)) for any i = −1, 0. (Figure 3). (See Remark 4.4.) The following lemmas follow easily from geometric arguments and we omit their proofs.

310

T.-S. Chiang and S.-J. Sheu z Y)TU
3 vj *

TU
3 vj,3 )3 , l* TU
3 vj )3
l*

Y)TU
3 vj,3 *

y Figure 2. X = X(ST ) satisfies AS

z Y)TU
3 vj,3 *

TU
3 vj )3 , l*

TU
3 vj,3 )3
l*

Y)TU
3 vj *

y Figure 3. X = X(ST ) satisfies DS

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Lemma 4.3. For l > r, the following conditions are equivalent. (a) z2 is to the left of (l, z1 ) and to the right of (r, z1 ) 1 (b) r ≤ xy21 −y ≤ l and y2 ≥ y1 , x1 ≥ x2 where z1 = (x1 , y1 ) and z2 = (x2 , y2 ). −x2 Remark 4.4. From Lemma 4.3, it is easy to see that condition AS is a generalization of [1] (p. 75) in a binomial model and is equivalent to ST −1 ui (1 − k) ≤ |X(ST −1 ui )X(ST −1 ui+1 )| ≤ ST −1 ui+1 (1 + k), i = 0, −1. Condition DS is equivalent to ST −1 ui (1 + k) ≤ |X(ST −1 ui+1 )X(ST −1 ui )| ≤ ST −1 ui+1 (1 − k), i = 0, −1. 1+k Obviously, condition DS can hold true only when 1−k < u. Also, decreasing (increasing) in Definitions 4.1 and 4.2 means non-increasing (non-decreasing).

Lemma 4.5. For two wedges (l1 , z1 , r1 ) and (l2 , z2 , r2 ) with l1 > l2 and r1 > r2 , if z2 is to the left of (l1 , z1 ) and to the right of (r2 , z1 ), then conv(l1 , z1 , r1 ) ∩ conv(l2 , z2 , r2 ) = conv(l1 , z, r2 ) where z ∈ (l1 , z1 ) ∩ (z2 , r2 ). Lemma 4.6. For three points z1 , z2 and z3 in R2 , if |z1 z2 | = l and |z2 z3 | = r, then min(r, l) ≤ |z1 z3 | ≤ max(r, l). The next two lemmas describe the intersection of three and four wedges respectively. Lemma 4.7. For three wedges conv(li , zi , ri ) with l1 > l2 > l3 and r1 > r2 > r3 , if zi+1 is to the left of (li , zi )

and to the right of

(ri+1 , zi ), i = 1, 2,

(4.1)

then 3 @ i=1

Moreover,

conv(li , zi , ri ) = conv(l1 , z, r3 ) where z ∈ (l1 , z1 , r1 ) ∩ (l3 , z3 , r3 ). 3 i=1

conv(li , zi , ri ) = conv(l1 , z1 , r1 )



conv(l3 , z3 , r3 ).

Lemma 4.8. Suppose there are (li , zi , ri ), i = 1, . . . , 4 satisfying l1 > l2 > l3 > l4 and r1 > r2 > r3 > r4 with zi+1 to the left of (li , zi ) and to the right of (ri+1 , zi ) for each i. Let 3 @

conv(li , zi , ri ) = conv(l1 , w2 , r3 )

and

conv(li , zi , ri ) = conv(l2 , w4 , r4 )

as in lemma 4.7

i=1 4 @ i=2

where w2 ∈ (z3 , r3 ) ∩ (l1 , z1 ) and w4 ∈ (z4 , r4 ) ∩ (l2 , z2 ). Then r3 ≤ |w4 w2 | ≤ l2 .

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Let X be a contingent claim satisfying AS in a trinomial model. For each ω ∈ Ω and 0 ≤ t ≤ T , we define the binomial process Siω,t , i = 0, . . . , T − t with S0ω,t = ωt and d = u1 . When restricted to {Siω,t }0≤i≤T −t , X naturally defines a contingent claim X ω,t on this binomial model at time T − t. To be precise, we have ω,t b ω,0 X ω,t (STω,t (STω,0 ). By Theorem −t ) = X(ST −t ). When t = 0, we use X to denote X ω,t 3.3, there exists a unique portfolio process Zi , i = 1, . . . , T − t + 1, perfectly replicating X ω,t . (Condition (3.12) in Theorem 3.3 holds because of condition AS.) We first claim that for each t, H t (ω) = conv(St (ω)u(1 + k), Z1ω,t , St (ω)u−1 (1 − k)).

(4.2)

It then follows that Rt (ω) = conv(St (ω)(1 + k), Z1ω,t , St (ω)(1 − k)).

(4.3)

For t = T − 1 and ω ∈ Ω, by definition, H T −1 (ω) = ∩i=−1,0,1 conv(li , X(ST −1 (ω)ui ), ri ) where li = ST −1 (ω)ui (1 + k), ri = ST −1 (ω)ui (1 − k). Condition AS implies that (4.1) in Lemma 4.7 holds and hence H T −1 (ω) = ∩i=−1,1 conv(li ,X(ST −1 (ω)ui ),ri ). Therefore, H T −1 (ω) = conv(ST −1 (ω)u(1 + k), z, ST −1 (ω)u−1 (1 − k)) and RT −1 (ω) = conv(ST −1 (ω)(1 + k), z, ST −1 (ω)(1 − k)) where z ∈ (ST −1 (ω)u, X(ST −1 (ω)u)) ∩ (X(ST −1 (ω)u−1 ), ST −1 (ω)u−1 ). But it is obvious that z = Z1ω,T −1 . Let ω  ∈ Ω be such that ω  (T − 2) = ω(T − 2) and ω  (T − 1)u = ω(T − 1). We need to show that 

ST −1 (ω  )(1 − k) ≤ |Z1ω ,T −1 Z1ω,T −1 | ≤ ST −1 (ω)(1 + k),

(4.4)

(i.e., (2) in Definition 4.2 in order for an induction to proceed.) Since conv(ST −1 (ω)u(1 + k), Z1ω,T −1 , ST −1 (ω)u−1 (1 − k)) = ∩i=−1,0,1 conv(ST −1 (ω)ui (1 + k), X(ST −1 (ω)ui ), ST −1 (ω)ui (1 − k)) and 

conv(ST −1 (ω  )u(1 + k), Z1ω ,T −1 , ST −1 (ω  )u−1 (1 − k)) = ∩i=−1,0,1 conv(ST −1 (ω  )ui (1 + k), X(ST −1 (ω  )ui ), ST −1 (ω  )ui (1 − k)) = ∩i=−2,−1,0 conv(ST −1 (ω)ui (1 + k), X(ST −1 (ω)ui ), ST −1 (ω)ui (1 − k)), Lemma 4.8 thus implies (4.4). We next consider t=T-2. Let ω−1 , ω0 and ω1 ∈ Ω be such that ω−1 (T − 2) = ω0 (T − 2) = ω1 (T − 2) and ST −1 (ω−1 ) = ST −1 (ω0 )u−1 ,

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ST −1 (ω1 ) = ST −1 (ω0 )u. Then, H T −2 (ω) = ∩i=−1,0,1 conv(ST −2 (ω)ui (1 + k), Z1ωi ,T −1 , ST −2 (ω)ui (1 − k)) = conv(ST −2 (ω)u1 (1 + k), Z1ω1 ,T −1 , ST −2 (ω)u1 (1 − k)) ∩ conv(ST −2 (ω)u−1 (1 + k), Z1 −1 ω

,T −1

, ST −2 (ω)u−1 (1 − k))

= conv(ST −2 (ω)u(1 + k), z, ST −2 (ω)u−1 (1 − k)). for some z ∈ R2 . By the uniqueness in Theorem 3.3, z = Z1ω,T −2 . An induction thus completes the proof of (4.2) and (4.3). Now, since both p(X) and p(X b ) are determined by Z1ω,0 , obviously p(X) = p(X b ). We summarize the above as follows. Theorem 4.9. Let X = (x(ST ), y(ST )) be a contingent claim satisfying condition AS on a trinomial model. Then there exists an adapted process {Zt } such that H t (ω) = conv(St (ω)u(1 + k), Zt (ω), St (ω)u−1 (1 − k)), t = 0, . . . , T − 1 and Rt (ω) = conv(St (ω)(1 + k), Zt (ω), St (ω)(1 − k)), t = 1, . . . , T. Moreover, if X b is the contingent claim in the corresponding binomial model with d = u1 , then p(X) = p(X b ). Obviously, any portfolio process super replicating X starting from Z0 is optimal. We next consider contingent claims satisfying condition DS and omit the proofs of first two lemmas. Lemma 4.10. For two wedges (l1 , z1 , r1 ) and (l2 , z2 , r2 ), if l1 > r1 > l2 > r2 and l2 < |z1 z2 | < r1 , then conv(l1 , z1 , r1 ) ∩ conv(l2 , z2 , r2 ) = conv(l1 , z1 , r1 , w, l2 , z2 , r2 ) for some w ∈ (z1 , r1 ) ∩ (l2 , z2 ). Lemma 4.11. For three wedges (li , zi , ri ), i = 1, 2, 3, if l1 > r1 > l2 > r2 > l3 > r3 and l2 < |z1 z2 | < r1 , l3 < |z2 z3 | < r2 , then 3 @

conv(li , zi , ri ) = conv(l1 , z1 , r1 , w1 , l2 , z2 , r2 , w2 , l3 , z3 , r3 )

i=1

for some w1 ∈ (z1 , r1 ) ∩ (l2 , z2 ) and w2 ∈ (z2 , r2 ) ∩ (l3 , z3 ). Theorem 4.12. For a contingent claim X = X(ST ) satisfying condition DS in a 1 2 trinomial model with u > 1+k 1−k , there are two adapted processes Zt and Zt such that H t (ω) = conv(St (ω)u(1 + k), X(St (ω)u), St (ω)u(1 − k), Zt1 (ω), St (ω)(1 + k), X(St (ω)), St (ω)(1 − k), Zt2 (ω), St (ω)u−1 (1 + k), X(St (ω)u−1 ), St (ω)u−1 (1 − k)), 0 ≤ t ≤ T − 1

(4.5)

Rt (ω) = conv(St (ω)(1 + k), X(St(ω)), St (ω)(1 − k)), 1 ≤ t ≤ T,

(4.6)

and

314

T.-S. Chiang and S.-J. Sheu

where wt1 ∈ (X(St (ω))u, St (ω)u(1 − k)) ∩ (St (ω)(1 + k), X(St (ω))) wt2

∈ (X(St (ω)), St (ω)(1 − k)) ∩ (St (ω)u

−1

(1 + k), X(St (ω)u

and −1

)).

(4.7)

Moreover, X(S0 ) determines the price of X where S0 is the initial price. Proof. Let ω−1 , ω0 and ω1 ∈ Ω such that ω−1 (T − 1) = ω0 (T − 1) = ω1 (T − 1) = ω(T − 1)and ST (ω1 ) = ST −1 (ω)u, ST (ω0 ) = ST −1 (ω), and ST (ω−1 ) = ST −1 (ω)u−1 . By definition, H T −1 (ω) = ∩i=−1,0,1 conv(ST −1 (ω)ui (1 + k),X(ST (ωi )),ST −1 (ω)ui (1 − k)) = conv(ST −1 (ω)u(1 + k),X(ST (ω1 )),ST −1 (ω)u(1 − k),ZT1 −1 (ω),ST −1 (ω)(1 + k), X(ST (ω0 )),ST −1 (ω)(1 − k),ZT2 −1 (ω),ST −1 (ω)u−1 (1 + k), X(ST (ω−1 )),ST −1 (ω)u−1 (1 − k)). The last equality follows from Lemma 4.11. (Condition DS implies that Lemma 4.11 applies here.) Also, ZT1 −1 (ω) ∈ (X(ST (ω1 )), ST −1 (ω)u(1 − k)) ∩ (ST −1 (ω)(1 + k), X(ST (ω0 )) and ZT2 −1 (ω) ∈ (X(ST (ω0 )), ST −1 (ω)(1 − k)) ∩ (ST −1 (ω)u−1 (1 + k), X(ST (ω−1 ))). By Remark 2.10, RT −1 (ω) = conv(ST −1 (ω)(1 + k), X(ST −1 (ω)), ST −1 (ω)(1 − k)). Since {X(ST −1 )} again satisfies condition DS, an induction thus concludes (4.5), (4.6) and (4.7). It follows from Remark 2.11 that X(S0 ) determines the price of X.  Corollary 4.13. Let X be a short European call option in a trinomial model with 1+k K K u ≥ 1−k . If no terminal price ST falls in the interval ( 1+k , 1−k ) where K is the strike price, then X satisfies condition DS. Proof. All we need to check is ST −1 (1 + k) < |X(ST −1 u)X(ST −1 )| < ST −1 u(1 − k)

(4.8)

when ST −1 < K < ST −1 u. But |X(ST −1 u)X(ST −1 )| = K, thus (4.8) holds if and K K only if ST −1 < 1+k and 1−k < ST −1 u.  Remark 4.14. The condition imposed on K in Corollary 4.13 is similar to that in binomial models for short European call options in [2, p. 278]. But the optimal super replicating portfolio process is not unique in trinomial models.

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References [1] Bensaid, B., Lesne, J., Pages, H. and Scheinkman, H. (1992), Derivative Asset Pricing with Transaction Costs, Mathematical Finance 2, 63–86. [2] Boyle, P. and Vorst, T. (1992), Option Pricing in Discrete Time with Transaction Costs, J. of Finance 47, 271–293. [3] Cvitani´c, J., Pham, H. and Touzi, N. (1999), A Closed-form Solution to the Problem of Super-replication under Transaction Costs, Finance and Stochastics 3, no. 1, 35– 54. [4] Cox, J.C., Ross, S.A. and Rubinstein, M. (1979), Option Pricing: A Simplified Approach, J. of Financial Economics 7, 229–263. [5] Davis, M.H.A., Panas, V.J. and Zariphopoulou, T. (1992), European Options Pricing with Transaction Costs, SIAM J. of Control and Optimization 31, 470–498. [6] Kocinski, M. (2001), Hedging in the CRR Model under Concave Transaction Costs, Demonstratio Math. 34, 497–512. [7] Kocinski, M. (2004), Hedging of the European Option in Discrete Time under Proportional Transcation Costs, Math. Method Oper. Reserch 59, 315–328. [8] Koehl, P.E., Pham, H. and Touzi, N. (1999), Hedging in Discrete Time under Transaction Costs and Continuous-time Limit, J. Appl. Probab. 36, no. 1, 163–178. [9] Kusuoka, S. (1995), Limit Theorem on Option Replication with Transaction Costs, Annals of Applied Probability 5, 198–221. [10] Leland, H.E. (1985), Option Pricing and Replication with Transcation Costs, J. of Finance 40, 1283–1301. [11] Rutkowski, M. (1998), Optimality of Replication in the CRR Model with Transaction Costs, Applicationes Mathematicae 25, 29–53. [12] Stettner, L. (1997), Option Pricing in CRR Model with Proportional Transaction Costs: A Cone Transformation Approach, Applicationes Mathematicae 24, 475–514. Tzuu-Shuh Chiang and Shuenn-Jyi Sheu Institute of Mathematics Academia Sinica Taipei, Taiwan e-mail: [email protected] [email protected]

Progress in Probability, Vol. 65, 317–330 c 2011 Springer Basel AG


Completeness and Hedging in a L´evy Bond Market Jos´e M. Corcuera Abstract. In this paper we analyze the completeness problem in a bond market where the short rate is driven by a non-homogeneous L´evy process. Even though it is known that under certain conditions we have a kind of uniqueness of the risk neutral measure, little is known about how to hedge in this market. We elucidate that perfect replication formulas are not, in general, possible to obtain and an approximate hedging, in an L2 sense, is then the appropriate approach. Mathematics Subject Classification (2000). 60H30, 91B28, 60G51. Keywords. Bond market, term structures, arbitrage, market completeness, L´evy processes.

1. Introduction Fixed income markets can be described as markets where agents trade contracts in which deterministic flows of money are prescribed at certain times in the future. The problem is how to price these contracts and to hedge the associated risk. The risk is due to the fact that the value of the money changes with time. One unit now is, normally, more valuable that one unit tomorrow. We can describe this by a money account process . : T BT = exp r(t)dt , T ≥ 0 0

that indicates that one unit today (t = 0) is equivalent, it has the same value, to Bt units tomorrow, at time t = T . This evolution is governed by r(t), 0 ≤ t ≤ T, the interest rate, a random process, usually positive, defined on a stochastic basis (Ω, F , P, (Ft )). Unfortunately, since r is random, we do not know the price of the money in the future. Even though we might have a perfect model for r(t), This work is supported by the MCI Grant MTM2009-08218.

318

J.M. Corcuera

0 ≤ t ≤ T, it is not obvious how to price, at time 0, one unit at time T . One could think that this price is 1/BT but its value is not observed at time 0. So, to have a model for r(t), 0 ≤ t ≤ T, is not sufficient to price the so-called T zero coupon bond, a contract which guarantees the holder one unit at time T . Let P (t, T ) denote a possible price of a T -zero coupon bond at time t. Arbitrage free prices can be obtained by doing 

Bt  P (t, T ) := EQ Ft , T ≥ t, BT  where Q is any probability measure equivalent to P . Fixing the real evolution of the short interest rates is not enough for explaining the prices of the T -zero coupon bonds, but if, additionally, we choose a risk-neutral measure Q we have a model for the prices of T -zero coupon bonds for every T ≥ 0. Then the question is if by doing this we are fixing the prices of the rest of contracts, or derivatives, in this market and if we can hedge against any risk. This is the completeness problem. ˜ equivalent Note that if we have another contract with payoff X at T¯ and we take Q to P, EQ˜ (X|Ft ) would give an arbitrage free price, so in order to ensure that the prices P (t, T ), T ≥ t, match the rest of prices we need that 



Bt  Bt  EQ F = E Ft , 0 ≤ t ≤ T ∧ T¯ , ˜ t Q BT  BT  ˜ on FT¯ . This can be read as the uniqueness of the risk neutral implies that Q = Q, (or martingale) measure and, as we shall see later, this implies that we can hedge approximately, in an L2 -sense, against any risk by using self-financing portfolios with bonds with different maturities. Another way of modelling the market, with horizon T¯, is to give a dynamics for P (t, T ), 0 ≤ t ≤ T ∧ T¯, or equivalently for the forward rates f (t, T ) defined as ∂ log P (t, T ) , 0 ≤ t ≤ T ∧ T¯, ∂T and to look if the model is arbitrage free and complete. If there is Q such that the discounted value of the bond prices, which we denote P˜ (t, T ), are Q-martingales with respect to the filtration (Ft ) we will have that 

Bt  P (t, T ) = EQ Ft , 0 ≤ t ≤ T ∧ T¯ , BT  f (t, T ) = −

and if Q is equivalent to P the model will be free of arbitrage, with respect to the information (Ft ). Moreover if this Q is unique we will have that 



Bt  Bt  EQ Ft = EQ˜ Ft , 0 ≤ t ≤ T ∧ T¯ , BT  BT  ˜ on FT¯ and we will be able to replicate in an L2 -sense. So, implies that Q = Q both approaches, once we have a martingale measure Q, are equivalent. In this paper we follow the first approach by skipping, by construction, the existence of Q and by taking an appropriate risk neutral measure in such a way that

Completeness and Hedging in a L´evy Bond Market

319

we can have an approximate complete model with jumps. The paper is organize as follows, in the second section we define the model and study is basic properties. In the third section we make precise the notion of (perfect) completeness and we analyze when the model is perfectly complete. The last section is devoted to study the approximate, L2 or quasi-completeness of the model.

2. The model We assume that only the interest rate is fixed exogenously and under the historical probability P . There is not any other primary asset in the market. So, any equivalent measure can serve to price derivatives, and a zero coupon bond is a derivative where the underlying is the bank account. Then we have a bank account process (Bt )t≥0 that evolves as % t & Bt = exp r(s)ds , 0

where r(t) is the so-called short-rate. We consider a dynamics of the form  t r(t) = µ(t) + γ(s, t)dLs ,

(1)

0

here L is a non-homogeneous L´evy process, or a process with independent increments and absolutely continuous characteristics (σs2 , υs , as ):  t  t  t Lt = as ds + σs dWs + x1{|x|≤1} (J(dx, ds) − υs (dx)ds) 0 0 0 R  + ∆Ls 1{|∆Ls|>1} s≤t





t

=

as ds + 0

+



 t

t

σs dWs + 0

0

|x|≤1

BsP xdM

∆Ls 1{|∆Ls|>1} .

s≤t

The coefficients µ(t) and γ(s, t) are assumed to be deterministic and c` adl` ag. We also assume that the filtration generated by L and r is the same. Fix T > 0 and consider a T -zero-coupon bond. This is a contract that guarantees the holder 1 monetary unit at time T . Write P (t, T ), 0 ≤ t ≤ T, for the price of this contract. We know that P (T, T ) = 1, take Q equivalent to P (the historical probability) and structure preserving (with respect to L) and define  .  :   T  P˜ (t, T ) = EQ exp − r(s)ds  Ft .  0 This discounted price, whatever Q, equivalent to P , we choose, does not produce arbitrage.

320

J.M. Corcuera

From here we have that  .  : T ˜ P (t, T ) = EQ exp − r(s)ds | F t .  = exp −

0

T

µ(s)ds

0

.  = exp −

:

 × EQ

:

T

µ(s)ds

.  exp −



.



0

γ(u, s)ds dLu u

T

: | Ft

:

Γ(T )u dLu

| Ft ,

T

Γ(T )t = −





T

0

where


·



0

× EQ exp

0

Under Q ,

T

γ(t, s)ds. t

Γ(T )u dLu is still a process with independent increments so . : T

EQ exp

| Ft

Γ(T )u dLu 0

%

t



&

.

Γ(T )u dLu EQ exp

= exp

:

T

Γ(T )u dLu

0

,

t

therefore

%

&

t

P˜ (t, T ) ∝ exp

Γ(T )u dLu , 0

* ) and, since P˜ (t, T )

is, by construction, a Q -martingale, we can write

0≤t≤T

!
" t exp 0 Γ(T )u dLu 3 !
"4 . P˜ (t, T ) = P (0, T ) t EQ exp 0 Γ(T )u dLu

(2)

2.1. The forward rates The previous dynamics of P˜ (t, T ) for all maturities, T > t, can be described in terms of the instantaneous forward rates, by using the definition .  :  T t P˜ (t, T ) = exp − r(s)ds − f (t, s)ds . 0

t

Then f (t, T ) = −∂T log P˜ (t, T )



= f (0, T ) + A(T )t +

t

γ(s, T )dLs , 0

Completeness and Hedging in a L´evy Bond Market where

321

3 !
"4 t ∂T EQ exp 0 Γ(T )u dLu 3 !
"4 . A(T )t = t EQ exp 0 Γ(T )u dLu

In Eberlein, Jacod and Raible (2005), the authors assume that the forward rates are given and then study the arbitrage and completeness problems. In fact they start by assuming that (under P )  t  t f (t, T ) = f (0, T ) + α(s, T )ds + γ(s, T )dLs , 0

0

and they look for martingale measures. They obtain that their existence implies
t some constraints for the process 0 α(s, T )ds. This was already observed by Heath, Jarrow and Morton in 1992 in the case when L is continuous. The constraint for the existence of “structure preserving” martingale measures is then  t α(s, T )ds = A(T )t , 0

for some measure structure preserving Q. We have considered that Q is a given structure preserving equivalent measure, we construct the bond market with this Q and deduce the form of the forward rates which are compatible with this prices. Then, by construction, the market is arbitrage free. So, the relevant problem is the completeness of the market. We take Q and we check if this market is complete by trying to demonstrate that any (Q-squared integrable) contingent claim can be replicated by a self-financing portfolio.

3. Completeness of the market According to Eberlein, Jacod and Raible (2005) completeness is important for two reasons: • Completeness amounts to the fact that any claim can be priced in a unique way: by taking the expectation of the discounted value of the claim with respect to the (unique) equivalent martingale measure. This means, in our context, that if we choose different, structure preserving equivalent measures Q, we obtain different bond prices. • Completeness is also related with hedging and is in fact “equivalent” to the property that any square integrable (discounted) contingent claim Y can be written as  T Y = E(Y ) + Hsj dP˜ (s, Tj ), (3) Tj ∈J

0

322

J.M. Corcuera and this is equivalent to the martingale representation property   t E(Y |Ft ) = E(Y ) + Hsj dP˜ (s, Tj ). Tj ∈J

0

Instead of looking if a system of bond prices implies a unique Q we can use the martingale representation theorems for studying if and how we can replicate any contingent claim. We have seen that P˜ (t, T ) = P (0, T ) exp{Z¯t }   where Z¯t 0≤t
B ν¯t (B) = vt , Γ(T )t where {vt , t ≥ 0} are the L´evy measures of the Q non-homogeneous L´evy process L. ¯ Then, by Itˆ o’s formula and the L´evy-Itˆo representation of Z,   x  c¯2 ¯t dP˜ (t, T ) = P˜ (t−, T ) γ¯t + t + e − 1 − x1{|x|<1 ν¯t (dx))dt + c¯t dW 2 R

 B ¯t + (ex − 1) dM R



 B ¯ t + (ex − 1) dM ¯t = P˜ (t−, T ) c¯t dW R

 ) * ¯t + BtQ = P˜ (t−, T ) Γ(T )tσt dW eΓ(T )t x − 1 dM R

 ) * BP = P˜ (t−, T ) Γ(T )tσt dWt + eΓ(T )t x − 1 dM t R

 ) * + P˜ (t−, T ) Γ(T )t σt Gt + eΓ(T )t x − 1 Ht (x)νt (dx) dt. R

Where, for fixed t, Gt is a real constant and Ht (x) is Borel function. Then, Gt and Ht (x), that are assumed to be in L2 (νt (dx)), we have a risk neutral measure and corresponding bond prices determined by the mappings (fixed t and ω) : ˜

KtP

: (R, L2 (νt (dx)) → C0 [0, ∞)

 ) * Γ(T )t x ˜ (G, H(x)) → P (t−, T ) Γ(T )t σt G + e − 1 H(x)νt (dx) , R

where, C0 (0, ∞) denotes the space of continuous functions vanishing at infinity. So, we assume that

 ) * Γ(T )t x ˜ P (t−, T ) Γ(T )t σt G + e − 1 H(x)νt (dx) → 0. R

T →∞

Completeness and Hedging in a L´evy Bond Market

323

The reason is the well-known duality C0∗ [0, ∞) = M[0, ∞), the space of measures in B([0, ∞)). Uniqueness of the martingale measure implies the injectivity of these mappings a.e. We could also restrict the maturity times to be less than, say T¯ and the dual will be M[0, T ]. In Eberlein, Jacod and Raible (2005) the authors consider a bond market in the interval I := [0, T¯ ] and with zero-coupon bonds with maturities Tj ∈ J where J is a dense set in I, and they prove the uniqueness of Q. That is there is not another equivalent measure that gives the same bond prices. Equivalently this means that we should be able to replicate (in an L2 -sense) any Q-square integrable contingent claim. In fact, incompleteness, in L2 (Q) sense, implies that there is X ≥ 0 , bounded, such that it cannot be replicated and consequently, by taking X − projR X Z= , 2||X||∞ where R is the closure of the subspace of r.v. of the form (3), we have that dP ∗ := (1 + Z)dQ provides another martingale measure. Instead of looking if a system of bond prices implies the uniqueness of the martingale measure Q we can use the martingale representation theorems for studying if and how we can replicate any contingent claim. In Bj¨ ork, Kabanov and Runggaldier (1997) the authors consider measurevalued self-financing portfolios. A portfolio is a pair {gt, ht (dT )}, where • g is a predictable process (units of the risk-free asset) • ht (ω, ·) is a signed finite on B((t, +∞)) • ht (·, A) is a predictable process, for every A ∈ B((t, +∞)). ht (dT ) is interpreted as the “number” of bonds with maturities in the interval [T, T + dT ]. The value of this portfolio at time t is given by  ∞
t rs ds 0 Vt = gt e + P (t, T )ht (dT ), t

and the self-financing condition is given by  ∞ dV˜t = dP˜ (t, T )ht(dT ) t

 )  ∞ * ¯t + BtQ ht (dT ). = P˜ (t−, T ) Γ(T )t σt dW eΓ(T )t x − 1 dM R

t

˜ and Suppose now that we want to replicate X ∈ L2 (Q), then V˜T = X ˜ t ) = EQ (V˜T |Ft ) = V˜t . Mt := EQ (X|F It is well known that Mt , as every L2 (Q)-martingale, has an integral representation of the form  t  t ¯ BQ , Mt = γs dWs + ϕ(x, s)dM 0

0

R

s

324

J.M. Corcuera


where γs2 + R ϕ2 (x, s)ν(dx) < ∞ a.e. Then, the replicating self-financing portfolio should satisfy 

∞

γt = 

t

∞

ϕ(x, t) =

P˜ (t−, T )Γ(T )t σt ht (dT ) ) * P˜ (t−, T ) eΓ(T )t x − 1 ht (dT ).

(4)

t

This system, for fixed t and ω, corresponds to the mappings ˜

KtP∗ : M[0, ∞) → R × L2 (νt (dx))  ∞

 ∞ ) * Γ(T )t x ˜ ˜ h(dT ) →

P (t−, T )Γ(T )tσt h(dT ), P (t−, T ) e − 1 h(dT ) . t

t

˜

˜

It is easy to see that KtP∗ and KtP are dual, so

)

˜

KerKtP

*⊥

) * ˜ = cl Im KtP∗ .

˜

Then, uniqueness of the martingale measure (that is KtP is injective) implies that this equation can be solved in a dense set of R × L2 (νt (dx)). ˜

A very important point is that KtP∗ is a compact operator, then if L2 (ν(dx)) ˜ is not a finite-dimensional space, KtP∗ cannot be onto L2 (ν(dx)) because then ) ˜ *−1 ) ˜ *−1 ˜ KtP∗ would be bounded and the identity KtP∗ KtP∗ would be compact. So, the set where we can solve the system (4) is a proper subset of R × L2 (νt (dx)) if the support of νt is infinite. This means that a perfect (a.s.) replication of an L2 (Q) contingent claim is in general not possible. We could consider L´evy measures νt (dx) with a countable number of different jump sizes and portfolios with a countable number of different bonds, then γt =

 k≥1

ϕ(x, t) =



P˜ (t−, Tk )Γ(Tk )t σt ht (Tk ) ) * P˜ (t−, Tk ) eΓ(T )t x − 1 ht (Tk ),

k≥1

but the situation does not change: we cannot replicate (a.s.) in general an L2 (Q) contingent claim. If we have only n different jump sizes: x1 , x2 , . . . , xn we will have a system 

∞

γt = 

t

ϕ(xi , t) = t

∞

P˜ (t−, T )Γ(T )t σt ht (dT ) ) * P˜ (t−, T ) eΓ(T )t xi − 1 ht (dT ), i = 1, . . . , n,

Completeness and Hedging in a L´evy Bond Market

325

and it is reasonable to look for a measure ht (dT ) concentrated in n + 1 points, T1 (t), . . . , Tn+1 (t), γt = σt

n+1 

Γ(Ti (t))t Git

i=1

ϕ(xj , t) =

n+1 )

* eΓ(Ti (t))t xj − 1 Git , j = 1, . . . , n,

i=1

where Git is the discounted amount invested at t in bonds with maturity Ti (t). So n+1  Aji (t)Git = φj , j = 1, . . . , n + 1, i=1

where A1i (t) = Γ(Ti (t))t , Aji (t) = e

i = 1, . . . , n + 1; φ1 = γt /σt

Γ(Ti (t))t xj−1

− 1; φj = ϕ(xj−1 , t)

j = 2, . . . , n + 1, i = 1, . . . , n + 1.

And we will have a solution provided that A is non singular. Let T¯ be the horizon of the market, it can be shown that if Γ(T )t is analytic in T and t we can choose any maturity times bigger than T¯ and independent of t. A way of obtaining such Tk (t) is the following. Take T1 = T > T¯ and assume now that for any k ≥ 2 there is Tk (t), such that kΓ(T )t = Γ(Tk (t))t , then it is easy to see that det A = Γ(T )t Πnj=1 (eΓ(T )t xj − 1)2 Πn1≤i<j≤n (eΓ(T )t xj − eΓ(T )t xi ) = 0, ∀t ∈ [0, T ]. If we consider affine term structure models, where Γ (T )t is given by Γ (T )t = a (T ) b (t) + c (t) , with a (·) strictly increasing and C 1 function and b (t) = 0, we have Tk (t) = a−1 (k(a(T ) − a(t)) + a(t)). In particular this is true for the Vasiˇcek model Γ(T )t = C(1 − e−a(T −t) ), and the Ho-Lee model Γ(T )t = C(T − t).

(5)

326

J.M. Corcuera

4. L2 -completeness 4.1. Power-jump processes We have seen that a perfect replication is in general not possible, then, instead, we can try to hedge in an approximate way. Here and along the paper to hedge in an approximate way means that we have a sequence of self-financing portfolios whose L2 limit replicates the contingent claim. See Corcuera, Nualart and Schoutens (2005) and Corcuera and Guerra (2009) for more details. First of all we can rewrite our model as dP˜ (t, T ) = dZt , P˜ (t−, T ) and where (Zt )0≤t
=

|x|>1 Γ(T )2t σt2 ,



νt (G) =

R

) * 1G eΓ(T )t x − 1 υt (dx) ,

G ∈ B (R) .

Note that the L´evy measures νt have support in (−1, +∞). Set  (i) Zt = (∆Zs )i , i ≥ 2, 0<s≤t

where ∆Zs = Zs − Zs− . Define the Q-martingales (i)

Ht

(i)

(i)

= Zt − EQ (Zt ),

i = 1, 2, . . . ,

(1)

with Zt = Zt . Assume that {νt }t∈[0,T¯ ]

satisfies, for some ε > 0 and λ > 0,  exp(λ|x|)νt (dx) < ∞.

sup t∈[0,T¯ ]

(6)

(−ε,ε)c

Then it is known that any Q-square-integrable martingale Mt can be expressed as ∞  t  ¯ (n) , βsn dH Mt = M0 + s n=1

0

¯ (n) are the orthogonal version of the H (n) defined previously and the β i where H are predictable processes. Then we can complete our market (with finite horizon T¯ < T ), M , with a (i) series of additional assets, Y (i) = {Yt , 0 ≤ t ≤ T¯ } , based on the above-mentioned processes:
t (i) (i) Yt = e 0 rs ds Ht , i ≥ 2, 0 ≤ t ≤ T¯ , the so-called “power-jump” assets.

Completeness and Hedging in a L´evy Bond Market

327

Proposition 1. The market model, MQ , obtained by enlarging the market M with the power-jump assets is complete, in the sense that any square-integrable contingent claim X ∈ L2 (Q) can be replicated by a self-financing portfolio. Proof. Let X be a square-integrable (with respect to Q) contingent claim. Consider
T¯ the squared-integrable martingale Mt := EQ (e− 0 rs ds X|Ft). We can write dMt =

∞ 

(n)

¯ βtn dH t

∞

= βt1

n=1

dP˜t  n ¯˜ (n) + βt dYt , P˜t− n=2

where Pt := P (t, T ). Then if we take a self-financing portfolio: ((φit )i≥1 )0≤t≤T , where φ1 denotes the number of units of the stock, and (φi )i≥2 the number of power-jump assets of different order, we will have that the discounted value of this portfolio evolves as ∞  (n) dV˜t = φ1t dP˜t + φnt dY¯˜t . n=2

φ1t

βt1 ˜ Pt−

φit

βti

So, by taking = and = we obtain the replicating portfolio. See Corcuera, Nualart and Schoutens (2005) and Corcuera and Guerra (2009) for more details.  4.2. Power payoffs In certain cases we can obtain hedging formulas directly, by using the Itˆ o formula. In fact, assume that the discounted price of a contingent claim at time t can be written as F˜ (s, P˜s ), F˜ smooth, where Pt := P (t, T ) has a single maturity time T , then by the Itˆ o formula   +∞  ˜ ˜ ∂ F ∂ F ˜ s, dF˜ (s, P˜s ) = dP˜s + F˜ (s, P˜s− (1 + y)) − F˜ (s, P˜s− ) − y P˜s− dM ∂ P˜s ∂ P˜s −∞ ˜ s is the compensated random measure associated with Z. where dM Then if we assume now that F˜ (s, P˜s− (1 + y)) is analytical in y we have  +∞  ∂ F˜ ˜ 1 ∂ k F˜ ˜ ˜ ˜s dF (s, Ps ) = dPs + y k dM k ∂ P˜s −∞ k≥2 k! ∂y |y=0 =

 1 ∂ k F˜ ∂ F˜ ˜ dPs + dY˜ (k) . k! ∂y k |y=0 s ∂ P˜s

(7)

k≥2




¯ T

For instance, consider derivatives with payoff e 0 rs ds P˜Tk¯ , k ≥ 2. Then its discounted price will be given by ) ˜ *

) ˜ *

PT¯ k PT¯ k F˜ (k) (t, P˜t ) = EQ (P˜Tk¯ |Ft ) = P˜tk EQ |Ft = P˜tk EQ P˜t P˜t (k) k ¯ ˜ = ϕ (t, T , T )Pt .

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Then, by using (7), we have that these derivatives can be replicated by using the power-jump assets

k k F˜ (k) (t, P˜t− ) ˜  ˜ (k) ˜ k ˜ (i) (k) ˜ ˜ dF (t, Pt ) = F (t, Pt− ) dPt + dYt . ˜ i Pt− i=2 Define F˜ (1) (t, P˜t ) = P˜t , and (1)

dY˜t

=

dP˜t . P˜t−

Then we can write dF˜ (k) (t, P˜t ) =



k ˜ (i) (k) ˜ F (t, Pt− ) dYt i i=1

k 

and, by a combinatorial argument, k

 k 1 (k) ˜ dYt = (−1)k−i dF˜ (i) (t, P˜t− ). (i) ˜ ˜t− ) i F (t, P i=1 The market model, MQ , obtained by enlarging the market M with the derivatives F (k) , k ≥ 2 is complete, in the sense that any square-integrable contingent claim X ∈ L2 (Q) can be replicated by a self-financing portfolio. 4.3. Completion with bonds In our market with horizon T¯ we have considered only the bond with maturity T , power-jump assets or derivatives with power payoffs, but obviously we can consider bonds with different maturity time to complete the market. Then the question is if these bonds are sufficient to complete the market. We know that, to complete the market, it is sufficient to produce discounted payoffs with values P˜ (T¯ , T )k , k ≥ 2 at time T¯. Let us consider again affine term structure models such that Γ (T )t is given by Γ (T )t = a (T ) b (t) + c (t) , (8) where a (·) is a strictly increasing function and b (·) = 0. Proposition 2. If the term structure model satisfies (8), then the discounted price function of the contingent claim with payoff X (k) is given by ) *   F+(k) t, P+(t, T ) = ψ (k) t, T , T P+(t, Tk (t))Bt1−k , where ψ (k) is a deterministic function and Tk (t) = a−1 (k(a (T ) − a(t)) + a(t)) .

Completeness and Hedging in a L´evy Bond Market

329

Proof. Using the formula deduced for P+(t, T ) in (2), we have that  t P (t, T ) ∝ exp{(a(T ) − a(t)) b(u)dLu }.

(9)

0

So,



t

P (t, T ) ∝ exp{k (a(T ) − a(t))

b(u)dLu},

k

0

and taking Tk (t) = a−1 (k (a(T ) − a(t)) + a(t)) we have that  t P (t, T )k ∝ exp{(a(Tk ) − a(t)) b(u)dLu } ∝ P (t, Tk ). 0

Finally P (t, T )k P+ (t, T )k = ∝ P+(t, Tk )Bt1−k . Btk



4.4. The hedging portfolios The idea is to use the fact that ) *   F+ (k) t, P+ (t, T ) = ψ (k) t, T , T P+ (t, Tk (t))Bt1−k is a martingale and from this, we obtain ) *   dF+ (k) t, P+ (t, T ) = ψ (k) t, T , T Bt1−k dP+ (t, Tk (t)) and

(k) dF+s dP+(s, Tk (s)) = . (k) P+ (s, Tk (s)) F+s

We have finally (k)

dY˜t

=

k

 k dP+(s, Ti (s)) (−1)k−i , i P+ (s, Ti (s)) i=1

which gives us a way of obtaining the replicating self-financing portfolio and the corresponding hedging formula. 4.5. Another set of basic bonds In our market with horizon T we have considered first only the bond with maturity T > T and we want to complete the market with bonds with maturity times T ∗ > T¯. Then the question is if these bonds are sufficient to complete the market. First we see that any measurable function h can be approximate by a linear combination of P (t, T ∗ ) with T ∗ > T¯ . In fact if EQ [h(P (t, T ))P (t, T ∗ )] = 0, for all T ∗ > T , we have that h(P (t, T )) = 0 a.s. since, by (9), #  t

%  t &$ ∗ EQ [h(P (t, T ))P (t, T )] = EQ g b(s)dLs exp λ b(s)dLs = 0, 0

0

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J.M. Corcuera

with g(x) = h(Ce(a(T )−a(t))x ), for all λ = a(T ∗ ) − a(t). Then, in particular, this implies that  P (t, T )k = λkj (t)P (t, Tjk (t)) j∈J

for certain countable set of maturity times {Tjk (t), j ∈ J} in (T¯ , ∞) and ) * *k  )   P (t, T )k F+(k) t, P+(t, T ) = ϕ(k) t, T , T P+(t, T ) = ϕ(k) t, T , T Btk    = ϕ(k) t, T , T B 1−k λk (t)P˜ (t, T k (t)). t

j

j

j∈J

Consequently the result follows as in the previous subsection. Remark. Note that if in the expression



¯t + dMt = γt dW

R

BQ , ϕ(x, s)dM s

ϕ(·, t) is analytic for fixed t, then we can write a.s. that  dMt = ak dY˜s(k) , k≥1

and the corresponding contingent claim will be replicated in a perfect (a.s.) sense. This was already observed, by using different arguments, in Bj¨ ork, Kabanov and Runggaldier (1997).

References [1] Bjork, T., Kabanov, Y. and Runggaldier, W. (1997) Bond market structure in the presence of marked point processes, Math. Finance 7(2), 211–239 [2] Corcuera, J.M., Guerra (2010) Dynamic complex hedging in additive markets. Quantitative Finance 10(9), 1023–1037. [3] Corcuera, J.M., Nualart, D. and Schoutens, W. (2005) Completion of a L´evy Market by Power-Jump-Assets. Finance and Stochastics 9(1), 109–127. [4] Eberlein, E., Jacod, J. and Raible, S. (2005) L´evy term structure models: Noarbitrage and completeness, Finance and Stochastics, 9, 67–88. [5] Heath, D, Jarrow, R. and Morton, A. (1992) Bond Pricing and the Term Structure of Interest Rates, Econometrica, 60, 77–106. Jos´e M. Corcuera Universitat de Barcelona Gran Via de les Corts Catalanes 585 E-08007 Barcelona, Spain e-mail: [email protected]

Progress in Probability, Vol. 65, 331–346 c 2011 Springer Basel AG


Asymptotically Efficient Discrete Hedging Masaaki Fukasawa Abstract. The notion of asymptotic efficiency for discrete hedging is introduced and a discretizing strategy which is asymptotically efficient is given explicitly. A lower bound for asymptotic risk of discrete hedging is given, which is attained by a simple discretization scheme. Numerical results for delta hedging in the Black-Scholes model are also presented. Mathematics Subject Classification (2000). 60F25, 60H05. Keywords. Asymptotic efficiency, discrete hedging, stopping time, Riemann sum.

1. Introduction This article considers hedging a derivative by dynamically rebalancing a portfolio of the underlying asset of the derivative and a risk-free asset under a realistic condition that the rebalancing is limited to occur discretely. Under the condition, a future payoff cannot be hedged in general even in the Black-Scholes nor the other complete market models. In incomplete market models, hedge-error is decomposed into two parts: one due to the incompleteness of the market and the another due to the fact that we cannot rebalance a portfolio continuously. We deal with the difference between what can be hedged by continuous rebalancing and what can be realized by discrete rebalancing. Therefore, we are not concerned with the portfolio selection in the usual sense but the selection of discretization schemes for a given continuous-time portfolio strategy. We present a lower bound for the asymptotic mean squared error. We introduce the notion of asymptotic efficiency for discrete hedging and constructs a discretization scheme which is efficient in that it attains the lower bound asymptotically. This problem is reduced to analyzing discretization error of stochastic integrals. Here, rebalancing times are corresponding to a partition for the Riemann sum associated with a stochastic integral. Naturally, we suppose that the rebalancing times are increasing stopping times. In the case that the times are equidistant, Rootz´en (1980) gave the convergence rate and the asymptotic distribution of the discretization error. Bertsimas, Kogan and Lo (2000), Hayashi and Mykland (2005)

332

M. Fukasawa

treated the same problem in the context of hedge-error. Tankov and Voltchkova (2009) extends the results to discontinuous semimartingales. Gobet and Temam (2001), Geiss (2002) among others gave asymptotic estimates in the Lp sense under the Black-Scholes model. However, from practical viewpoints, it seems not natural to suppose that the rebalancing times are deterministic. They should be determined adaptively to, for example, the number of shares currently held, the number of shares that should be held according to a given continuous-time strategy, and the current price of the underlying asset. Nevertheless, it remains an open problem to construct an optimal strategy of rebalancing times. Martini and Patry (1999) considered the minimization of L2 hedge-error under the risk-neutral measure in the Black-Scholes model. They analyzed the optimal stopping times given a fixed number of transactions: their existence, uniqueness and numerical construction. We take a different approach to the discrete hedging problem taking the following aspects into consideration; 1) there is no need to fix the number of transactions in practice as far as it is finite, 2) it is not so clear which probability measure we should take in the L2 minimization problem, and 3) an explicit construction of a discretization scheme is preferable even if it is only an approximation of the optimal solution. We develop an asymptotic theory to tackle the discrete hedging problem. A rigorous formulation is given in Section 2. The notion of asymptotic efficiency for discrete hedging is introduced and a discretizing strategy which is asymptotically efficient is constructed in Section 3. Numerical results for delta hedging in the Black-Scholes model are also presented in Section 4.

2. Formulation 2.1. Notation and definitions Here we give a rigorous formulation of our problem. For the sake of brevity, we suppose that the risk-free rate is always 0 throughout this article. Suppose that an asset price process Y is a continuous semimartingale on a stochastic basis (Ω, F , {Ft }, P ). Denote by E the expectation with respect to the measure P . If we write G · F for adapted processes G and F , it stands for the stochastic integral or the Stieltjes integral of G with respect to F . Fix a stopping time T which stands for the maturity of a derivative. Since we are interested in the discretization of hedging strategy, we suppose that a hedging strategy X defined on [0, T ] is given and the stochastic integral  T X ·Y = Xs dYs (1) 0

is what we should replicate. Due to our restriction on rebalancing, what we can realize is of the form  T ∞  n n ¯ n (Yτ n ∧T − Yτ n ∧T ), X ·Y = Xs dYs = X (2) j j+1 j 0

j=0

Asymptotically Efficient Discrete Hedging

333

where we define X n as n ¯ jn , s ∈ [τjn ∧ T, τj+1 Xsn = X ∧ T)

¯ n which are Fτ n -measurable for each j ≥ 0. Here, τ n are with random variables X j j j stopping times such that 0 = τ0n < τ1n < · · · < τjn < · · · ,

a.s.,

and that for any stopping time τ such that τ < T , it holds Nτn := max{j ≥ 0; τjn ≤ τ } < ∞, τjn

a.s.

(3)

Nτn

Note that stands for jth rebalancing time and is the number of transactions up to time τ . The problem treated here is that even if Xs is given explicitly for any s ∈ [0, T ], we can only rebalance our portfolio discretely in time. For example, in the Black-Scholes world dYt = µYt dt + σYt dWt ,

(4)

the future payoff f (YT ) can be hedged by continuously rebalancing as  T f (YT ) = P (0, Y0 ) + Xs dYs 0

with



Xt = ∂y Pf (t, Yt ), Pf (t, y) =

√ f (y exp(−σ 2 (T − t)/2 + σ T − tz))φ(z)dz,

% 2& 1 z φ(z) = √ exp − . 2 2π Although the hedging strategy X is given explicitly here, we can only have the Riemann sum (2) in practice. The stochastic part of the corresponding hedge-error is then given by Z n = X · Y − X n · Y. ¯ n) Our aim here is to estimate Z n and to construct an optimal sequence of (τjn , X j in some sense. The simplest strategy (stopping times) is equidistant one; where

τjn = jhn , j = 0, 1, . . . , for a positive constant hn > 0. We can consider, for example, ! " n τ0n = 0, τj+1 = inf t > τjn ; |Xt − Xτjn |2 = hn on the other hand, which was studied in Karandikar (1995). We want to choose a strategy among these many candidates. Here we explain the role of the artificial perturbation variable n. By the n nature of the problem, the durations of transactions τj+1 − τjn can be considered sufficiently small relative to [0, T ]. For example, if the maturity is one year T = 1 n and if the portfolio is rebalanced approximately every weekday, then τj+1 − τjn ≈ n n n n 1/250. Therefore, dealing with a sequence τ = {τj }j with τj+1 − τj → 0 as

334

M. Fukasawa

n → ∞, we can expect that the distribution of Z n is approximated well by its limit as n → ∞. This asymptotic argument enables us to derive a simple lower bound of risk which is attained by a simple strategy as we will see in the next section. The precise description of the asymptotic condition is given later. We call ¯ n )}j a discrete hedging strategy. the sequence {π n }n with π n = {(τjn , X j ¯ n = Xτ n for each j ≥ 0. Put The simplest and most natural is to take X j

j

ˆn · Y Zˆ n = X · Y − X with ˆ n = Xτ n , s ∈ [τ n ∧ T, τ n ∧ T ). X s j j+1 j ˆ n . Note that there is no need We say X n is natural and π n is natural if X n = X to take such a special form of X n from practical point of view. Nevertheless, we will see that our lower bound of asymptotic risk is attained by a natural strategy. 2.2. Discrete hedging risk Here we shall introduce a risk criterion in terms of the stochastic process Z n . At the maturity T , the hedge-error is ZTn , therefore the simplest criterion would be E[|ZTn |2 ]. It is however not clear which measure E we should take. Martini and Patry (1999) and many preceding studies on hedging in incomplete markets took an equivalent martingale measure as E. If it is the case, Z n = (X − X n) · Y is a local martingale, so that E[|ZTn |2 ] = E[Z n T ] provided that Z n T is integrable. Notice that Z n  can be interpreted as the integrated variance of Z n , so that it also serves as a risk criterion. Taking this into consideration, we introduce E[Z n T ] (5) as our risk criterion. We stress that E is not necessarily an equivalent martingale measure here. We will see later that an asymptotically efficient strategy with respect to the risk defined as (5) does not depend explicitly on E. In other words, we do not need to estimate the drift parameter under E. This is important because statistical estimation of the drift parameter for the physical measure is not stable. Now, we fix our stochastic basis (Ω, F, {Ft }, P ) and introduce several definitions. Definition 1. An adapted process ϕ is said to be locally bounded on [0, T ) if there exists a sequence of stopping times σ m with σ m < T , σ m → T a.s. such that ϕσm ∧· is bounded. Definition 2. An adapted process M is said to be a local martingale on [0, T ) if there exists a sequence of stopping times σ m with σ m < T , σ m → T a.s. such that Mσm ∧· is a bounded martingale.

Asymptotically Efficient Discrete Hedging

335

Definition 3. An adapted process ϕ is said to be F -continuous for an increasing continuous adapted process F if ϕ is continuous and for any stopping times τ1 , τ2 with τ1 < τ2 Fτ1 = Fτ2 , ϕ is constant on the interval [τ1 , τ2 ]. Let us assume the following structure condition: Condition S. Y is a one-dimensional continuous semimartingale and there exist 1. a continuous adapted process M which is a local martingale on [0, T ), 2. an adapted process ψ which is locally bounded on [0, T ) 3. an M -continuous adapted process κ which is strictly positive on [0, T ) such that it holds on [0, T ) that X = X0 + ψ · M  + M, Y  = κ2 · X. The above structure condition holds, for example, in the preceding BlackScholes model (4) with M = (σY Γ) · W, κ = Γ−1 , Γt = ∂y2 Pf (t, Yt ) provided that the payoff f is a convex function whose right derivative f  is not a continuous singular function. Here we use the fact that ∂y2 Pf is a strictly positive continuous function on [0, T ) × (0, ∞) if the Stieltjes measure defined by the right continuous increasing function f  has a discrete or absolutely continuous component. A sufficient condition is that f is a convex function which is piecewise C 2 . See Section 4 for the details. Note that we allow ΓT = 0 as well as ΓT = ∞, so that the call and put payoffs are included. If f is not convex, then Γ can reach 0 in [0, T ) so the third condition may fail. It also remains for further research to include the case Y is multi-dimensional and discontinuous.

3. Main results: asymptotic efficiency The notion of the asymptotic efficiency for discrete hedging and an asymptotically efficient strategy are given in this section. It should be noted first that our risk (5) converges to 0 as n → ∞ if, for example, n sup |τj+1 ∧ T − τjn ∧ T | → 0, j≥0

sup 0≤j≤NTn

¯ jn − Xτ n | → 0 |X j

in probability under suitable integrability conditions. The more frequently is the portfolio rebalanced, the less hedge-error does it result in. To make the problem realistic, we introduce a cost for rebalancing. Let us consider minimizing C(E[Nτn ], E[Z n τ ]) for a given stopping time τ with τ ≤ T and for a given function C : [0, ∞)2 → [0, ∞) which is increasing in both variables. Suppose for a while that we are given such an inequality that E[Nτn ]E[Z n τ ] ≥ K

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M. Fukasawa

for any discrete hedging strategy, where K > 0 is a constant depending on τ . Then apparently it holds that C(E[Nτn ], E[Z n τ ]) ≥ C(E[Nτn ], K/E[Nτn]). Therefore, if there exists a discrete hedging strategy {π n }n which attains the lower bound K for each n and τ and E[Nτn ] → ∞ (n → ∞), then the solution of the minimization problem is given by π n for a certain n which minimizes C(E[Nτn ], K/E[Nτn]). In the following, we realize this idea in an asymptotic sense. Let us introduce a class of discrete hedging strategies which seems sufficiently ¯ n )}j,n satisfying large. Denote by T the class of discrete hedging strategies {(τjn , X j the following condition. Condition T . There exists a sequence of stopping times σ m with σ m < T , σ m → T , a.s., (m → ∞) such that for each m, • it holds that n ∧σ m − Xτ n ∧σ m | → 0, sup |Xτj+1 (6) j j≥0

in probability as n → ∞, • there exists a deterministic sequence Km,n with Km,n → ∞ (n → ∞) such that n ∧σ m − Xτ n ∧σ m |Fτ n ∧σ m ], E[Nσnm ] sup E[Xτj+1 j j j≥0

(7)

Km,n sup |Xτjn∧σm − Xτnjn ∧σm | j≥0

are uniformly bounded in n, • the sequences E[Nσnm ]Zˆ n σm , E[Nσnm ]Z n σm are uniformly integrable in n.

(8)

Note that Z n  = (X − X n )2 · Y . Condition T is satisfied if, for example, it holds that dXt = gt dt for such an adapted process g that both g and 1/g are locally bounded on [0, T ) and that n ¯ jn − Xτ n |2 ≤ ahn a.s. sup |τj+1 ∧ T − τjn ∧ T | ≤ ahn , NTn ≤ a/hn , sup |X 0≤j≤NTn

j≥0

j

with a constant a and a sequence hn → 0. In fact (6) and (7) follow immediately. To see (8), notice that there exists a localizing sequence σ m of stopping times such that E[Z n σm ] ≤ Cm hn + E[Zˆ n σm ]  T ˆ n m )2 ]dt ≤ Cm hn , ≤ Cm hn + Cm E[(Xt∧σm − X t∧σ 0

where Cm is a generic constant depending on m. The convergence (6) is weaker than the usual high frequency assumption n sup |τj+1 ∧ T − τjn ∧ T | → 0 j≥0

Asymptotically Efficient Discrete Hedging

337

ˆn · Y in probability. In such an asymptotic situation, the natural Riemann sum X converges to the stochastic integral X · Y . It is therefore not a serious restriction ¯ n − Xτ n | → 0 as in (7). The first quantities of (7) and (8) are related to suppose |X j j each other and their uniform properties serve as a condition on the regularity of the sequence of stopping times τjn . The last one of (8) actually controls the difference ˆ n . Besides, the integrability condition on Z n  is preferable in between X n and X terms of minimizing the hedge-error. Theorem A. For any {π n }n ∈ T and any stopping time τ with τ ≤ T a.s., it holds lim inf E[Nτn ]E[Z n τ ] ≥ n→∞

1 |E[κ · Xτ ]|2 . 6

Proof. There exists a sequence of stopping times σ m with σ m < T , σ m → T , a.s.(m → ∞) such that for each m, Mσm ∧· is a bounded local martingale on [0, τ ] and the adapted processes Yσm ∧· , Xσm ∧· , κσm ∧· ,

1 , ψσm ∧· κσm ∧·

are bounded on [0, τ ]. Without loss of generality, we assume that the statements of Condition T hold with the same localizing sequence σ m . By monotonicity, it suffices to show for each m 1 lim inf E[Nσnm ∧τ ]E[Z n σm ∧τ ] ≥ |E[κ · Y σm ∧τ ]|2 . n→∞ 6 Hence, we can suppose without loss of generality that M itself is a bounded martingale on [0, τ ] and that Y , X, κ, 1/κ, ψ theirselves are bounded on [0, τ ] as well as that (6), (7), (8) are satisfied for σ m ≡ τ . Now, define uniformly bounded adapted processes κn and ψ n on [0, τ ] as n κns = κτjn , ψsn = ψτjn , s ∈ [τjn ∧ τ, τj+1 ∧ τ ).

By Itˆo’s formula, we have  t n Z t = (Xs − Xsn )2 dY s 0  t  t n 2 n 2 = (Xs − Xs ) |κs | dXs + (Xs − Xsn)2 (κ2s − |κns |2 )dXs 0

0

∞ " 1 2 ! n 4 n 4 n ∧t − X n n = κτjn (Xτj+1 ) − (X − X ) n τj ∧t τj ∧t τj ∧t 6 j=0  t  2 t n2 n 3 − |κ | (Xs − Xs ) dXs + (Xs − Xsn )2 (κ2s − |κns |2 )dXs . 3 0 s 0

Let us see

# lim E

n→∞



$

τ

(Xs −

E[Nτn ] 0

Xsn )2 (κ2s

−

|κns |2 )dXs

= 0.

(9)

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M. Fukasawa

In fact, putting



n = sup |κ2s − |κns |2 |, V n = E[Nτn ] 0≤s≤τ

τ

(Xs − Xsn )2 dXs , 0

we have



τ

(Xs − Xsn )2 |κ2s − |κns |2 |dXs ≤ n V n .

E[Nτn ] 0

By the assumptions, n is uniformly bounded and V n is uniformly integrable. By Karatzas and Shreve (1991), 3.4.5 (iv), the M -continuity of κ implies that n → 0 in probability under (6). Since n is bounded, n V n is uniformly integrable, which implies E[n V n ] → 0. Similarly, we can show # τ $ n n 2 n 3 E[Nτ ]E |κs | (Xs − Xs ) dXs 0 # τ $ n n 2 n 3 = E[Nτ ]E |κs | (Xs − Xs ) ψs dXs → 0. 0

Here we have used the M -continuity of X instead of κ. So far, we have lim inf E[Nτn ]E[Z n τ ] n→∞   ∞ ! "  1 n 2 n 4 n 4 . n ∧τ − X n = lim inf E[Nτ ]E  κτjn (Xτj+1 τj ∧τ ) − (Xτjn ∧τ − Xτjn ∧τ ) n→∞ 6 j=0 Let us denote Ej [·] = E[·|Fτjn ∧τ ] and put ˇ j = Xτ n ∧τ − Ej [Xτ n ∧τ ], αj = Ej [Xτ n ∧τ ] − X nn , βj = Xτ n ∧τ − X nn . X τj ∧τ τj ∧τ j+1 j+1 j+1 j Then

 ∞ !  E κ2τ n (Xτ n

j+1 ∧τ

j

j=0



=E  =E

− Xτnjn ∧τ )4 − (Xτjn ∧τ

j



n

Nτ 

 " − Xτnn ∧τ )4 

ˇ j + αj )4 − βj4 ) κ2τjn ((X

j=0 n

Nτ 

,

4

ˇ j4 ] + 4αj Ej [X ˇ j3 ] + 6α2j Ej [X ˇ j2 ] + α4j − βj  . κ2τjn Ej [X

j=0

Let us see that



 n  Nτ  , lim E[Nτn ]E  κ2τjn α4j − βj4  = 0.

n→∞

j=0

In fact, putting n ∧τ ] − Xτ n ∧τ = αj − βj , α ˆ j = Ej [Xτj+1 j

Asymptotically Efficient Discrete Hedging it suffices to observe that  n  Nτ  lim E[Nτn ]E  κ2τjn α ˆ 4j  = 0, n→∞

j=0

 n  Nτ  lim E[Nτn ]E  κ2τjn α ˆ 2j βj2  = 0,

n→∞

j=0

 lim E[Nτn ]E 

n→∞

 lim E[Nτn ]E 

n→∞



n

Nτ 

κ2τjn |ˆ αj |3 |βj | = 0,

j=0



n

Nτ 

339

(10)

κ2τjn |ˆ αj ||βj |3  = 0.

j=0

n ∧τ − Xτ n ∧τ . Let us prove (10). Let C be a generic constant and put Qj = Xτj+1 j  n   n  Nτ Nτ   E[Nτn ]E  κ2τjn α ˆ 4j  ≤ CE[Nτn ]E  |Ej [Qj ]|4 

j=0

j=0

 n  Nτ  C CE[Xτ ] ≤ E Ej [Qj ] = . E[Nτn ]2 E[Nτn ]2 j=0

We have used Condition S and (7) for the first and second inequalities respectively. The right-hand side converges to 0 since by (6), 0<

Xτ ≤ sup Qj → 0, lim inf E[Nτn ] ≥ E[lim inf Nτn ] = ∞. n→∞ n→∞ Nτn j≥0

Next, again by Condition S and (7), we have   n 2  n   n    Nτ Nτ Nτ      |E[Nτn ]|2 E  κ2τjn |ˆ αj |3 |βj | ≤ C|E[Nτn ]|2 E  |Ej [Qj ]|5  E  Qj βj2    j=0 j=0 j=0 ≤ as well as



E[Nτn ]E 

n

Nτ 

CE[Xτ ] (E[Z n τ ] + E[Zˆ n τ ]) → 0, E[Nτn ]2



 n  Nτ  κ2τjn α ˆ 2j βj2  ≤ CE  Qj βj2  ≤ C(E[Z n τ ] + E[Zˆ n τ ]) → 0.

j=0

j=0

Here we have used that n  τj+1  ∧τ βj2 Qj ≤ 2 (Xt − Xτnjn ∧τ )2 dXt + 2 τjn ∧τ

n τj+1 ∧τ

τjn ∧τ

(Xt − Xτjn ∧τ )2 dXt .

The last one of (10) follows from  n   n  Nτ Nτ   E[Nτn ]E  κ2τjn |ˆ αj ||βj |2  ≤ CE[Nτn ]E  Qj |βj |2  j=0

j=0

≤ and Condition T .

CE[Nτn ](E[Z n τ ]

+ E[Zˆ n τ ])

340

M. Fukasawa Now, since ˇ 4 ] + 4αj Ej [X ˇ 3 ] + 6α2 Ej [X ˇ 2] Ej [X j j j j   ˇ 3 ] 2 2 |Ej [X ˇ 3 ]|2 Ej [X j j 4 2 ˇ j ] + 6Ej [X ˇ j ] αj + = Ej [X − 2 ˇ 2] ˇ 3 3Ej [X E [ X j j j]

and ˇ 4] − Ej [X j

ˇ 3 ]|2 2 |Ej [X j ˇ 2 ]|2 ≥ |Ej [X j ˇ 2] 3 Ej [X j

by Lemma B below, we have

 n  Nτ  1 ˇ j2 ]|2  . lim inf E[Nτn ]E[Z n τ ] ≥ lim inf E[Nτn ]E  κ2τjn |Ej [X n→∞ n→∞ 6 j=0

On the other hand, by Cauchy-Schwarz and Jensen’s inequalities, it holds that   n 2   2     Nτ Nτn   5     1 2 2 n E  ˇ   ˇ    κτjn Ej [Xj ]  = E 1 + Nτ 5 κτjn Ej [Xj ]   n 1 + Nτ j=0     j=0   2  Nτn  1    ˇ 2]  n Ej [X ≤E[1 + Nτn ]E (1 + Nτn ) κ  τ j j  1 + Nτn  j=0  n  Nτ  ˇ 2 ]|2  . ≤E[1 + Nτn ]E  κ2τjn |Ej [X j j=0

It follows then that

  n 2   Nτ    1 n n 2  ˇ   n lim inf E[Nt ]E[Z τ ] ≥ lim inf E κτj Ej [Xj ]  . n→∞ n→∞ 6   j=0

Notice that  n   n Nτ Nτ !   ˇ j2 ] = E  E κτ n Ej [X κτ n (Xτ n j

j=0

j+1 ∧τ

j

"



− Xτjn ∧τ )2 − α ˆ 2j 

j=0

and that by Condition S and (7),  n   n  Nτ Nτ   CE[Xτ ] E κτjn α ˆ 2j  ≤ CE  |Ej [Qj ]|2  ≤ → 0. E[Nτn ] j=0 j=0 It remains to observe that  ∞  2 n n n κτj (Xτj+1 ∧τ − Xτj ∧τ ) − κ · Xτ = 2 j=0

0

τ

κns (Xs − Xsn )dXs − (κ − κn ) · Xτ ,

Asymptotically Efficient Discrete Hedging and that

#

$

τ

κns (Xs

E

341

−

Xsn)ψs dXs

→ 0, E[(κ − κn ) · Xτ ] → 0.

0

These convergences follow from the bounded convergence theorem because κ and X are M -continuous.  ˇ be a random variable with E[X] ˇ = 0 and E[X ˇ 4 ] < ∞. Then Lemma B. Let X ˇ 4] ˇ 3 ]|2 E[X |E[X − ≥ 1. ˇ 2 ]|2 ˇ 2 ]|3 |E[X |E[X ˇ is a Bernoulli random variable. The equality holds if and only if X Proof. This is known as Pearson’s inequality or Kurtosis-Skewness inequality. The proof is as follows. ˇ 3 ]|2 = |E[X( ˇ X ˇ 2 − E[X ˇ 2 ])]|2 ≤ E[X ˇ 2 ]E[(X ˇ 2 − E[X ˇ 2 ])2 ] |E[X ˇ 2 ](E[X ˇ 4 ] − |E[X ˇ 2 ]|2 ). = E[X



Definition 4. A discrete hedging strategy {π n }n is said to be asymptotically efficient if there exists a sequence of stopping times σ m with σ m < T , σ m → T a.s. such that for each m, it holds lim E[Nσnm ]E[Z n σm ] =

n→∞

1 |E[κ · Xσm ]|2 . 6

¯ n )}j ; for a deterministic Theorem C. Consider the following strategy π n = {(τjn , X j sequence hn with hn → 0, n ¯ jn = Xτ n . τ0n = 0, τj+1 = inf{t > τjn ; |Xt − Xτjn |2 ≥ hn /κτjn }, X j

Then, {π n }n ∈ T and {π n }n is asymptotically efficient. Proof. Let us prove that {π n }n ∈ T . By Condition S, there exists a sequence of stopping times σ m with σ m < T , σ m → T a.s. such that for each m, Mσm ∧· is a bounded martingale and Yσm ∧· , Xσm ∧· , κσm ∧· ,

1 κσm ∧·

, ψσm ∧·

are bounded. Suppose that there exists a set Ω ⊂ Ω such that n ∧σ m − Xτ n ∧σ m | > 0 lim sup sup |Xτj+1 j

n→∞

j≥0

on Ω . Then for each ω ∈ Ω , there exists a subsequence (sk , tk ) of the sequence of n two points (τjn (ω) ∧ σ m (ω), τj+1 (ω) ∧ σ m (ω)) such that inf |Xtk (ω) − Xsk (ω)| > 0. k

342

M. Fukasawa

Since (sk , tk ) is a sequence in the compact set [0, σ m (ω)] × [0, σm (ω)], there exists a further subsequence (ˆ sk , tˆk ) which converges a point (a, b). By the construction, (ω) < X we have Xa b (ω). This implies also a < b. On the other hand, tˆk = inf{t > sˆk ; |Xt (ω) − Xsˆk (ω)|2 ≥ hnk /κsˆk (ω)} with some subsequence nk of n. Hence a < b implies that X(ω) is constant on [a, b]. Thus we obtain P (Ω ) = 0. Let us show (8). By definition of τjn and the boundedness of 1/κ there exists a constant am such that 2 n ∧t − Xσ m ∧τ n ∧t | ≤ am hn . sup |Xσm ∧τj+1 j

t≥0,j≥0

This implies that Z n σm = (X − X n )2 · Y σm ≤ hn a2m Xσm . Besides, by the boundedness of κ, there exists a constant Nσnm ≤ am h−1 n

Nσnm



n (Xτjn − Xτj−1 )2 ≤ am h−1 n

j=1

∞ 

am

(11)

with

2 n ∧σ m − Xτ n ∧σ m ) , (Xτj+1 j

(12)

j=0

so that there exists a constant am with Hence E[Nσnm ]Z n σm

 E[Nσnm ] ≤ am h−1 (13) n (E[Xσm ] + am ). n n m ˆ = E[Nσm ]Z σ is uniformly bounded. To see (7), put

n ∧σ m − Xτ n ∧σ m |Fτ n ∧σ m ] Dj = E[Xτj+1 j j

and observe that by Itˆ o’s formula,  Dj ≤ hn 1/κ·∧σm ∞ − 2E

n τj+1 ∧σm

τjn ∧σm

≤ hn 1/κ·∧σm ∞ + 2 ψ·∧σm ∞

5

(Xs − Xτjn )ψs dXs |Fτjn ∧σm

hn /1κ·∧σm ∞ Dj .

Hence for sufficiently large n, Dj /hn ≤

1/κ·∧σm ∞ 5 , 1 − 2 ψ·∧σm ∞ hn /1κ·∧σm ∞

which is sufficient for (7) with the aid of (13). So far we proved {π n }n ∈ T . Now, let us prove the asymptotic efficiency. n n n We have that h−1 n Z σm and h Nσm are uniformly integrable by (11) and (12) respectively. Hence it suffices to show 1 n n n h−1 n Z σm → κ · Xσm , h Nσm → κ · Xσm 6 in probability. By (9) and the definition of τjn , the first convergence follows from ∞

∞

1 2 1 4 2 n ∧σ m − Xτ n ∧σ m ) = hn n ∧σ m − Xτ n ∧σ m ) + op (hn ). κτjn (Xτj+1 κτ n (Xτj+1 j j 6 j=0 6 j=0 j

Asymptotically Efficient Discrete Hedging The second convergence follows from ∞  2 −1 n ∧σ m − Xτ n ∧σ m ) + op (h Nσnm = h−1 κτjn (Xτj+1 n n ). j

343



j=0

Here we make several remarks on the asymptotic efficiency. First, we do not claim that the discretization scheme given in Theorem C attains the lower bound of Theorem A with τ = T . What we showed is that there exists a sequence of increasing stopping times σ m with σ m → T such that the lower bound at τ = σ m is finite and asymptotically attained by the scheme. We are working with a general framework where we cannot show in general the lower bound is finite for any stopping time τ ≤ T . In addition, we cannot claim that NTn < ∞ a.s. for the scheme although it is easy to check Nσnm < ∞ a.s. for each m. These limitations are however not a problem in practice; practitioners are never interested in rebalancing their hedging portfolios up to the last moment of the maturity. An asymptotically efficient hedging strategy should be understood as such a discretization scheme that any other scheme does not have a better asymptotic performance in terms of our risk criterion. It should be noted also that we cannot answer how much hn should be small so that this asymptotic result may apply. To study this point, we give numerical results in the next section.

4. The Black-Scholes model In this final section, we review our results in the Black-Scholes model (4) and present numerical results. For an European option payoff f (YT ), put  √ Xt = ∂y Pf (t, Yt ), Pf (t, y) = f (y exp(−σ 2 (T − t)/2 + σ T − tz))φ(z)dz as before. The hedging strategy X is what is called Delta in practice. Lemma D. For any Borel function f of polynomial growth, Pf is a C ∞ function on [0, T ) × (0, ∞). If in addition f is a convex function whose right derivative f  is not a continuous singular function, then ∂y2 Pf > 0 on [0, T ) × (0, ∞). Proof. The first assertion is due to the smoothness of φ. If f is convex, then Pf is convex as well. So ∂y2 Pf ≥ 0 is trivial. Recall that the right derivative f  is an increasing right continuous function so there corresponds a Stieltjes measure. It is well known that such a measure is decomposed into an absolutely continuous part, a continuous singular part and a discrete part. This decomposition of the Stieltjes measure induces a decomposition of f of the form ∞  f (y) = fa (y) + fs (y) + fd (y), fd (y) = aj (y − bj )+ , j=0

where fa is a convex function whose right derivative is absolutely continuous, fs is a convex function whose right derivative is continuous singular and aj ≥ 0, bj are constants. The assumption is equivalent to saying at least one of fa and fd is

344

M. Fukasawa

not null. If fa is not null, then ∂y2 Pf (y) ≥ ∂y2 Pfa (y) = P∂y2 fa (y) > 0. If fd is not null, then ∂y2 Pf (y) ≥ ∂y2 Pfd (y) > 0 because the Gamma of the Black-Scholes price for call option is strictly positive on [0, T ) × (0, ∞).  Now we can apply the results in the preceding section with M = (σΓY ) · W, κ = Γ−1 , Γt = ∂y2 Pf (t, Yt ). Note that Γ is what is called Gamma in practice. By Theorem C, an asymptotically efficient strategy is given by n ¯ jn = Xτ n . τ0n = 0, τj+1 = inf{t > τjn ; |Xt − Xτ n |2 ≥ hn Γτ n }, X j

j

j

It is easy to see that n ¯ jn = Xτ n τ0n = 0, τj+1 = inf{t > τjn ; |Yt − Yτjn |2 ≥ hn /Γτjn }, X j

also defines an asymptotically efficient strategy. It is quite nice that we can construct these strategies by only calculating Delta and Gamma. It remains to choose a constant hn . The optimal hn depends on the choice of C in the last section. Once C is given, it is elementary to calculate the optimal one. It also will be feasible to tune hn by an empirical rule in practice. Table 1 presents a simulation result (10000 repetitions) for hedging call options with strike price K = 80, 90, 100, 110, 120 in the Black-Scholes model with µ = 0.1, σ = 0.3, T = 1.0, Y0 = 100. The columns ∆2 /Γ, Karandikar, equidistant stand for τ0n = 0,

n τj+1 = inf{t > τjn ; |Xt − Xτjn |2 ≥ 0.05Γτjn },

τ0n = 0,

n τj+1 = inf{t > τjn ; |Xt − Xτjn | ≥ 0.03},

τjn = j/200

5 respectively. For each of ZTn , NTn , NTn ZTn , the table presents mean, variance and maximum of absolute values. The mean E[NTn ] and the variance Var[ZTn ] are important in the present context. The asymptotically efficient strategy ∆2 /Γ is superior to the other two as easily seen. This implies that an asymptotically efficient strategy has a nice performance even in a realistic situation E[NTn ] ≈ 200. We generated Brownian motions with equidistant time step 10−6 , which seems enough small to deal with passage times. This value 10−6 is however a lower n bound of τj+1 − τjn in our simulation, and so we have NTn ≤ 106 , which means that what we are observing are not exactly the same as what we considered in the previous section. In addition, we treated Nσnm instead of NTn . Nevertheless, the numerical results show at least that the schemes obtained by our asymptotic analysis certainly have a potential to improve the hedging technique in financial practice.

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Asymptotically Efficient Discrete Hedging 345

Vcdng30 Jgfikpi Ecnn Qrvkqpu kp vjg Dncem.Ujqngu oqfgn

346

M. Fukasawa

References [1] Rootz´en, H. (1980): Limit distributions for the error in approximations of stochastic integrals, Ann. Probab. 8, no. 2, 241–251 [2] Bertsimas, D.; Kogan, L.; Lo, A.W. (2000): When is time continuous? Journal of Financial Economics 55, no. 2, 173–204 [3] Hayashi, T.; Mykland, P.A. (2005): Evaluating Hedging Errors: An Asymptotic Approach, Math. Finance 15, no. 2, 309–343 [4] Tankov, P.;Voltchkova, E. (2009): Asymptotic analysis of hedging errors in models with jumps, Stoch. Proc. Appl. 119, 2004–2027 [5] Gobet, E.; Temam, E. (2001): Discrete time hedging errors for options with irregular payoffs, Finance Stoch. 5, no. 3, 357–367 [6] Geiss, S. (2002) Quantitative approximation of certain stochastic integrals. Stoch. Stoch. Rep. 73, no. 3-4, 241–270 [7] Martini, C.; Patry, C. (1999): Variance Optimal Hedging in the Black-Scholes model for a given Number of Transactions, INRIA Rapport de recherche no. 3767 [8] Karandikar, R.L. (1995): On pathwise stochastic integration, Stochastic Process. Appl. 57, no. 1, 11–18 [9] Karatzas, I.; Shreve, S.E. (1991): Brownian motion and stochastic calculus. Second edition, Graduate Texts in Mathematics, 113. Springer-Verlag, New York Masaaki Fukasawa Center for the Study of Finance and Insurance Osaka University 1-3, Machikaneyama-cho Toyanaka-shi Osaka, 560-8531, Japan e-mail: [email protected]

Progress in Probability, Vol. 65, 347–359 c 2011 Springer Basel AG


Efficient Importance Sampling Estimation for Joint Default Probability: The First Passage Time Problem Chuan-Hsiang Han Abstract. Motivated from credit risk modeling, this paper extends the twodimensional first passage time problem studied by Zhou (2001) to any finite dimension by means of Monte Carlo simulation. We provide an importance sampling method to estimate the joint default probability, and apply the large deviation principle to prove that the proposed importance sampling is asymptotically optimal. Our result is an alternative to the interacting particle systems proposed by Carmona, Fouque, and Vestal (2009). Mathematics Subject Classification (2000). 60F10, 65C05. Keywords. Structural-form model, probability of joint default, efficient importance sampling, large deviation theory.

1. Introduction Estimation of the joint default probability under a structural-form models emerged pretty early in the presence of stochastic financial theory. In models of Black and Scholes [3] and Merton [15], default can only happen at expiration of debt when its issuer’s asset value is less than the debt value. Black and Scholes modeled the asset value process by a geometric Brownian motion, then Merton incorporated an additional compound Poisson jump term. Black and Cox [4] generalized these models by allowing that default can occur at any time before the expiration of debt. They considered a first passage time problem for the geometric Brownian motion in one dimension. Zhou [16] extended this one-dimensional geometric Brownian motion to the jump-diffusion model as Merton did. Later in Zhou [17], the joint default for two-dimensional geometric Brownian motions was treated. A comprehensive technical review can be found in [5]. In this paper, we focus on generalizing the joint default from a two-dimensional first passage time problem studied in [17] to higher dimensions through a

348

C.-H. Han

Monte Carlo study. A high-dimensional setup of the first passage time problem under correlated geometric Brownian motions is the following. We assume that each firm value process, Sit 1 ≤ i ≤ n, has the dynamics, dSit = µi Sit dt + σi Sit dWit ,

(1)

where σi is a constant volatility and Brownian motions Wi s are correlated by d Wi , Wj t = ρij dt with ·, ·t the quadratic variation at time t. Each firm also has a constant default barrier Bi for 1 ≤ i ≤ n, and its default happens at the first time when the asset value Sit falls below the barrier level. Therefore, default time τi of the ith firm is defined as τi = inf{t ≥ 0 : Sit ≤ Bi }.

(2)

Let F be the filtration generated by all Si· , i = 1, . . . , n, under a probability measure P . At time 0, the joint default probability with a terminal time T is defined by DP = E {Πni=1 I (τi ≤ T ) |F0 } .

(3)

In general, there is no closed-form solution for the probability of joint default (3) in high dimension, so one has to rely on numerical methods. Using a deterministic approach such as numerical PDE (partial differential equation) or binomial tree to calculate the joint default probability typically suffers from the curse of dimensionality. Due to the property of dimension independence, Monte Carlo simulation becomes a feasible approach. Moreover, it is crucial to reduce the variance of Monte Carlo estimators for convergence improvement. Recently, Carmona, Fouque, and Vestal [8] study a first passage time problem and estimate the loss density function for a credit portfolio under a stochastic volatility model. They use interacting particle systems for variance reduction. Alternatively, we propose an efficient Monte Carlo method which incorporates importance sampling for variance reduction in order to accurately estimate joint default probabilities under the classical Black-Cox model in high dimension. Han et al. [12] apply the same importance sampling for risk management applications with empirical studies and backtesting under stochastic volatility models. There are many ways to analyze the variance of an importance sampling estimator. In a strong sense, one can minimize the variance, say over a parametrized space, by solving an optimization problem. See for example [1, 2], in which authors proposed an adaptive scheme, namely Robbins Monro algorithms, to solve a class of importance sampling estimators. They utilize constant change of drift in high dimension and solve nontrivial optimization problems. See also [10] for using an asymptotic result to approximate the optimal change of measure. In a weak sense, one seeks to minimize asymptotically the rate of variance, instead of the variance itself, of an importance sampling estimator. To demonstrate fundamental ideas, we provide an introduction in Section 2 that include constructions of importance sampling schemes for the standard normal random variable, and an asymptotic

Efficient Importance Sampling Estimation

349

variance analysis of our proposed estimator by means of the large deviation principle. Details about the general definition of rate of variance and its analysis can be found in Chapter 5 of [7]. In this entire paper, we adopt variance analysis in the weak sense. By means of the large deviation principle, we show that the proposed importance sampling estimator for the one-dimensional first passage time problem has a zero variance rate. That implies that our proposed importance sampling scheme is asymptotically optimal. The organization of this paper is as follows: Section 2 provides a fundamental understanding of importance sampling in a simple Gaussian random variable model with a spatial scale. Approximation to variance rate of the efficient importance sampling estimator can be obtained by an application of Cramer’s theorem. Section 3 explores the high-dimensional first passage time model, known as the Black-Cox model. We show that the proposed importance sampling method is asymptotically optimal (or called efficient) in one dimension by an application of Freidlin-Wentzell theorem. Then we conclude in Section 4.

2. Efficient importance sampling for a one-dimensional toy model We start from a simple and static model to estimate the probability of default defined by P1c := E{I (X > c)}, where the standard normal random variable X ∼ N (0, 1) and c > 0 stand for the loss of a portfolio and its loss threshold, respectively. Of course, P1c is simply a tail probability and admits a closed-form solution N (−c), where N (x) denotes the cumulative normal integral function. Using basic Monte Carlo method to estimate P1c is not accurate enough particularly when the loss threshold c is large. This is because the variance of I (X > c) is of the same order of the default probability P1c when a rare event occurs. For example, E{(I (X > c))2 } − (P1c )2 ≈ P1c ( 1) when c is large enough. This implies that the relative error of the basic Monte Carlo estimator is rather large. A problem of rare event simulation arises when the spatial scale c is sufficiently large. Applying importance sampling as a variance reduction method to treat rare event simulation has been extensively studied [7]. A general procedure, known as exponential change of measure, for constructing an importance sampling scheme in order to estimate P1c is reviewed below. Assume that the density function of a real-valued random variable X is f (x) > 0 for each x ∈ %. One can change the probability measure by incorporating a likelihood function f (x)/fµ (x) so that the default probability P1c can be evaluated under the new probability measure Pµ : % & f (X) E {I(X > c)} = Eµ I(X > c) , fµ (X)

350

C.-H. Han

where the new density function of X is fµ (x) > 0 for each x ∈ %. The twist or tilted probability measure refers to the choice of a new density fµ (x) =

exp(µ x) f (x) , M (µ)

(4)

where M (µ) = E[exp(µ X)] denotes the moment generating function of X. We ! " 2 c define the leading term of the estimator variance as P2 (µ) := Eµ I(X > c) ff 2 (X) . (X) µ

By substituting fµ (x) given above into P2c (µ), one can obtain the following results: % & f (X) c P2 (µ) = E I(X > c) fµ (X) = M (µ)E {I(X > c) exp(−µ X)} ≤ M (µ) exp(−µ c),

(5)

where µ and c are assumed positive numbers for this upper bound to hold. To minimize the logarithm of this upper bound, its first-order condition satisfies d ln(M (µ) exp(−µ c)) M  (µ) = −c dµ M (µ) = 0. 



(µ ) Let µ solve for M M (µ ) = c. It follows that the expected value of X under the new probability measure
Pµ is exactly the loss threshold c. This is confirmed by evaluating Eµ (X) = x fµ (x) dx and substituting fµ (x) defined in (4) so that

M  (µ ) M (µ ) = c.

Eµ (X) =

(6)

Remark that this whole procedure of exponential change of measure can be extended to high dimension. From the simulation point of view, the final result shown in (6) is appealing because the rare event of default under the original probability measure is no longer rare under this new measure Pµ . In addition, even in the situation that moment generating functions are difficult or impossible to find, one can still possibly use other change of measure techniques to fulfill (6), i.e., “the expected value of a defaultable asset is equal to its debt value” in financial terms. We will see such example in Section 3 for high-dimensional Black-Cox model. In the concrete case of X being a standard normal random variable, the minimizer µ is exactly equal to c. This result is derived from a direct calculation of c = M  (µ )/M (µ ) given that the moment generating function of X is M (µ) =    exp(µ2 /2). The twist or tilted √ density function fµ (x) = exp(µ x) f (x)/M (µ ) 2 becomes exp(−(x − c) /2)/ 2 π. Hence, random samplings are generated from X ∼ N (c, 1) under this new density function, instead of X ∼ N (0, 1) under the

Efficient Importance Sampling Estimation original measure. The default probability P1c can be explicitly expressed by  ∞ 2 2 1 e−x /2 P1c = √ I (x > c) −(x−c)2 /2 e−(x−c) /2 dx e 2π −∞ ! " c2 /2−cX = Ec I (X > c) e ,

351

(7)

and its second moment becomes

! " 2 P2c (c) := Ec I (X > c) ec −2cX .

Naturally, one can ask an optimization problem which minimizes variance of all possible √ importance sampling estimators associated with fµ (x) = exp(−(x − µ)2 /2)/ 2 π for each µ ∈ %. That is, given that % & f (X) P1c = Eµ I (X > c) fµ (X) ! " 2 = Eµ I (X > c) eµ /2−µX , (8) we seek to minimize its second moment ! " 2 P2c (µ) = Eµ I (X > c) eµ −2µX .

(9)

The associated minimizer guarantees the minimal variance within the µ-parametrized measures but solving the minimizer via (9) requires numerical computation. In Section 2.2, we compare variance reduction performance between the optimal estimator (8) by minimizing (9), and the efficient estimator (7). We will see that in our numerical experiments these two estimators reach the same level of accuracy, but the computing times are very different. Efficient estimators perform more effectively then the optimal estimators. 2.1. Asymptotic variance analysis by large deviation principle It is known that the number of simulation to estimate a quantity in a certain level 2 of accuracy should be proportional to P2c (µ)/ (P1c ) − 1. See for example Section 2 4.5 in [7]. If the decay rate of P2c (µ) and (P1c ) are the same asymptotically, we say that the asymptotic variance rate is zero and the corresponding importance sampling estimator is asymptotically optimal or efficient. In this section, we aim to prove that when the density parameter µ is particularly chosen as the default loss threshold c, the variance reduced by the importance sampling scheme defined in (7) is asymptotically optimal by an application of Cramer’s theorem in large deviation theory. Recall that Theorem 1 (Cramer’s theorem [7]). Let {Xi } be real-valued i.i.d. random variables under P and EX1 < ∞. For any x ≥ E {X1 }, we have

1 Sn lim ln P ≥ x = − inf Γ∗ (y), (10) n→∞ n y≥x n

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, where Sn = ni=1 Xi denotes the sample sum of size n, Γ(θ) = ln E eθX1 denotes the cumulant function, and Γ∗ (x) = supθ∈ [θ x − Γ(θ)]. From this theorem and the moment generating function E {exp(θX)} = exp(θ2 /2) for X being a standard normal univariate, we obtain the following asymptotic approximations. Lemma 1 (Asymptotically Optimal Importance Sampling). When c approaches infinity, the variance rate of the estimator, defined in (7), approaches zero. That is, 1 1 ln P2c (c) = 2 lim 2 ln P1c = −1. c→∞ c c2 Therefore, this importance sampling is asymptotically optimal or efficient. lim

c→∞

Proof. Given that Xi , i = 1, 2, . . . are i.i.d. one-dimensional standard normal random variables, it is easy to obtain n

1 x2 i=1 Xi lim ln P ≥x =− , n→∞ n n 2 ) n * 2 i=1 Xi or equivalently P ≥ x ≈ exp(−n x2 ) by an application of Theorem 1. n √ Introduce a rescaled default probability P (X ≥ nx) for n large, then n

 √  √ i=1 Xi √ P X ≥ nx = P nx ≥ n n

i=1 Xi =P ≥x , n in which each √ random variable Xi has the same distribution as X. Hence, let 1  c := nx, the approximation to the first moment of I(X ≥ c) or default probability P1c is obtained:  √  1 1 lim √ ln P X ≥ nx = − , 2 n→∞ ( nx) 2 or equivalently c2 ). 2 Given the second moment defined in (9), it is easy to see that P (X ≥ c) ≈ exp(−

(11)

2

P2c (µ) = E−µ {I (X > c)} eµ

2

= E0 {I (X > µ + c)} eµ . The first line is obtained by changing measure via dPµ /dP−µ , and the second line shifts the mean value of X ∼ N (−µ, 1) to 0. With the choice µ = c, we get   2 P2c (c) = E0 {I (X > 2 c)} ec and its approximation P2c (c) ≈ exp −c2 can be easily obtained from the same derivation as in (11). As a result, we verify that

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the decay rate of the second moment P2c (c) is two times the decay rate of the probability P1c , i.e., lim

c→∞

1 1 log P2c (c) = 2 lim 2 log P1c = −1. c→∞ c c2

(12) 

We have shown that the proposed importance sampling scheme defined in (7) is efficient. This zero variance rate should be understood as an optimal variance reduction in an asymptotic sense because the variance rate cannot be less than zero. Remark. This lemma can be generalized to any finite dimension and possible extended to other generalized distribution such as Student’s t. We refer to [14] for further details with applications in credit risk. For our thematic topic of estimating joint default probabilities under highdimensional Black-Cox model, this technique unfortunately does not work because the moment generating function of multivariate first passage times is unknown. We overcome this difficulty by considering a simplified problem, namely change measure for the joint distributions of underlying processes at maturity rather than their first passage times. Remarkably, we find that this measure change can still be proven asymptotically optimal for the original first passage time problem. Details can be found in Section 3. 2.2. Numerical results Table 1 demonstrates performance of two importance sampling schemes, including optimal estimator and efficient estimator, to estimate the default probability P (X > c) for X ∼ N (0, 1) with various loss threshold values c. In Column 2, exact solutions of N (−c) are reported. In each column of simulation, Mean and SE stand for the sample mean and the sample standard error, respectively. IS(µ = c) represents the scheme in (7) using the pre-determined choice of µ = c suggested from the asymptotic analysis in Lemma 1, while IS(µ = µ ) represents the optimal scheme in (7) using µ = µ , which minimizes P2c (µ) numerically. We observe that standard errors obtained from these two importance sampling schemes are comparable in terms of the same order of accuracy, while the computing time is not. From the last row, the optimal importance sampling scheme IS(µ = µ ) takes about 50 times more than the efficient importance sampling scheme IS(µ = c). These numerical experiments are implemented in Matlab on a laptop PC with 2.40GHz Intel Duo CPU T8300. There have been extensive studies and applications, see Chapter 9 of [11] for various applications in risk management, by using this concept of minimizing an upper bound of the second moment under a parametrized twist probability, then construct an importance sampling scheme. However, it remains to check whether this scheme is efficient (with zero variance rate) or not.

354

C.-H. Han Table 1. Estimation of default probability P (X > c) with different loss threshold c when X ∼ N (0, 1). The total number of simulation is 10,000. DP

IS(µ = c)

Basic MC

IS(µ = µ )

c

true

Mean

SE

Mean

SE

Mean

SE

1 2 3 4

0.1587 0.0228 0.0013 3.17e-05

0.1566 0.0212 1.00E-03 –

0.0036 0.0014 3.16E-04 –

0.1592 0.0227 0.0014 3.13E-05

0.0019 3.49E-04 2.53E-05 6.62E-07

0.1594 0.0225 0.0014 3.11E-05

0.0018 3.37E-04 2.51E-05 6.66E-07

time

0.004659

0.020904

1.060617

3. Efficient importance sampling for high-dimensional first passage time problem In this section, we review an importance sampling scheme developed by Han and Vestal [13] for the first passage time problem (3) in order to improve the convergence of Monte Carlo simulation. In addition, we provide a variance analysis to justify that the importance sampling scheme is asymptotic optimal (or efficient) in one dimension. The basic Monte Carlo simulation approximates the joint default probability defined in (3) by the following estimator DP ≈

N * 1  n ) (k) Πi=1 I τi ≤ T , N

(13)

k=1

(k)

where τi denotes the kth i.i.d. sample of the ith default time defined in (2) and N denotes the total number of simulation. By the Girsanov Theorem, one can construct an equivalent probability measure P˜ defined by the following Radon-Nikodym derivative    T dP 1 T 2 ˜ = QT (h· ) = exp h(s, Ss ) · dWs − ||h(s, Ss )|| ds , (14) 2 0 dP˜ 0 where we denote by ) Ss = (S1s , .*. . , Sns ) the state variable (asset value process) ˜ ˜ 1s , . . . , W ˜ ns the vector of standard Brownian motions, revector and Ws = W spectively. The function h(s, Ss ) is assumed to satisfy Novikov’s condition such
˜ t = Wt + t h(s, Ss )ds is a vector of Brownian motions under P˜ . that W 0 The importance sampling scheme proposed in [13] is to select a constant vector h = (h1 , . . . , hn ) which satisfies the following n conditions E˜ {SiT |F0 } = Bi , i = 1, . . . , n.

(15)

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These equations can be simplified by using the explicit log-normal density of SiT , so we deduce the following sequence of linear equations for hi ’s: µi ln Bi /Si0 − , i = 1, . . . , n. (16) σi σi T If the covariance matrix Σ = (ρij )1≤i,j,≤n is non-singular, the vector h exists uniquely so that the equivalent probability measure P˜ is uniquely determined. The joint default probability defined in (3) becomes Σij=1 ρij hj =

˜ {Πni=1 I (τi ≤ T ) QT (h)|F0 } . DP = E

(17)

Equation (15) requires that, under the new probability measure P˜ , the expectation of asset’s value at time T is equal to its debt level. When the debt level B of a company is much smaller than its initial asset value S0 (see examples in Table 2), or returns of any two names are highly negative correlated (see examples in Table 3), joint default events are rare. By the proposed importance sampling scheme, random samples drawn under the new measure P˜ cause more defaults than those samples drawn under P . Table 2. Comparison of single-name default probability by basic Monte Carlo (BMC), exact solution, and importance sampling (IS). The number of simulation is 104 and an Euler discretization for (1) is used by taking time step size T /400, where T is one year. Other parameters are S0 = 100, µ = 0.05 and σ = 0.4. Standard errors are shown in parenthesis. B

BMC

50 0.0886 (0.0028)

Exact Sol

IS

0.0945

0.0890 (0.0016) −5

20

– (–)

7.7310 ∗ 10

1

– (–)

1.3341 ∗ 10−30

7.1598 ∗ 10−5 (2.3183 ∗ 10−6 ) 1.8120 ∗ 10−30 (3.4414 ∗ 10−31 )

Table 2 and Table 3 illustrate numerical results for estimating the (joint) default probabilities of a single-name case and a three-name case. The exact solution of the single name default probability 1−2µ/σ2 S0 + − 1 − N (d2 ) + N (d2 ) (18) B 2

± ln(S0 /B)+(µ−σ /2)T √ with d± can be found in [8]. This result is obtained from 2 = σ T the distribution of the running minimum of Brownian motion. However, there is no closed-form solution for the joint default probability of three names in Table 3. It is worth noting the drastic difference of joint default probabilities caused by different correlations. Furthermore, Table 4 shows the capability of proposed importance sampling method to treat the first passage-time problems in high dimension.

356

C.-H. Han Table 3. Comparison of three-name joint default probability by basic Monte Carlo (BMC), and importance sampling (IS). The number of simulation is 104 and an Euler discretization for (1) is used by taking time step size T /100, where T is one year. Other parameters are S10 = S20 = S30 = 100, µ1 = µ2 = µ3 = 0.05, σ1 = σ2 = 0.4, σ3 = 0.3 and B1 = B2 = 50, B3 = 60. Standard errors are shown in parenthesis. ρ

BMC

IS

0.3

0.0049(6.9832 ∗ 10−4 )

0.0057(1.9534 ∗ 10−4 )

0

3.0000 ∗ 10−4 (1.7319 ∗ 10−4 )

6.4052 ∗ 10−4 (6.9935 ∗ 10−5 )

−0.3

−(−)

2.2485 ∗ 10−5 (1.1259 ∗ 10−5 )

Table 4. Comparison of multiname joint default probabilities by basic Monte Carlo (BMC), and importance sampling (IS) under highdimensional Black-Cox model. Let n denote the dimension, the total number of firms. The number of simulation is 3 ∗ 104 and an Euler discretization for (1) is used by taking time step size T /100, where T is one year. Other parameters are S0 = 100, µ = 0.05, σ = 0.3, ρ = 0.3, and B = 50. Basic MC n 2 5 10 15 20 25 30 35 40 45 50

Mean

SE

Importance Sampling Mean

1.1E-03 3.31E-04 1.04E-03 6.36E-06 2.90E-07 9.45E-09 1.15E-09 2.06E-10 6.76E-11 1.35E-11 6.59E-12 3.25E-12 6.76E-13

SE 2.83E-05 3.72E-07 2.66E-08 1.16E-09 1.98E-10 3.84E-11 2.36E-11 2.89E-12 1.58E-12 1.08E-12 2.26E-13

3.1. Asymptotic variance analysis by large deviation principle We provide a theoretical verification to show that the importance sampling developed above is asymptotic optimal for the one-dimensional first passage time problem under the geometric Brownian motion. This problem has also been considered in Carmona et al. [8].

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Our proof is based on the Freidlin-Wentzell theorem [6, 9] in large deviation theory in order to approximate the default probability and the second moment of the importance sampling estimator defined in (17). We consider the scale ε = −1 − (ln(B/S0 )) being small or equivalently 0 < B  S0 , the current asset value S0 is much larger than its debt value B. Our asymptotic results show that the second moment approximation is the square of the first moment (or default probability) approximation. Therefore, we attain the optimality of variance reduction in an asymptotic sense, i.e., the proposed importance sampling scheme is efficient. Theorem 2 (Efficient Importance Sampling). Let St denote the asset value following the log-normal process dSt = µSt dt + σ St dWt with the initial value S0 , and B denote the default boundary. We define the default probability and its importance sampling scheme by %

& ε P1 = E I min St ≤ B 0≤t≤T %

& ˜ = E I min St ≤ B QT (h) , 0≤t≤T

where the Radon-Nykodym derivative Q(h) is defined in (14). The second moment of this estimator is denoted by %

& P2ε (h) = E˜ I min St ≤ B Q2T (h) . 0≤t≤T

By the choice of h = (µ T + 1/ε) /(σ T ) with the scale ε defined from −1/ε = ln(B/S0 ), the expected value ST under P˜ is B. That is, E˜ {ST } = B. When ε is small enough or equivalently B  S0 , we obtain a zero variance rate, i.e., ) * 2 limε→0 ε2 ln P2ε (h)/ (P1ε ) = 0, so that the importance sampling scheme is efficient. Proof. Recall that the one-dimensional default probability is defined by # $ ) * 2 µ− σ2 t+σWt P inf St = S0 e ≤B 0≤t≤T #

$ σ2 =E I inf ε µ − t + εσWt ≤ −1 , 0≤t≤T 2

(19)

where we have use the strictly monotonicity of the logarithmic transformation . For small parameter ε, the default and we introduce a scaling ln (B/S0 ) = −1 ε probability will be small in financial intuition because the debt to asset value, B/S0 , is small. By an application of Freidlin-Wentzell Theorem [6, 9], it is easy to prove that the rate function of (19) is 2σ12 T . That is the rescaled default probability # $

) * 2 −1 µ− σ2 t+σWt P inf St = S0 e ≤ B ≈ exp 2 2 , (20) 0≤t≤T ε 2σ T

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C.-H. Han

when ε is small. Recall that under the*measure change defined in (17), the price ) 2

˜ µ− σ −σh t+σW

t 0 2 dynamics becomes St = S0 e with h = µσ − ln σB/S T . The second ε moment P2 (h) becomes #

$ 2 ˜ ˜ I E inf St ≤ B e2 h WT −h T 0≤t≤T #

$ ) * 2 ˆt 2 µ− σ2 +σh t+σW ˆ I =E inf S0 e ≤B eh T 0≤t≤T #



$ 1 σ2 1 )2 T , ˆ I ˆ t ≤ −1 × e( σr + εσT =E inf ε 2µ − + t + εσ W 0≤t≤T 2 T where the measure change dPˆ /dP˜ is defined by QT (2h) for the second line and we incorporate the same scaling ln (B/S0 ) = −1 ε to rescale our problem for the last line above. By Freidlin-Wentzell theorem, the rate function of the expectation is 2 . Consequently, the approximation σ2 T #

$

2 −1 ˜ ˜ I E inf St ≤ B e2 h WT −h T ≈ exp 2 2 (21) 0≤t≤T ε σ T

is derived. By limε→0 ε2 ln P2ε (h) = 2 limε→0 ε2 ln P1ε , we confirm that the variance rate of this importance sampling is asymptotically zero so that this scheme is efficient.  Remark. The same result can be obtained from a PDE argument studied in [13].

4. Conclusion Estimation of joint default probabilities under a first passage time problem in the structural form model are tackled by importance sampling. Imposing “the expected asset value at debt maturity equals to its debt value” as a condition, importance sampling schemes can be uniquely determined within a family of parametrized probability measures. This approach overcomes the hurdle of exponential change of measure that requires existence of the moment generating function of first passage times. According to the large deviation principle, our proposed importance sampling scheme is asymptotically optimal or efficient in a rare event simulation. Acknowledgment Work supported by NSC 97-2115-M-007-002-MY2, Taiwan. We are also grateful to one anonymous referee and Professor Nicolas Privault. Other acknowledgments: NCTS, National Tsing-Hua University; TIMS, National Taiwan University; CMMSC, National Chiao-Tung University.

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References [1] B. Arouna,“Robbins Monro algorithms and variance reduction in finance,” Journal of Computational Finance, 7 (2), (Winter 2003/04). [2] B. Arouna, “Adaptive Monte Carlo method, a variance reduction technique,” Monte Carlo Methods Appl, 10 (1) (2004), 1–24. [3] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, 81 (1973), 637–654. [4] F. Black and J. Cox, “Valuing Corporate Securities: Some Effects of Bond Indenture Provisions,” Journal of Finance, 31(2) (1976): 351–367. [5] T.R. Bielecki and M. Rutkowski, Credit Risk: Modeling, Valuation and Hedging, Springer 2002. [6] J.A. Bucklew, Large Deviation Techniques in Decision, Simulation, and Estimation, Wiley-Interscience, Applied Probability and Statistics Series, New York, 1990. [7] J.A. Bucklew, Introduction to rare event simulation, Springer, 2003. [8] R. Carmona, J.P. Fouque, and D. Vestal, “ Interacting Particle Systems for the Computation of Rare Credit Portfolio Losses,” Finance and Stochastics, 13(4), 2009, 613–633. [9] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 2/e, Springer, 1998. [10] J.-P. Fouque and C.-H. Han, “Variance Reduction for Monte Carlo Methods to Evaluate Option Prices under Multi-factor Stochastic Volatility Models,” Quantitative Finance, Volume 4, number 5 (2004) 1–10. [11] P. Glasserman, Monte Carlo Methods for Financial Engineering, Springer-Verlag, New York, 2003. [12] C.-H. Han, W.-H. Liu, Z.-Y. Chen, “An Improved Procedure for VaR/CVaR Estimation under Stochastic Volatility Models,” submitted, 2010. [13] C.-H. Han and D. Vestal, “Estimating the probability of joint default in structural form by efficient importance sampling,” working paper, Natoinal Tsing-Hua University, 2010. [14] C.-H. Han and C.-T. Wu, “Efficient importance sampling for estimating lower tail probabilities under Gaussian and Student’s distributions,” preprint, National TsingHua University, 2010. [15] R. Merton, “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,” The Journal of Finance, 29 (1974), 449–470. [16] C. Zhou, “The Term Structure of Credit Spreads with Jump Risk”, Journal of Banking & Finance, Vol. 25, No. 11, (November 2001), pp. 2015–2040. [17] C. Zhou, “An Analysis of Default Correlations and Multiple Defaults,” The Review of Financial Studies, 14(2) (2001), 555–576. Chuan-Hsiang Han Department of Quantitative Finance National Tsing Hua University Hsinchu, Taiwan, 30013, ROC e-mail: [email protected]

Progress in Probability, Vol. 65, 361–411 c 2011 Springer Basel AG


Market Models of Forward CDS Spreads Libo Li and Marek Rutkowski Abstract. The paper re-examines and generalizes the construction of several variants of market models for forward CDS spreads, as first presented by Brigo [10]. We compute explicitly the joint dynamics for some families of forward CDS spreads under a common probability measure. We first examine this problem for single-period CDS spreads under certain simplifying assumptions. Subsequently, we derive, without any restrictions, the joint dynamics under a common probability measure for the family of one- and two-period forward CDS spreads, as well as for the family of one-period and co-terminal forward CDS spreads. For the sake of generality, we work throughout within a general semimartingale framework. Mathematics Subject Classification (2000). 60H30, 91B70. Keywords. Credit default swap, market model, LIBOR.

1. Introduction The market model for forward LIBORs was first examined in papers by Brace et al. [6] and Musiela and Rutkowski [23]. Their approach was subsequently extended by Jamshidian in [16, 17] to the market model for co-terminal forward swap rates. Since then, several papers on alternative market models for LIBORs and other families of forward swap rates were published. Since modeling of (non-defaultable) forward swap rates is not presented here, the interested reader is referred, for instance, to Galluccio et al. [14], Pietersz and Regenmortel [25], Rutkowski [26], or the monographs by Brace [5] or Musiela and Rutkowski [24] and the references therein. To the best of our knowledge, there is relatively scarce financial literature in regard either to the existence or to methods of construction of market models for forward CDS spreads. This apparent gap is a bit surprising, especially when confronted with the market practitioners approach to credit default swaptions, which hinges on a suitable variant of the Black formula. The standard argument which underpins the validity of this formula is the postulate of lognormality of

362

L. Li and M. Rutkowski

credit default swap (CDS) spreads, as discussed, for instance, in Brigo and Morini [11], Jamshidian [18], Morini and Brigo [22], or Rutkowski and Armstrong [28]. In the commonly used intensity-based approach to default risk, this crucial property of forward CDS spreads fails to hold, however (see, e.g., Bielecki et al. [2]), and thus a need for a novel modeling approach arises in a natural way. Recently, attempts have been made to search for explicit constructions of market model for forward CDS spreads in papers by Brigo [8, 9, 10] and Schl¨ ogl [29] (related issues were also studied in Lotz and Schl¨ ogl [20] and Sch¨ onbucher [30]). The present work is inspired by these papers, where under certain simplifying assumptions, the joint dynamics of a family of CDS spreads were derived explicitly under a common probability measure and a construction of the model was provided. Our main goal will be to derive the joint dynamics for certain families of forward CDS spreads in a general semimartingale setup. Our aim is to derive the joint dynamics of a family of CDS spreads under a common probability. Firstly, we will derive the joint dynamics of a family of single period CDS spreads under the postulate that the interest rate is deterministic. In the second part, we will derive the joint dynamics of a family of single period CDS spreads under the assumption that the interest rate and default indicator process are independent. Lastly, without any simplifying assumptions, we will derive both the joint dynamics of a family of one- and two-period CDS spreads and a family of one-period and co-terminal CDS spreads. We would also like to mention that, although not presented here, the joint dynamics of a family of one-period and co-initial CDS spreads can also be derived using the techniques developed in Subsections 5.3 and 5.4. For each market model, we will also present both the bottom-up approach and the top-down approach to the modeling of CDS spreads. In the bottom-up approach, one usually starts with a credit risk model and, relying on the assumption that the Predictable Representation Property (PRP) holds, one shows the existence of a family of ‘volatility’ processes for a family of forward CDS spreads and derives their joint dynamics under a common probability measure. On the other hand, in the top-down approach, for any given in advance family of ‘volatilities’, we focus on the direct derivation of the joint dynamics for a given family of forward CDS spreads under a common probability measure. It is fair to point out, however, that in this paper we do not provide a fully developed credit risk model obtained through the top-down approach, since the construction of the default time consistent with the derived dynamics of forward CDS spreads is not studied. In Subsection 5.5, we make an attempt to identify the set of postulates that underpin the top-down approach to a generic model of forward spreads and we derive the joint dynamics of forward spreads in a fairly general set-up. We emphasize the fact that this construction is only feasible for a judiciously chosen family of forward spreads. The work concludes by a brief discussion of the most pertinent open problems that need to be addressed in the context of top-down models.

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2. Forward credit default swaps Let (Ω, G, F, Q) be a filtered probability space, where F = (Ft )t∈[0,T ] is the reference filtration, which is assumed to satisfy the usual conditions. We work throughout within the framework of the reduced-form (i.e., intensity-based) methodology. Let us first take the perspective of the bottom-up approach, that is, an approach in which we specify explicitly the default time using some salient probabilistic features, such as the knowledge of its survival process or, equivalently, the hazard process. We thus assume that we are given the default time τ defined on this space in such a way that the F-survival process Gt = Q(τ > t | Ft ) is positive. It is well known that this goal can be achieved in several alternative ways, for instance, using the so-called canonical construction of the random time for a given in advance F-adapted intensity process λ. We denote by G = (Gt )t∈[0,T ] the full filtration, that is, the filtration generated by F and the default indicator process Ht = 1{τ ≤t} . Formally, we set Gt = σ(Ht , Ft ) for every t ∈ R+ , where H = (Ht )t∈[0,T ] is the filtration generated by H. It is well known that for any 0 ≤ t < u ≤ T and any Q-integrable, Fu measurable random variable X the following equality is valid (see, for instance, Chapter 5 in Bielecki and Rutkowski [1] or Chapter 3 in Bielecki et al. [4]) EQ (1{τ >u} X | Gt ) = 1{τ >t} G−1 t EQ (Gu X | Ft ).

(1)

Finally, we assume that an underlying default-free term structure model is given and we denote by β(t, u) = Bt Bu−1 the default-free discount factor over the time period [t, u] for 0 ≤ t ≤ u ≤ T , where in turn B = (Bt , t ∈ [0, T ]) represents the savings account. By assumption, the probability measure Q will be interpreted as the risk-neutral measure. The same basic assumptions underpinning the bottom-up approach will be maintained in Section 5 where alternative variants of market models are presented. Let T = {T0 < T1 < · · · < Tn } with T0 ≥ 0 be a fixed tenor structure and let us write ai = Ti − Ti−1 . We observe that it is always true that, for every i = 1, . . . , n, Q ( τ > Ti−1 | Ft ) ≥ Q ( τ > Ti | Ft ) . (2) When dealing with the bottom-up approach, we will make the stronger assumption that the following inequality holds, for every i = 1, . . . , n, Q ( τ > Ti−1 | Ft ) > Q ( τ > Ti | Ft ) .

(3)

We are in a position to formally introduce the concept of the forward credit default swap. To this end, we will describe the cash flows of the two legs of the stylized forward CDS starting at Ti and maturing at Tl , where T0 ≤ Ti < Tl ≤ Tn . We denote by δj ∈ [0, 1) the constant recovery rate, which determines the size of the protection payment at time Tj , if default occurs between the dates Tj−1 and Tj . Definition 2.1. The forward credit default swap issued at time s ∈ [0, Ti ], with the unit notional and the Fs -measurable spread κ, is determined by its discounted

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payoff, which equals Dti,l = Pti,l − κAi,l t for every t ∈ [s, Ti ], where in turn the discounted payoff of the protection leg equals Pti,l =

l 

(1 − δj )β(t, Tj )1{Tj−1 <τ ≤Tj }

(4)

j=i+1

and the discounted payoff of the fee leg (also known as the premium leg) per one unit of the spread equals Ai,l t =

l 

aj β(t, Tj )1{τ >Tj } .

(5)

j=i+1

Remark 2.1. It should be stressed that in the specification of the two legs we have, in particular, deliberately omitted the so-called accrual payment, that is, a portion of the fee that should be paid if default occurs between two tenor dates, say Tj−1 and Tj . The interested reader is referred to Brigo [8, 9, 10], Brigo and Mercurio [12], or Rutkowski [27] for more details. Specifications (4)–(5) mean that we decided to adopt here the postponed running CDS convention proposed by Brigo [8, 9, 10]. This particular choice of convention is motivated by the fact that it appears to be the most convenient for constructing market models of forward CDS spreads. The value (or the fair price) of the forward CDS at time t is based on the risk-neutral formula under Q applied to the discounted future payoffs. Note that we only consider here the case t ∈ [s, Ti ], although an extension to the general case where t ∈ [s, Tl ] is readily available as well. Definition 2.2. The value of the forward credit default swap for the protection buyer equals, for every t ∈ [s, Ti ], Sti,l (κ) = EQ (Dti,l | Gt ) = EQ (Pti,l | Gt ) − κ EQ (Ai,l t | Gt ).

(6)

In the second equality in (6) we used the definition of the process Di,l and the postulated property that the spread κ is Fs -measurable, and thus also Gt measurable for every t ∈ [s, Ti ]. Let us observe that Ai,l = 1{τ >Ti } Ai,l and t t i,l i,l i,l i,l Pt = 1{τ >Ti } Pt so that also Dt = 1{τ >Ti } Dt . Using formula (1), it is thus straightforward to show that the value at time t ∈ [s, Ti ] of the forward CDS satisfies i,l +i,l Sti,l (κ) = 1{τ >t} G−1 (7) t EQ (Dt | Ft ) = 1{τ >t} St (κ), i,l i,l i,l +t , where we denote where the pre-default price satisfies S+t (κ) = P+t − κA i,l P+ti,l = G−1 t EQ (Pt | Ft ),

+i,l = G−1 EQ (Ai,l | Ft ). A t t t

(8)

More explicitly, the pre-default value at time t ∈ [0, Ti] of the fee leg per one unit of spread, that is, of the defaultable annuity is given by +i,l = A t

l  j=i+1

aj G−1 t EQ (β(t, Tj )1{τ >Tj } | Ft ).

(9)

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Similarly, the pre-default value at time t ∈ [0, Ti ] of the protection leg equals P+ti,l =

l 

(1 − δj )G−1 t EQ (β(t, Tj )1{Tj−1 <τ ≤Tj } | Ft ).

(10)

j=i+1

Since the forward CDS is terminated at default, the concept of the fair (or par) forward CDS spread is only meaningful prior to default. It is also possible, and in fact more convenient, to introduce the notion of the pre-default fair forward CDS spread, which can be formally defined for any date t ∈ [0, T0 ]. To this end, we use the following definition, in which the issuance date s of a forward CDS is irrelevant, and thus it defines the F-adapted process κi,l = (κi,l t , t ∈ [0, Ti ]). Definition 2.3. For any date t ∈ [0, Ti ], the pre-default fair forward CDS spread at +i,l i,l time t is the Ft -measurable random variable κi,l t such that St (κt ) = 0. For brevity, the pre-default fair forward CDS spread will simply be called i,l +i,l +i,l forward CDS spread in what follows. Recall that S+ti,l (κi,l t ) = Pt − κt At , where +i,l and P+ i,l are given by (9) and (10), respectively. The forward the processes A spread for the CDS starting at Ti and maturing at Tl is thus given as the ratio of pre-default values of the fee and protection legs, that is, κi,l t =

P+ti,l , +i,l A t

∀ t ∈ [0, Ti].

(11)

Within the bottom-up approach, the main goal is to analyze the dynamics of the forward CDS spread κi,l for a given in advance specification of the default time. In particular, we are interested in the existence of a martingale measure for this process and explicit computation of the ‘volatility’ process for κi,l . The latter task appears to be rather difficult and thus it can only be performed for some relatively simple stochastic models of default intensity (see Bielecki et al. [3] who examine the hedging of a credit default swaption in the CIR default intensity model). Moreover, an explicit control of the volatility process for forward CDS spread is not feasible. Therefore, the valuation of options on a forward CDS through closedform solutions is a very difficult task within the classic reduced-form approach, in which one usually specifies the model by selecting the default intensity process. By contrast, in the top-down approach we postulate that the volatilities for a given family of forward CDS spreads are predetermined (and thus they can be chosen, for instance, as deterministic functions) and we aim to derive the joint dynamics of these processes under a common probability measure that, typically, is chosen to be a martingale measure for one of these processes. Since this work focuses on the concept of a market model, which is obtained in this way, it is clear that we will be mainly interested in the top-down approach. However, a detailed analysis of the bottom-up approach is also useful, since it provides important guidance about the salient features of forward CDS spreads.

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Before we proceed to the main topic of this work, we will present and examine certain abstract semimartingale setups that will prove useful in Section 5. It is notable that ‘abstract’ results established within some particular setup may be later used to obtain dynamics within different ‘applied’ frameworks. For instance, Setup (A) introduced in Subsection 4.1 covers simultaneously the cases of (defaultfree) forward LIBORs under stochastic interest rates and one-period forward CDS spreads under deterministic interest rates.

3. Preliminaries We need first to recall some definitions and results from stochastic calculus, which will be used in what follows. In this section, the multi-dimensional Itˆ o integral should be interpreted as the vector stochastic integral (see, for instance, Shiryaev and Cherny [31]). We first quote a version of Girsanov’s theorem (see, e.g., Br´emaud and Yor [7] or Theorem 9.4.4.1 in Jeanblanc et al. [19]). Let (Ω, G, F, P) be a given filtered probability space. Note that it is not postulated in Section 4 that the σfield F0 is trivial. For arbitrary real-valued semimartingales X and Y , we denote by [X, Y ] their quadratic covariation; it is known to be a finite variation process. + and P be equivalent probability measures on (Ω, FT ) with Proposition 3.1. Let P the Radon-Nikod´ym density process +  dP Zt = (12)  , ∀ t ∈ [0, T ]. dP Ft If M is a (P, F)-local martingale then the process  1 B Mt = Mt − d[Z, M ]s (0,t] Zs + F)-local martingale. is a (P, Let Mloc (P, F) (M(P, F), respectively) stand for the class of all (P, F)-local martingales ((P, F)-martingales, respectively). Assume that Z is a positive (P, F)+ martingale such that EP (Z0 ) = 1. Also, let an equivalent probability measure P + F), which is be given by (12). Then the linear map ΨZ : Mloc (P, F) → Mloc (P, defined by the formula  1 ΨZ (M )t = Mt − d[Z, M ]s , ∀ M ∈ Mloc (P, F), Z (0,t] s is called the Girsanov transform associated with the Radon-Nikod´ ym density process Z. By the symmetry of the problem, the process Z −1 := 1/Z is the Radon+ The corresponding Girsanov Nikod´ ym density process of P with respect to P. + transform ΨZ −1 : Mloc (P, F) → Mloc (P, F) associated with Z −1 is thus given by the formula  + F). B)t = M Bt − B]s , ∀ M B ∈ Mloc (P, ΨZ −1 (M Zs d[Z −1 , M (0,t]

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+ be a probability measure equivalent to P on (Ω, FT ) with the Proposition 3.2. Let P + F)-local martingale N + , there Radon-Nikod´ym density process Z. Then, for any (P, ? exists a (P, F)-local martingale N such that  1 + ? ? ]s . Nt = Nt − d[Z, N (13) Z (0,t] s + Z. Then the process N ? is given by the formula Let us set L = N   1 Ls− ? + Nt = N0 + dLs − 2 dZs . Z (0,t] s− (0,t] Zs−

(14)

? given by (14) belongs to From Proposition 3.2, it follows that the process N −1 + −1 + ? the set (ΨZ ) (N ). In fact, we have that N = (ΨZ ) (N ), as the following result shows. + be any (P, + F)-local martingale and let the process N ? be given Lemma 3.1. Let N + ? by formula (14) with L = NZ. Then the process N is also given by the following expression  ?t = N +t − + ]s . N Zs d[Z −1 , N (15) (0,t]

+ F) is bijective and the Corollary 3.1. The linear map ΨZ : Mloc (P, F) → Mloc (P, −1 + inverse map (ΨZ ) : Mloc (P, F) → Mloc (P, F) satisfies (ΨZ )−1 = ΨZ −1 . The following definition is standard (see, for instance, Jacod and Yor [15]). Definition 3.1. We say that an Rk -valued (P, F)-martingale M = (M 1 , . . . , M k ) has the predictable representation property (PRP) with respect to F under P, if an arbitrary (P, F)-local martingale N can be expressed as follows  Nt = N0 + ξs · dMs , ∀ t ∈ [0, T ], (0,t]

for some R -valued, F-predictable, M -integrable process ξ = (ξ 1 , . . . , ξ k ). We then say that M = (M 1 , . . . , M k ) is the spanning (P, F)-martingale. k

The following well-known result examines the predictable representation property under an equivalent change of a probability measure. For the sake of completeness, we provide the proof of this result. Proposition 3.3. Assume that the PRP holds under P with respect to F with the + be a probability measure spanning (P, F)-martingale M = (M 1 , . . . , M k ). Let P equivalent to P on (Ω, FT ) with the Radon-Nikod´ym density process Z. Then the + with respect to F with the spanning (P, + F)-martingale M B= PRP holds under P 1 k B B (M , . . . , M ), where for every l = 1, . . . , k,  1 Btl = Mtl − M d[Z, M l ]s . Z s (0,t]

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+ F)-local martingale N + there exists an Rk -valued, F-predictable, Hence for any (P, + + B M -integrable process ξ such that under P  +t = N +0 + Bs . N ξ+s · dM (0,t]

? ∈ Mloc (P, F) with N ?0 = 0 such that Proof. By Proposition 3.2, there exists N  1 +t − N +0 = ΨZ (N ? )t = N ?t − ? ]s . N d[Z, N (0,t] Zs Since the PRP is assumed to hold under P with respect to F with the spanning martingale M = (M 1 , . . . , M k ), there exists some Rk -valued, F-predictable, M integrable process ξ = (ξ 1 , . . . , ξ k ) such that   ξs +t − N +0 = N ξs · dMs − d[Z, M ]s (0,t] (0,t] Zs  = ξs · dΨZ (M )s . (0,t]

B = ΨZ (M ). Note that the Therefore, the claim holds by setting ξ+ = ξ and M + B + process ξ is M -integrable under P, since it is ΨZ (M ) integrable under P.  For the definitions and properties of the stochastic exponential (also known as the Dol´eans-Dade exponential) and stochastic logarithm, we refer to Section 9.4.3 in Jeanblanc et al. [19]. Definition 3.2. The stochastic exponential E(X) of a real-valued semimartingale X is the unique solution Y to the following stochastic differential equation  Yt = 1 + Ys− dXs . (16) (0,t] c

Let X be the continuous martingale part of X, that is, the unique continuous local martingale X c such that X0c = X0 and X − X c is a purely discontinuous semimartingale. Then the unique solution to equation (16) is given by the following expression  c c 1 Yt = E(X)t = eXt −X0 − 2 [X ,X ]t (1 + ∆Xs )e−∆Xs . 0<s≤t

The map X → E(X) can be inverted if almost all paths of the process E(X) and its left-continuous version E(X)− do not hit zero (see, e.g., Choulli et al. [13]). Recall that a subset of Ω × R+ is called evanescent if its projection on Ω has null probability. Let S be the space of all real-valued semimartingales. Consider the following subspaces:  , S0 = X ∈ S  {∆X = −1} is evanescent and  , S1 = Y ∈ S  {Y Y− = 0} is evanescent .

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Definition 3.3. The stochastic logarithm of Y ∈ S1 is defined by the formula  1 L(Y ) = dYs . (17) (0, · ] Ys− It is clear that for any non-zero constant semimartingale Y = c = 0, the stochastic logarithm vanishes, that is, L(Y ) = L(c) = 0. We also have the following result, which shows, in particular, that the map E : S0 ∩{X0 = 0} → S1 ∩{Y0 = 1} is a bijection and its inverse is given by the map L : S1 ∩{Y0 = 1} → S0 ∩{X0 = 0}. Proposition 3.4. (i) For every X ∈ S0 , the equality (L ◦ E)(X) = X − X0 holds. (ii) For every Y ∈ S1 , the equality Y = Y0 (E ◦ L)(Y ) holds. The following result summarizes the crucial properties of the stochastic logarithm. Proposition 3.5. (i) For any two semimartingales Y 1 and Y 2 belonging to S1 , the product Y 1 Y 2 belongs to S1 and L(Y 1 Y 2 ) = L(Y 1 ) + L(Y 2 ) + [L(Y 1 ), L(Y 2 )] so that L(Y 1 Y 2 )c = L(Y 1 )c + L(Y 2 )c .

(18)

(ii) For every Y ∈ S1 , the process 1/Y belongs to S1 and L(1/Y ) = −L(Y ) + [Y, 1/Y ]. Therefore, L(1/Y )c = −L(Y )c .

(19)

(iii) For every Y ∈ S1 the continuous martingale part L(Y ) of the stochastic logarithm L(Y ) satisfies 

c  1 1 L(Y )c = dYs = dYsc . (20) (0, · ] Ys− (0, · ] Ys c

Consequently, for any continuous local martingale M , we have that  1 [L(Y )c , M ]t = d[Y c , M ]s . Y (0,t] s

(21)

4. Abstract semimartingale setups The goal of this section is to examine a few alternative abstract semimartingale setups and to establish some generic results that will be subsequently used in Section 5 to derive the joint dynamics of CDS spreads under various sets of assumptions. In fact, the results of this section can also be used to deal with modeling of nondefaultable forward swap rates, but we do not deal with this issue in this work,

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since it was treated elsewhere (see, for instance, Galluccio et al. [14], Jamshidian [16] or Rutkowski [26]). Let (Ω, G, F, P) be a given filtered probability space. We consider an Fadapted stochastic processes with the time parameter t ∈ [0, T ] for some fixed T > 0. We postulate that the PRP holds with respect to F under P with the spanning (P, F)-martingale M = (M 1 , . . . , M k ). It is well known (see Proposition 3.3) that the PRP is preserved under an + be any probability measure equivequivalent change of measure. Specifically, let P alent to P on (Ω, FT ), with the Radon-Nikod´ ym density process Zt =

+  dP  , dP Ft

∀ t ∈ [0, T ].

+ F)-martingale N + can be expressed under P + as Then an arbitrary (P,  +t = N +0 + N ξ+s · dΨ(M )s , ∀ t ∈ [0, T ], (0,t]

+ F)-martingale Ψ(M ) := (Ψ(M 1 ), . . . , + where the Rk -valued (P, for some process ξ, k Ψ(M )) satisfies, for l = 1, . . . , k,  1 Ψ(M l )t = Mtl − d[Z, M l ]s , ∀ t ∈ [0, T ]. Z s (0,t] The following auxiliary result will prove to be useful in the sequel. Of course, the probability measure Pi introduced in part (i) in Lemma 4.1 is simply defined by setting Pi (A) = EP (ZTi 1A ) for every A ∈ FT . Recall that M c = (M 1 , . . . , M k )c = (M 1,c , . . . , M k,c ). Lemma 4.1. (i) Let (Ω, G, F, P) be a filtered probability space and let for any i = 1, . . . , n the process Z i be a positive (P, F)-martingale with EP (Z0i ) = 1. Then, for each i = 1, . . . , n, there exists a probability measure Pi equivalent to P on (Ω, FT ) with the Radon-Nikod´ym density process given by dPi   = Zti , dP Ft

∀ t ∈ [0, T ].

(22)

(ii) Let us assume, in addition, that the PRP holds with respect to F under P with the spanning (P, F)-martingale M = (M 1 , . . . , M k ). Then, for any i = 1, . . . , n, the spanning (Pi , F)-martingale Ψi (M ) = (Ψi (M 1 ), . . . , Ψi (M k )) is given by the following expression, for every l = 1, . . . , k,  1 Ψi (M l )t = Mtl − d[Z i , M l ]s (23) i Z (0,t] s

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or, equivalently,

 ' ( Ψi (M )t = Mt − L(Z i )c , M c t − 0<s≤t

1 ∆Zsi ∆Ms . Zsi

(24)

The equality Ψi (M )c = M c holds for every i = 1, . . . , n. Proof. The first part of the lemma is rather clear. The second part in Lemma 4.1 + = Pi except for equality (24). To derive (24) follows from Proposition 3.1 with P from (23), we start by recalling that  [Z i , M l ]t = [Z i,c , M l,c ]t + ∆Zsi ∆Msl . 0<s≤t

By combining this equality with (23), we obtain   1 1 Ψi (M l )t = Mtl − d[Z i,c , M l,c ]s − ∆Zsi ∆Msl . i i Z Z (0,t] s s 0<s≤t

Formula (21) yields, for every l = 1, . . . , k,  ' ( i c l,c L(Z ) , M t =

(0,t]

1 d[Z i,c , M l,c ]s , Zsi

i

and thus (24) follows. The equality Ψ (M )c = M c is an immediate consequence of (24).  In what follows, we will write [M c ] and to denote the variation processes for the Rk -valued process M c , that is, ( ' 1,c (  ' 1,c M , M 1,c . . . M , M k,c  .. .. .. [M c ] =  . ' . ' k,c. 1,c ( ( M ,M ... M k,c , M k,c

matrix of quadratic   .

(25)

4.1. Setup (A) Throughout this subsection, we will work under the following standing assumptions, referred to as Setup (A) in the sequel. Let (Ω, G, F, P) be a given filtered probability space. Assumptions 4.1. We postulate that: (i) the process X = (X 1 , . . . , X n ) is F-adapted, (ii) the processes 1 + aj X j , j = 1, . . . , n belong to S1 , where a1 , . . . , an are nonzero constants, (iii) for every i = 0, . . . , n, the process Z X,i , which is given by the formula (by the usual convention, Z X,n = 1) ZtX,i =

n  j=i+1

is a positive (P, F)-martingale,

(1 + aj Xtj ),

(26)

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(iv) the PRP holds under (P, F) with the spanning (P, F)-martingale M = (M 1 , . . . , M k ). The following lemma reveals an interesting feature of Setup (A). Lemma 4.2. For every i = 1, . . . , n, the process Z X,i X i is a (P, F)-martingale. Proof. We first observe that ZtX,n−1 = 1 + an Xtn where an is a non-zero constant. Hence X n is a (P, F)-martingale and, since Z X,n = 1, this also means that Z X,n X n is a (P, F)-martingale. Note also that (26) yields, for i = 1, . . . , n − 2, ZtX,i = (1 + ai+1 Xti+1 )ZtX,i+1 = ZtX,i+1 + ai+1 Xti+1 ZtX,i+1 , where, by assumption, the processes Z X,i and Z X,i+1 are (P, F)-martingales. Since ai+1 is a non-zero constant, we conclude that the process ZtX,i+1 Xti+1 is a (P, F)martingale as well.  Let the probability measures Pi , i = 1, . . . , n be defined as in Lemma 4.1, with the Radon-Nikod´ ym density processes Z i , i = 1, . . . , n given by Zti = ci ZtX,i = ci

n 

(1 + aj Xtj ),

(27)

j=i+1

where the positive constants c1 , . . . , cn are chosen in such a way that EP (Z0i ) = 1 for i = 1, . . . , n. Note that, in particular, we have that Pn = P since Z n = 1. Lemma 4.3. For every i = 1, . . . , n, the process X i is a (Pi , F)-martingale and it admits the following representation under Pi  Xti

=

X0i

 ξsi

+ (0,t]

· dΨ (M )s = i

X0i

+

k 

ξsi,l dΨi (M l )s ,

(28)

(0,t] l=1

where ξ i = (ξ i,1 , . . . , ξ i,k ) is an Rk -valued, F-predictable process and the (Pi , F)martingale Ψi (M ) is given by (24), so that the equality Ψi (M )c = M c holds for every i = 1, . . . , n. Proof. Let us fix i = 1, . . . , n. It follows easily from Lemma 4.2 that the process X i is a (Pi , F)-martingale. Furthermore, in view of part (ii) in Lemma 4.1, the PRP holds with respect to F under Pi with the spanning (Pi , F)-martingale Ψi (M ). This immediately implies the stated property.  Remark 4.1. It is worth stressing that part (iv) in Assumptions 4.2 is only used to deduce the existence of processes ξ 1 , . . . , ξ n such that representation (28) holds for processes X 1 , . . . , X n . In Section 5, when dealing with the construction of market models, we will not assume that the PRP holds, and we will postulate instead that

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for a given family of forward CDS spreads the counterpart of representation (28) is valid for some ‘volatility’ processes, which can be chosen arbitrarily. A similar remark applies to other abstract setups considered in the foregoing subsections. The following result underpins, on the one hand, the market model of forward LIBORs (see, for instance, Jamshidian [17]) and, on the other hand, the market model of one-period forward CDS spreads under the assumption that interest rates are deterministic (see Section 5.1 below). In the proof of Proposition 4.1, part (iv) in Assumptions 4.2 is only used indirectly through formula (28). Proposition 4.1. For every i = 1, . . . , n, the semimartingale decomposition of the (Pi , F)-martingale Ψi (M ) under the probability measure Pn = P is given by  n   1 aj ξsj · d [M c ]s i Ψ (M )t = Mt − − ∆Zsi ∆Ms . (29) j i Z 1 + a X (0,t] s s j j=i+1 0<s≤t

More explicitly, for every i = 1, . . . , n and l = 1, . . . , k,  n k   aj Ψi (M l )t = Mtl − ξsj,m d[M l,c , M m,c ]s j 1 + a X (0,t] j s m=1 j=i+1  1 − ∆Zsi ∆Msl . Zsi

(30)

0<s≤t

For every i = 1, . . . , n, the dynamics of the process X i with respect to the (P, F)martingale M are k n k    aj dXti = ξti,l dMtl − ξti,l ξtj,m d[M l,c , M m,c ]t (31) j j=i+1 1 + aj Xt l,m=1 l=1  i,l 1 i ∆Z ξt ∆Mtl . t Zti l=1 k

−

Proof. We will apply part (ii) of Lemma 4.1. We start by noting that (17), (18), (20) and (26) yield

c  n n  n    aj dXsj aj dXsj,c L(Z i )ct = L(1 + aj X j )ct = = , j 1 + aj Xsj (0,·] 1 + aj Xs− t j=i+1 j=i+1 j=i+1 (0,t] where we also used part (ii) in Assumptions 4.1. In view of (24) and (28), we have that    aj dXsj,c aj ξsj aj ξsj j c = · dΨ (M ) = · dMsc , s j j j (0,t] 1 + aj Xs (0,t] 1 + aj Xs (0,t] 1 + aj Xs where the second equality follows from the fact that the equality Ψj (M )c = M c holds for every j = 1, . . . , n (see Lemma 4.1). Consequently,   n n   ' ( aj ξsj aj ξsj · d[M c ]s i c c c c L(Z ) , M t = · dMs , M = . j 1 + aj Xsj (0, · ] 1 + aj Xs j=i+1 j=i+1 (0,t] t

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Combining the last formula with formula (24) in Lemma 4.1, we obtain the desired equality (29). Formula (30) is an immediate consequence of (29). Finally, in order to derive the dynamics of X i under P, it suffices to combine (28) with (30).  4.2. Setup (B) In this subsection, we work under the following extension of Assumptions 4.1, which is hereafter termed Setup (B). We assume that we are given two independent filtrations, denoted as F1 and F2 , and we set F = F1 ∨ F2 , meaning that for every t we have that Ft = σ(Ft1 , Ft2 ). We consider the Rn -valued processes X and Y given on the filtered probability space (Ω, G, F, P). Assumptions 4.2. We postulate that: (i) the processes X = (X 1 , . . . , X n ) and Y = (Y 1 , . . . , Y n ) are adapted to the filtrations F1 and F2 , respectively, (ii) the processes 1 + aj X j , j = 1, . . . , n and 1 + bj Y j , j = 1, . . . , n belong to S1 , where a1 , . . . , an and b1 , . . . , bn are non-zero constants, (iii) for every i = 0, . . . , n, the processes Z X,i and Z Y,i , which are given by (by convention, Z X,n = Z Y,n = 1) ZtX,i =

n  j=i+1

(1 + aj Xtj ),

ZtY,i =

n 

(1 + bj Ytj ),

j=i+1

are positive (P, F)-martingales, (iv) the PRP holds under (P, F) with the spanning (P, F)-martingale M = (M 1 , . . . , M k ). The following auxiliary lemma is rather standard and thus its proof is omitted. Lemma 4.4. Assume that F1 and F2 are independent filtrations and let F = F1 ∨F2 . (i) Let X and Y be real-valued stochastic processes adapted to the filtrations F1 and F2 , respectively. Then we have that, for every 0 ≤ s ≤ t, EP (Xt Yt | Fs ) = EP (Xt | Fs ) EP (Yt | Fs ) = EP (Xt | Fs1 ) EP (Yt | Fs2 ). (ii) Let M and N be real-valued martingales with respect to the filtrations F1 and F2 , respectively. Then the product M N is a (P, F)-martingale. Remark 4.2. It is enough to assume in condition (iii) in Assumptions 4.2 that the process Z X,i (Z Y,i , respectively) is a (P, F1 )-martingale (a (P, F2 )-martingale, respectively). Indeed, if the process Z X,i is a (P, F1 )-martingale then it also follows a (P, F)-martingale, by the assumed independence of filtrations F1 and F2 . Of course, the symmetric argument can be applied to the process Z Y,i . We are in a position to prove the following generalization of Lemma 4.2. Lemma 4.5. Under Assumptions 4.2 the following properties hold, for every i = 1, . . . , n, (i) the processes Z Y,i Z X,i X i and Z Y,i Z X,i Y i are (P, F)-martingales,

Market Models of Forward CDS Spreads (ii) let the process Z X,Y,i be defined by the formula n ZtX,Y,i := ci ZtX,i ZtY,i = ci (1 + aj Xtj )(1 + bj Ytj ), j=i+1

375

(32)

where ci is a positive constant; then Z X,Y,i is a positive (P, F)-martingale. Proof. To prove part (i), we first argue, as in the proof of Lemma 4.2, that the processes Z X,i X i and Z Y,i Y i are (P, F)-martingales. Since Z X,i X i (Z Y,i Y i , respectively) is F1 -adapted (F2 -adapted, respectively) we conclude that Z X,i X i (Z Y,i Y i , respectively) is also a (P, F1 )-martingale ((P, F2 )-martingale, respectively). We also observe that the process Z Y,i is a (P, F2 )-martingale and thus, in view of part (ii) in Lemma 4.4, we conclude that the product Z Y,i Z X,i X i is a (P, F)-martingale. By the same token, the product Z X,i Z Y,i Y i is a (P, F)-martingale. The proof of the second statement is based on similar arguments and thus it is omitted.  In what follows, it is assumed that the constants ci are chosen in such a way that EP (Z0X,Y,i ) = 1 for every i = 1, . . . , n. Let the probability measures Pi , i = 1, . . . , n be defined as in part (i) in Lemma 4.1 with the Radon-Nikod´ ym density processes Z i = Z X,Y,i . In particular, we have that Pn = P since, by the usual convention, Z X,Y,n = 1. The following result is a slight extension of part (ii) in Lemma 4.1. Lemma 4.6. For every i = 1, . . . , n, the processes X i and Y i are (Pi , F)-martingales and they have the following integral representations under Pi  i i Xt = X0 + ξsi · dΨi (M )s , (33) (0,t]



and Yti

=

Y0i

ζsi · dΨi (M )s ,

+

(34)

(0,t]

where ξ i = (ξ i,1 , . . . , ξ i,k ) and ζ i = (ζ i,1 , . . . , ζ i,k ) are Rk -valued, F-predictable processes and the Rk -valued (Pi , F)-martingale Ψi (M ) is given by.  ' ( 1 Ψi (M )t = Mt − L(Z X,Y,i )c , M c t − ∆ZsX,Y,i ∆Ms . (35) X,Y,i Z 0<s≤t s The equality Ψi (M )c = M c holds for every i = 1, . . . , n. Proof. In view of Lemma 4.5, the processes X i and Y i are manifestly (Pi , F)martingales. Moreover, by virtue of part (ii) in Lemma 4.1, the PRP holds with respect to F under Pi with the spanning (Pi , F)-martingale Ψi (M ). This in turn implies that equalities (33) and (34) hold for some suitably integrable processes ξ i and ζ i . Equality (35) follows immediately from (24). The last statement is a consequence of (35).  The following result will be employed in the joint modeling of forward LIBORs and forward CDS spreads.

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Proposition 4.2. Suppose that Assumptions 4.2 are satisfied. Then for every i = 1, . . . , n the semimartingale decomposition of the (Pi , F)-martingale Ψi (M ) under the probability measure Pn = P is given by the following expression   n n   aj ξsj · d[M c ]s bj ζsj · d[M c ]s Ψi (M )t = Mt − − (36) j 1 + aj X s 1 + bj Ysj j=i+1 (0,t] j=i+1 (0,t]  1 − ∆ZsX,Y,i∆Ms . X,Y,i Z 0<s≤t s The dynamics of X i and Y i with respect to the (P, F)-martingale M are given by dXti =

k 

ξti,l dMtl −

l=1

−

n 

n 

k 

aj

1 + aj Xtj j=i+1 k 

bj

j j=i+1 1 + bj Yt

ξti,l ξtj,m d[M l,c ,M m,c ]t

l,m=1

ξti,l ζtj,m d[M l,c ,M m,c ]t −

l,m=1

1

∆ZtX,Y,i X,Y,i

k 

Zt

ξti,l ∆Mtl

l=1

and dYti =

k 

ζti,l dMtl −

l=1

−

n 

n 

k 

aj

1 + aj Xtj j=i+1 k 

bj

1 + bj Ytj j=i+1

ζti,l ξtj,m d[M l,c ,M m,c ]t

l,m=1

ζti,l ζtj,m d[M l,c ,M m,c ]t −

l,m=1

1

∆ZtX,Y,i X,Y,i

k 

Zt

ζti,l ∆Mtl .

l=1

Proof. The proof is analogous to the proof of Proposition 4.1. We start by noting that n n   L(Z X,Y,i )ct = L(1 + aj X j )ct + L(1 + bj Y j )ct =

j=i+1  n  j=i+1

aj dXsj,c j (0,t] 1 + aj Xs

j=i+1 n 



bj dYsj,c

+

j=i+1

(0,t]

1 + bj Ysj

.

Recall that Ψj (M )c = M c for every j = 1, . . . , n. In view of (33) and (34), by similar computations as in the proof of Proposition 4.1, we thus obtain  n  ' ( aj ξsj X,Y,i c c c c L(Z ) ,M t= j · dMs , M 1 + a X (0, · ] s j j=i+1 t  n j  bj ζ s + · dMsc , M c j 1 + b Y (0, · ] s j j=i+1 t and this in turn yields

 n  ' ( X,Y,i c c L(Z ) ,M t = j=i+1

aj ξsj · d[M c ]s (0,t]

1+

aj Xsj

+

 n  j=i+1

(0,t]

bj ζsj · d[M c ]s 1 + bj Ysj

.

(37)

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By combining (35) with (37), we obtain (36). To establish the dynamics of processes X i and Y i under P, it suffices to substitute (36) into (33) and (34), respectively.  4.3. Setup (C) Throughout this subsection, we will work under the following set of assumptions, which are satisfied by the processes X and Y defined on a filtered probability space (Ω, G, F, P). Assumptions 4.3. We postulate that: (i) the processes X = (X 1 , . . . , X n ) and Y = (Y 2 , . . . , Y n ) are F-adapted, (ii) for every i = 1, . . . , n, the process Z X,i , which is given by the formula (by the usual convention, Z X,n = 1) ZtX,i = ci

n 

Ytj − Xtj

j=i+1

Xtj−1 − Ytj

,

(38)

is a positive (P, F)-martingale and c1 , . . . , cn are constants such that EP (Z0X,i ) = 1, (iii) for every i = 2, . . . , n, the process Z Y,i, which is given by the formula   X i−1 − Xti X,i ZtY,i = + ci ZtX,i + ZtX,i−1 = + ci ti−1 Zt , Xt − Yti

(39)

is a positive (P, F)-martingale and + c2 , . . . , + cn are constants such that EP (Z0Y,i ) = 1, (iv) for every i = 1, . . . , n, the process X i is a (Pi , F)-martingale, where the Radon-Nikod´ ym density of Pi with respect to P equals Z X,i , + i , F)-martingale, where the Radon(v) for every i = 2, . . . , n, the process Y i is a (P +i with respect to P equals Z Y,i, Nikod´ ym density of P (vi) the PRP holds under (P, F) with the spanning (P, F)-martingale M = (M 1 , . . . , M k ). When dealing with Setups (A) and (B), we explicitly stated the assumption that certain processes are in the space S1 , so that their stochastic logarithms are well defined. In the case of Setups (C) and (D), we make instead a general assumption that whenever the notion of the stochastic logarithm is employed, the underlying stochastic process is assumed to belong to the space S1 . +i are defined as in part (i) Note also that the probability measures Pi and P in Lemma 4.1, with the Radon-Nikod´ ym density processes Z X,i and Z Y,i , respecn tively. In particular, we have that P = P since Z X,n = 1. Lemma 4.7. For every i = 1, . . . , n, the process X i admits the following representation under Pi  Xti = X0i + ξsi · dΨi (M )s , (40) (0,t]

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where ξ i = (ξ i,1 , . . . , ξ i,k ) is an Rk -valued, F-predictable process and the Rk -valued (Pi , F)-martingale Ψi (M ) is given by  1 ' ( Ψi (M )t = Mt − L(Z X,i )c , M c t − ∆ZsX,i ∆Ms . (41) X,i Z 0<s≤t s +i For every i = 2, . . . , n, the process Y i has the following representation under P  + i (M )s , Yti = Y0i + ζsi · dΨ (42) (0,t]

where ζ i = (ζ i,1 , . . . , ζ i,k ) is an Rk -valued, F-predictable process and the Rk -valued + i , F)-martingale Ψ + i (M ) is given by (P  1 ' ( + i (M )t = Mt − L(Z Y,i )c , M c − Ψ ∆ZsY,i ∆Ms . (43) Y,i t Z s 0<s≤t + i (M )c = M c , i = 2, . . . , n are The equalities Ψi (M )c = M c , i = 1, . . . , n and Ψ valid. Proof. Let us fix i = 1, . . . , n. By parts (iv) and (vi) in Assumptions 4.3, the process X i is a (Pi , F)-martingale. In view of part (ii) in Lemma 4.1, the PRP holds with respect to F under Pi with the spanning (Pi , F)-martingale Ψi (M ). This yields the integral representation of X i under Pi . The proof of the second statement is similar, with parts (v) and (vi) in Assumptions 4.3 used.  Proposition 4.3. For every i = 1, . . . , n, the semimartingale decomposition of the (Pi , F)-martingale Ψi (M ) under the probability measure Pn = P is given by   n  (ζsj − ξsj ) · d[M c ]s (ξsj−1 − ζsj ) · d[M c ]s i Ψ (M )t = Mt − − Ysj − Xsj Xsj−1 − Ysj (0,t] (0,t] j=i+1  1 − ∆ZsX,i ∆Ms . X,i Z s 0<s≤t + i , F)-martingale For every i = 2, . . . , n, the semimartingale decomposition of the (P i n + (M ) under the probability measure P = P is given by Ψ   (ξsi−1 − ξsi ) · d[M c ]s (ξsi−1 − ζsi ) · d[M c ]s i + Ψ (M )t = Mt − + Xsi−1 − Xsi Xsi−1 − Ysi (0,t] (0,t]    n  (ζsj − ξsj ) · d[M c ]s (ξsj−1 − ζsj ) · d[M c ]s − − Ysj − Xsj Xsj−1 − Ysj (0,t] (0,t] j=i+1  1 − ∆ZsY,i ∆Ms . Y,i 0<s≤t Zs

Market Models of Forward CDS Spreads

379

Proof. Using (18), (20) and (38), we obtain L(Z X,i )ct =

n  ' j=i+1

=

n 

L(Y j − X j )ct − L(X j−1 − Y j )ct



dYsj,c − dXsj,c

(0,t]

j=i+1

Ysj − Xsj



(

dXsj−1,c − dYsj,c

− (0,t]

.

Xsj−1 − Ysj

Consequently, '

L(Z

X,i c

) ,M

c

( t

=

 n 

d[Y j,c , M c ]s − d[X j,c , M c ]s

j=i+1 (0,t]  n 

−

j=i+1

(0,t]

Ysj − Xsj d[X j−1,c , M c ]s − d[Y j,c , M c ]s Xsj−1 − Ysj

.

+ j (M )c = M c , which Using equations (40) and (42) and the equalities Ψj (M )c = Ψ were established in Lemma 4.7, we conclude that   n  ' ( (ζsj − ξsj ) · d[M c ]s (ξsj−1 − ζsj ) · d[M c ]s X,i c c L(Z ) , M t = − . j j Ys − X s Xsj−1 − Ysj (0,t] (0,t] j=i+1 By combining the formula above with (41), we obtain the desired expression for the process Ψi (M ). + i (M ), we first note that To establish the formula for Ψ L(Z Y,i )c = L(X i−1 − X i )c − L(X i−1 − Y i )c + L(Z X,i )c , which in turns implies that ' ( ' ( ' ( L(Z Y,i )c , M c = L(X i−1 − X i)c − L(X i−1 − Y i )c , M c + L(Z X,i )c , M c . Since the decomposition of the process [L(Z X,i )c , M c ] was already found above, it suffices to focus on the process L(X i−1 − X i )c − L(X i−1 − Y i )c . In view of (20), we have that   dXsi−1,c − dXsi,c dXsi−1,c − dYsi,c i−1 i c i−1 i c L(X − X )t − L(X − Y )t = − . Xsi−1 − Xsi Xsi−1 − Ysi (0,t] (0,t] By making again use of integral representations (40) and (42), we thus obtain ' ( L(X i−1 − X i )c − L(X i−1 − Y i )c , M c t   (ξsi−1 − ξsi ) · d[M c ]s (ξsi−1 − ζsi ) · d[M c ]s = − . Xsi−1 − Xsi Xsi−1 − Ysi (0,t] (0,t] After combining the decomposition of ' ( ' ( L(Z X,i )c , M c and L(X i−1 − X i )c − L(X i−1 − Y i )c , M c ,

380

L. Li and M. Rutkowski

we conclude that

 (ξsi−1 − ξsi ) · d[M c ]s (ξsi−1 − ζsi ) · d[M c ]s + i−1 Xs − Xsi Xsi−1 − Ysi (0,t] (0,t]    n  (ζsj − ξsj ) · d[M c ]s (ξsj−1 − ζsj ) · d[M c ]s − − j j Ys − Xs Xsj−1 − Ysj (0,t] (0,t] j=i+1  1 − ∆ZsY,i ∆Ms , Y,i Z 0<s≤t s

+ i (M )t = Mt − Ψ





which is the desired formula. 4.4. Setup (D)

The setup considered in this subsection is similar to Setup (C) and thus it will be presented rather succinctly. We work here under the following standing assumptions. Assumptions 4.4. We postulate that: (i) the processes X = (X 0 , . . . , X n−2 ) and Y = (Y 0 , . . . , Y n−1 ) are F-adapted, (ii) for every i = 0, . . . , n − 1, the process Z Y,i , which is given by the formula (by the usual convention, Z Y,0 = 1) ZtY,i = + ci

i−1 

Ytj − Xtj

j=0

Ytj+1 − Xtj

,

(44)

is a positive (P, F)-martingale and + c1 , . . . , + cn−1 are constants such that EP (Z0Y,i ) = 1, (iii) for every i = 0, . . . , n − 2, the process Z X,i , which is given by the formula Z X,i = ci

Y i+1 − Y i Y,i Z , Y i+1 − X i

(45)

is a positive (P, F)-martingale and c0 , . . . , cn−2 are constants such that EP (Z0X,i ) = 1, (iv) for every i = 0, . . . , n − 2, the process X i is a (Pi , F)-martingale, where the Radon-Nikod´ ym density of Pi with respect to P equals Z X,i , + i , F)-martingale, where the (v) for every i = 0, . . . , n − 1, the process Y i is a (P i + with respect to P equals Z Y,i , Radon-Nikod´ ym density of P (vi) the PRP holds under (P, F) with the spanning (P, F)-martingale M = (M 1 , . . . , M k ). + i are defined as in part (i) in Lemma 4.1, The probability measures Pi and P with the Radon-Nikod´ ym density processes Z X,i and Z Y,i , respectively. Hence 0 Y,0 + P = P since Z = 1.

Market Models of Forward CDS Spreads

381

Lemma 4.8. (i) For every i = 0, . . . , n − 2, the process X i admits the following representation under Pi  Xti = X0i + ξsi · dΨi (M )s , (46) (0,t]

where ξ = (ξ , . . . , ξ ) is an R -valued, F-predictable process and the Rk valued (Pi , F)-martingale Ψi (M ) equals  1 ' ( Ψi (M )t = Mt − L(Z X,i )c , M c t − ∆ZsX,i ∆Ms . X,i Z 0<s≤t s i

i,1

i,k

k

(ii) For every i = 0, . . . , n − 1, the process Y i has the following representation +i under P  + i (M )s , Yti = Y0i + ζsi · dΨ (47) (0,t]

where ζ = (ζ , . . . , ζ ) is an Rk -valued, F-predictable process and the Rk + i , F)-martingale Ψ + i (M ) equals valued (P  1 ' ( + i (M )t = Mt − L(Z Y,i )c , M c − Ψ ∆ZsY,i ∆Ms . Y,i t Z 0<s≤t s i

i,1

i,k

+ i (M )c = M c , The equalities Ψi (M )c = M c , i = 0, . . . , n − 2 and Ψ i = 0, . . . , n − 1 are valid. Proof. Let us fix i = 0, . . . , n − 2. By parts (iv) and (vi) in Assumptions 4.4, the process X i is a (Pi , F)-martingale. In view of part (ii) in Lemma 4.1, the PRP holds with respect to F under Pi with the spanning (Pi , F)-martingale Ψi (M ). This proves the first part of the lemma. The proof of the second statement is similar, with parts (v) and (vi) in Assumptions 4.4 used.  To derive the formulae of Proposition 4.4, it suffices to make a suitable change of the index and to permute the processes X i and Y j in the proof of Proposition 4.3. Proposition 4.4. For every i = 0, . . . , n − 2, the semimartingale decomposition of +0 = P is given by the (Pi , F)-martingale Ψi (M ) under the probability measure P   (ζsi+1 − ζsi ) · d[M c ]s (ζsi+1 − ξsi ) · d[M c ]s Ψi (M )t = Mt − + i+1 Ys − Ysi Ysi+1 − Xsi (0,t] (0,t]   i−1   (ζsj − ξsj ) · d[M c ]s (ζsj+1 − ξsj ) · d[M c ]s − − Ysj − Xsj Ysj+1 − Xsj (0,t] (0,t] j=0  1 − ∆ZsX,i ∆Ms . X,i 0<s≤t Zs

382

L. Li and M. Rutkowski

+i , F)-martinFor every i = 0, . . . , n − 1, the semimartingale decomposition of the (P i 0 + + gale Ψ (M ) under the probability measure P = P is given by   i−1   (ζsj − ξsj ) · d[M c ]s (ζsj+1 − ξsj ) · d[M c ]s i + Ψ (M )t = Mt − − j j Ys − Xs Ysj+1 − Xsj (0,t] (0,t] j=0  1 − ∆ZsY,i ∆Ms . Y,i Z 0<s≤t s

5. Market models for forward CDS spreads We are in a position to apply the abstract setups developed in Section 4 to the modeling of finite families of forward CDS spreads in various configurations. As mentioned in the introduction, we will first consider the simplest case of deterministic interest rate. Subsequently, we will proceed to the study of more general, and thus more practically relevant, market models. The aim in all models is to derive the joint dynamics of a given family of forward CDS spreads under a single measure by first finding expressions for all ratios of defaultable annuities (annuity deflated swap num´eraires) in terms of forward CDS spreads from a predetermined family. In this section, it will be assumed throughout that the σ-field F0 is trivial. Although this assumption is not necessary for our further developments, it is commonly used in financial modeling, since it fits well the interpretation of the σ-field F0 as representing the (deterministic) market data available at time 0. 5.1. Model of one-period CDS spreads Following the papers by Brigo [9, 10] and Schl¨ ogl [29], we will first examine the issue of properties and construction of a market model for a family of one-period forward CDS spreads under the assumption that the interest rate is deterministic. It is worth noting that the situation considered in this subsection is formally similar to the modeling of a family of discrete-tenor forward LIBORs (see, for instance, Brace [5], Jamshidian [16, 17] and Musiela and Rutkowski [23, 24]). However, we need also to examine here the issue of existence of a default time consistent with a postulated dynamics of forward CDS spreads. Needless to say that this issue does not arise in the problem of modeling default-free market interest rates, such as: forward LIBORs or, more generally, forward swap rates. 5.1.1. Bottom-up approach. By the bottom-up approach we mean a modeling approach in which one starts with some credit risk model (where the default time τ and the associated survival process G are defined) as well as a term structure of interest rates. All other quantities of interest are subsequently computed from these fundamentals, in particular, the joint dynamics of a family of forward CDS spreads. In this sense, we are building up starting from the bottom. We refer to Section 2 for the description of a generic forward CDS and the definition of the associated fair forward CDS spread. Recall that we consider a

Market Models of Forward CDS Spreads

383

tenor structure T = {T0 < T1 < · · · < Tn } with T0 ≥ 0 and we write ai = Ti −Ti−1 . We start by noting that, using formulae (9) and (10), one can derive the following expression for the pre-default forward CDS spread κi := κi−1,i corresponding to the one-period forward CDS starting at Ti−1 and maturing at Ti 1+? ai κit =

EQ (β(t, Ti )1{τ >Ti−1 } | Ft ) , EQ (β(t, Ti )1{τ >Ti } | Ft )

∀ t ∈ [0, Ti−1 ],

(48)

where we denote ? ai = ai /(1 − δi ) (recall that δi is a non-random recovery rate). In addition to the family of one-period forward CDS spreads κ1 , . . . , κn , we also consider in Subsections 5.1 and 5.2 the family of forward LIBORs L1 , . . . , Ln , which are defined by the formula 1 + ai Lit =

B(t, Ti−1 ) , B(t, Ti )

(49)

where for every T > 0 we denote by B(t, T ) the price at time t of the default-free unit zero-coupon bond maturing at time T . Throughout Subsection 5.1, we work under the standing assumption that the interest rate is deterministic, so that 1+? ai κit =

Q(τ > Ti−1 | Ft ) Q(τ > Ti | Ft )

∀ t ∈ [0, Ti−1 ],

(50)

It then follows immediately from (3) and (50) that the process κi is positive, for i = 1, . . . , n. We observe that the right-hand side in (50) can be represented as follows, for every i = 1, . . . , n and t ∈ [0, Ti−1 ], 1+? ai κit =

+i−2,i−1 +i−2,i−1 ai β(t, Ti ) A ai 1 A t t = . i +i−1,i +i−1,i ai−1 β(t, Ti−1 ) A a 1 + a L i−1 i A t t t

(51)

Formula (51) shows that 1 + ? ai κit is equal, up to a deterministic function, to the ratio of defaultable annuities corresponding to the forward CDS spreads κi−1 and κi . Since the interest rates are now assumed to be deterministic, we find it easier to work directly with (50), however. Let us mention that the case of random interest rates is examined in Subsection 5.2. Let us define the probability measure P equivalent to Q on (Ω, FTn ) by setting, for every t ∈ [0, Tn ], dP  Q(τ > Tn | Ft ) ηt = .  = dQ Ft Q(τ > Tn ) Lemma 5.1. For every i = 0, . . . , n, the process Z¯ κ,i , which is given by the formula Z¯tκ,i =

n  

 1+? aj κjt ,

∀ t ∈ [0, Ti],

j=i+1

is a (P, F)-martingale where, by the usual convention, we set Z¯tκ,n = 1.

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Proof. It suffices observe that from (50) we obtain Q(τ > Ti | Ft ) Z¯tκ,i = . (52) Q(τ > Tn | Ft ) It is now easily seen that the process Z¯ κ,i η is a (Q, F)-martingale, which in turn implies that the asserted property is valid.  In the next step, for any i = 1, . . . , n, we define the probability measure Pi equivalent to P by setting n    dPi  κ,i κ,i ¯ = Z := c Z  1+? aj κjt , (53)  i t t dP Ft j=i+1 >Tn ) where ci = Q(τ Q(τ >T) is the normalizing constant and the symbol  means that the equality holds up to a normalizing constant. Note that the equality Ztκ,n = 1 holds and thus Pn = P.

Lemma 5.2. For any fixed i = 1, . . . , n, the one-period forward CDS spread κi is a positive (Pi , F)-martingale. Proof. It is enough to show that 1 + ? ai κit is a (Pi , F)-martingale. To this end, it suffices to combine formulae (50) and (52).  The relationships between various martingale measures for the family of oneperiod forward CDS spreads are summarized in the following diagram dPn dQ

dPn−1 n

dPn−2 n−1

dP2 3

dP1 2

dP −−→ Pn−1 −− −−→ . . . −−dP −−→ P2 −−dP −−→ P1 Q −−−−→ Pn −−dP

where

dPn  Q(τ > Tn | Ft ) ,  = dQ Ft Q(τ > Tn ) dPi−1  Q(τ > Ti−1 | Ft ) ci (1 + ? ai κit )  ,  =? i dP Ft Q(τ > Ti | Ft)

Q(τ >Ti ) where ? ci = Q(τ >Ti−1 ) is the normalizing constant. Let us assume, in addition, that the PRP holds under P = Pn with the spanning (P, F)-martingale M . Then this property is also valid with respect to the filtration F under any probability measure Pi for i = 1, . . . , n with the spanning (Pi , F)-martingale Ψi (M ). Therefore, for every i = 1, . . . , n, the positive (Pi , F)martingale κi admits the following integral representation under Pi  i i κt = κ0 + κis− σsi · dΨi (M )s , (54) (0,t]

where σ is an R -valued, F-predictable process. For the sake of convenience, we will henceforth refer to σ i as the ‘volatility’ of the forward CDS spread κi . To sum up, we have just shown the existence of the volatility processes for one-period forward CDS spreads. It will be now not difficult to derive the joint i

k

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dynamics of these processes under a single probability measure, for instance, the martingale measure P = Pn . However, since the joint dynamics of the one-period CDS spreads are the same as in the top-down approach presented in the foregoing subsection, we do not provide the explicit formula here and we refer instead the reader to equation (60). 5.1.2. Top-down approach. As was explained above, in the standard bottom-up approach to modeling of CDS spreads, we start by choosing a term structure model and we combine it with the specification of the default time. Together with the assumption that the PRP holds, we can subsequently derive the joint dynamics and show the existence of a family of volatility processes. By contrast, in the top-down approach to a market model, we take as inputs a family of volatility processes and a family of driving martingales. We then derive the joint dynamics of a given family of forward CDS spreads under a single probability measure and show that they are uniquely specified by the choice of the volatility processes and driving martingales. Thus, instead of doing computation from a given credit and term structure model, one can directly specify the joint dynamics of a family of CDS spreads through the choice of the volatility processes and driving martingales. This justifies the name ‘top-down’ approach. In principle, it is also possible to construct the corresponding default time, but this issue is not dealt with in this paper. In this subsection, we will show how to apply the top-down approach in order to construct the model of one-period forward CDS spreads under the assumption that the interest rate is deterministic. To achieve this goal, we will work under the following standing assumptions, which are motivated by the analysis of the bottom-up approach. Assumptions 5.1. We are given a filtered probability space (Ω, G, F, P) and we postulate that: (i) the initial values of processes κ1 , . . . , κn are given, (ii) for every i = 1, . . . , n, the process Z κ,i , which is given by the formula (by the usual convention, Z κ,n = 1) n Ztκ,i = ci (1 + ? aj κjt ), (55) j=i+1

is a positive (P, F)-martingale, where ci is a constant such that Z0κ,i = 1, (iii) for every i = 1, . . . , n, the process κi is a (Pi , F)-martingale, where the RadonNikod´ ym density of Pi with respect to P equals Z κ,i , so that, in particular, n P = P, (iv) for every i = 1, . . . , n, the process κi has the following representation under (Pi , F)  i i +si , κt = κ0 + κis− σ +si · dN (56) (0,t]

+ i is an Rki -valued (Pi , F)-martingale and σ where N +i is an Rki -valued, i + F-predictable, N -integrable process.

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In view of part (iii) in Assumptions 5.1, the following definition is rather obvious. Definition 5.1. The probability measure Pi is called the martingale measure for the one-period forward CDS spread κi . We should make some important comments here. According to the top-down approach, the processes κ1 , . . . , κn are in fact not given in advance, but should be constructed. Also, the default time is not defined, so the forward CDS spreads cannot be computed using formulae (9), (10) and (11). Therefore, the volatility processes σ +1 , . . . , σ +n appearing in formula (56), as well as the driving martin1 n + + gales N , . . . , N , can be chosen arbitrarily by the modeler. Put another way, the volatility processes and driving martingales should now be seen as model inputs. As an output in the top-down construction of a market model, we obtain the joint dynamics of the forward CDS spreads κ1 , . . . , κn under a common probability measure. The form of the joint dynamics obtained below should be seen as a necessary condition for the arbitrage-free property of the model. Unfortunately, sufficient conditions for the arbitrage-free property are more difficult to handle. Remark 5.1. It is worth mentioning that assumption (56) can be replaced by the following one  + i, κit = κi0 + σ +si · dN s (0,t]

which, in fact, is more general than (56). All foregoing results can be easily adjusted to this alternative postulate. Similar comments apply to other variants of market models of forward CDS spreads (and forward LIBORs), which are examined in the foregoing subsections, so they will not be repeated in what follows. Lemma 5.3. Let k = k1 + · · · + kn . There exists an Rk -valued (P, F)-martingale M such that the process κi admits the following representation under the martingale measure Pi  κit = κi0 + κis− σsi · dΨi (M )s , (57) (0,t]

where σ is an R -valued, F-predictable process and the process Ψi (M ) is given by (24). i

k

Proof. Using the bijectivity of the Girsanov transform, for every i = 1, . . . , n one can establish the existence of an Rki -valued (P, F)-martingale N i such that + i = Ψi (N i ) = ΨZ κ,i (N i ). Hence (56) yields N  κit = κi0 + κis− σ +si · dΨi (N i )s . (0,t]

To complete the proof, it suffices to define the Rk -valued (P, F)-martingale M by setting M = (N 1 , . . . , N n ) and to formally extend each process Rki -valued, Fpredictable process σ +i to the corresponding Rk -valued, F-predictable process σ i by setting to zero all irrelevant components of σ i . 

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+ n = N n since Z κ,n = 1. Consequently, for i = n It is worth noting that N equation (57) becomes  κnt = κn0 + κns− σsn · dMs . (0,t]

Of course, this is consistent with the interpretation of Pn as the martingale measure for κn . By contrast, for i = 1, . . . , n − 1 it is expected that the dynamics (57) of κi under Pn will also have a non-zero ‘drift’ term, which is due to appear, since + i is not a (Pn , F)-martingale, in general. The main result of this the process N subsection, Proposition 5.1, furnishes an explicit expression for the drift term in the dynamics (57) of κi under Pn . + i are Brownian motions Remark 5.2. In practice, for instance, when the processes N under the respective martingale measures Pi , the actual dimension of the underlying driving (P, F)-martingale can typically be chosen to be less than k1 + · · · + kn . For more details on explicit constructions of market models for forward CDS spreads driven by correlated Brownian motions, we refer to papers by Brigo [10] and Rutkowski [27], where the common volatilities-correlations approach is presented. In view of Lemma 5.3, we may reformulate Assumptions 5.1 in the simpler, and more practically appealing, manner. Assumptions 5.2. We are given a filtered probability space (Ω, G, F, P) and we postulate that: (i) the initial values of processes κ1 , . . . , κn are given, (ii) for every i = 1, . . . , n, the process Z κ,i , which is given by the formula (by the usual convention, Z κ,n = 1) Ztκ,i = ci

n 

(1 + ? aj κjt ),

(58)

j=i+1

is a positive (P, F)-martingale, where ci is a constant such that Z0κ,i = 1, (iii) for every i = 1, . . . , n, the process κi has the following representation under (P, F)  κit = κi0 + κis− σsi · dΨi (M )s , (59) (0,t]

where M is an Rk -valued (P, F)-martingale, σ i is an Rk -valued, F-predictable process and the process Ψi (M ) is given by formula (24) with Z i = Z κ,i . By comparing Assumptions 5.2 with Assumptions 4.1 and Lemma 4.3, we see that the joint dynamics of κ1 , . . . , κn under P = Pn can now be computed by applying results obtained for Setup (A). Of course, in order to proceed, we need to make an implicit assumption that all relevant processes belong to the space S1 introduced in Section 3. The following lemma is a rather straightforward

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consequence of Proposition 4.1. Recall that for the Rk -valued (P, F)-martingale M we have that M c = (M 1 , . . . , M k )c = (M 1,c , . . . , M k,c ) and the matrix [M c ] is given by formula (25). Lemma 5.4. Under Assumptions 5.2, for every i = 1, . . . , n the semimartingale decomposition of the (Pi , F)-martingale Ψi (M ) under the martingale measure Pn = P is given by  n   1 ? aj κjs σsj · d[M c ]s Ψi (M )t = Mt − − ∆Zsκ,i ∆Ms , j κ,i 1 + ? a κ Z j s j=i+1 (0,t] 0<s≤t s where σj = (σ j,1 , . . . , σ j,k ). More explicitly, for every i = 1, . . . , n and l = 1, . . . , k, Ψi (M l )t =Mtl −

n  

k ? aj κjs 



1 σsj,m d[M l,c ,M m,c ]s − ∆Zsκ,i ∆Msl . j κ,i 1+? a κ Z j s m=1 j=i+1 (0,t] 0<s≤t s

The next result, which follows easily by combining formula (57) with Lemma 5.4, furnishes an explicit expression for the joint dynamics of the family of oneperiod forward CDS spreads under deterministic interest rates. An important conclusion from Proposition 5.1 is that the joint dynamics of one-period forward CDS spreads are uniquely specified, under Assumptions 5.1, by the choice of the driv+ 1, . . . , N + n and the volatility processes σ ing (Pi , F)-martingales N +1 , . . . , σ +n or, more practically, under Assumptions 5.2, by the choice of the driving (P, F)-martingale M under P and the volatility processes σ 1 , . . . , σ n . Proposition 5.1. Under Assumptions 5.2, the joint dynamics of forward CDS spreads κ1 , . . . , κn under the martingale measure P = Pn are given by the following n-dimensional stochastic differential equation driven by the (P, F)-martingale M : for every i = 1, . . . , n and t ∈ [0, Ti−1 ] dκit =

k 

κit− σti,l dMtl −

n 

? aj κit κjt

j=i+1

l=1

−

κit− Ztκ,i

1

k 

+? aj κjt l,m=1

∆Ztκ,i

k 

σti,l σtj,m d[M l,c , M m,c ]t

(60)

σti,l ∆Mtl .

l=1

We need also to address some consistency issues regarding the top-down methodology. In the top-down approach, we start with a given family of volatility processes related to some driving martingale and we derive the SDE (60). Therefore, it is necessary to establish the existence and uniqueness result for a solution to the SDE (60) governing the forward CDS spreads κi , i = 1, . . . , n under P. Moreover, we need to show that given a solution to (60) exists, the processes Z κ,i computed from the SDE (60) is indeed a (P, F)-martingale, as it was assumed.

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Proposition 5.2. Let σ i , i = 1, . . . , n be Rk -valued, F-predictable processes. Assume that the joint dynamics of processes κi , i = 1, . . . , n under P are given by (60) with Z κ,i given by (55). Then: (i) the processes κi , i = 1, . . . , n are well defined, specifically, the stochastic differential equation (60) admits a unique strong solution (κ1 , . . . , κn ), (ii) for every i = 1, . . . , n, the process Z κ,i given by (55) is a (P, F)-martingale, (iii) for every i = 1, . . . , n, the process κi is a (Pi , F)-martingale governed by (54), where in turn the probability measure Pi is given by (53). Proof. Part (i) is rather standard; it suffices to employ the classic result on stochastic differential equations with locally Lipschitz continuous coefficients. Part (ii) can be established by induction. It suffices to focus on processes Z κ,i given n by (55). From process  (60),nwe  deduce that κ is a (P, F)-martingale and thus theκ,i+1 κ,n−1 Zt = 1+? an κt is a (P, F)-martingale as well. Assume now that Z is a (P, F)-martingale so that, in view of (55) and (60), dZtκ,i+1 =

n  j=i+2

Note that



Ztκ,i+1 j j  κt σt · dMt . 1+? aj κjt

  κ,i+1 Ztκ,i = 1 + ? ai+1 κi+1 Zt . t

(61)

(62)

To show that Z κ,i is a (P, F)-martingale, it suffices to check that the drift term in the dynamics of this process vanishes. For this purpose, we apply the Itˆ o formula to the right-hand side in (55) and we employ formulae (60) and (61) in order to compute the covariation [κi+1 , Z κ,i+1 ]. Part (iii) is an immediate consequence of the postulated dynamics (60) and Lemma 5.4.  5.2. Model of LIBORs and one-period CDS spreads In this subsection, we relax the assumption that interest rate is deterministic. Our goal is to examine the joint dynamics of one-period forward CDS spreads κ1 , . . . , κn and forward LIBORs L1 , . . . , Ln associated with the tenor structure T = {T0 < T1 < · · · < Tn }. In general, this task seems to be quite difficult and thus we will examine the special case when an independence property holds. 5.2.1. Bottom-up approach. In the bottom-up approach, we will work under the assumption that the term structure model and the default intensity are independent under the risk-neutral probability measure, denoted as Q in what follows. For a fixed time horizon T = Tn , we work under the following assumptions. Assumptions 5.3. We are given a filtered probability space (Ω, G, F, Q) and we postulate that: (i) the risk-neutral default intensity process (λt )t∈[0,T ] generates the filtration Fλ = (Ftλ )t∈[0,T ] , (ii) the interest rate process (rt )t∈[0,T ] generates the filtration Fr = (Ftr )t∈[0,T ] ,

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(iii) the interest rate r and the default intensity λ are independent processes under Q, (iv) the (H)-hypothesis holds, meaning that the filtration F := Fr ∨Fλ is immersed in G. For the (H)-hypothesis (also known as the immersion property or the martingale invariance property) we refer, for instance, to Section 6.1 in [1] or Section 3.2 in [4]. Recall that, under the (H)-hypothesis, the risk-neutral default intensity λ is an F-predictable process such that the survival process G admits the following representation, for every t ∈ [0, T ],  t

Gt = Q(τ > t | Ft ) = exp − λu du . 0

It is thus easy to deduce from Assumptions 5.3 that the interest rate r and the survival process G are independent under the risk-neutral probability measure Q. In the bottom-up approach, our goal is to derive the joint dynamics of the family of one-period forward CDS spreads κ1 , . . . , κn , where (cf. (50))    EQ β(t, Ti )1{τ >Ti−1 }  Ft i   ,  1+? ai κt = (63) EQ β(t, Ti )1{τ >Ti }  Ft and the family of forward LIBORs L1 , . . . , Ln , which are given by (cf. (49)) 1 + ai Lit =

B(t, Ti−1 ) , B(t, Ti )

(64)

where B(t, T ) is the price at time t of the default-free, unit zero-coupon bond maturing at time T . We will now attempt to simplify the formula for the forward CDS spread κi . Lemma 5.5. Let Assumptions 5.3 be satisfied. Then for every i = 1, . . . , n, the CDS spread κi has the following representation, for every t ∈ [0, Ti−1 ], 1+? ai κit =

Q ( τ > Ti−1 | Ft ) . Q ( τ > Ti | F t )

(65)

Proof. Using (63) and the properties of conditional expectation, we obtain      EQ EQ β(t, Ti )1{τ >Ti−1 }  FTi  Ft i      1+? ai κ t = EQ EQ β(t, Ti )1{τ >Ti }  FTi  Ft =

EQ ( β(t, Ti )Q ( τ > Ti−1 | FTi ) | Ft ) . EQ ( β(t, Ti )Q ( τ > Ti | FTi ) | Ft )

In view of the immersion property postulated in part (iv) of Assumptions 5.3, we have that   Q ( τ > Ti−1 | FTi ) = Q τ > Ti−1 | FTi−1 = GTi−1 , and thus        EQ β(t, Ti )Q τ > Ti−1 | FTi−1  Ft EQ β(t, Ti )GTi−1  Ft i = . 1+? ai κt = EQ ( β(t, Ti )Q ( τ > Ti | FTi ) | Ft ) EQ ( β(t, Ti )GTi | Ft )

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Since the interest rate and default intensity are assumed to be independent under Q, using Lemma 4.4, we obtain the following chain of equalities    EQ β(t, Ti )GTi−1  Ftr ∨ Ftλ i   1+? ai κ t = EQ β(t, Ti )GTi | Ftr ∨ Ftλ    EQ ( β(t, Ti ) | Ftr ) EQ GTi−1  Ftλ   = EQ ( β(t, Ti ) | Ftr ) EQ GTi | Ftλ   Q τ > Ti−1 | Ftλ Q ( τ > Ti−1 | Ft )  =  = . Q ( τ > Ti | Ft ) Q τ > Ti | Ftλ This shows that equality (65) is valid.



From (65), we see that, under Assumptions 5.3, we once again obtain expression (50), as in the case of deterministic interest rates. Let us point out that it is sufficient to start from (65) and to follow the exact derivation for the case of deterministic interest rate, in order to obtain the joint dynamics of the family of one-period forward CDS spreads. We wish, however, to obtain a model in which the joint dynamics of forward LIBORs and one-period forward CDS spreads are specified. To this end, we first define the probability measure P equivalent to Q on (Ω, FTn ) by postulating that the Radon-Nikod´ ym density is given by    EQ β(t, Tn )1{τ >Tn }  Ft dP  ηt = . (66)  = dQ Ft EQ (β(t, Tn )1{τ >Tn } ) For every i, j = 1, . . . , n, we will now derive expressions for the ratios of defaultable annuities (i.e., annuity deflated swap num´eraires)    +i−1,i A ai EQ β(t, Ti )1{τ >Ti }  Ft t    = (67) +j−1,j aj EQ β(t, Tj )1{τ >Tj }  Ft A t in terms of one-period CDS spreads and forward LIBORs. Lemma 5.6. Under Assumptions 5.3, the annuity deflated swap num´eraire satisfies, for every i = 2, . . . , n, +i−2,i−1 A ai−1 t = (1 + ? ai κit )(1 + ai Lit ). i−1,i + ai At

(68)

Proof. In view of (9) and the assumed independence of the interest rate and the default intensity, we obtain    +i−2,i−1 A ai−1 EQ β(t,Ti−1 )1{τ >Ti−1 }  Ft a B(t,Ti−1 )Q( τ > Ti−1 |Ft ) t   = i−1  = . i−1,i  + a a B(t,Ti )Q( τ > Ti |Ft ) E β(t,T )1 F i i At Q i {τ >Ti } t The result now follows by substituting (64) and (65) into the formula above.



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Formula (68) furnishes a collection of positive processes, which will be used to define a family of equivalent probability measures Pi for i = 1, . . . , n such that the processes κi and Li are (Pi , F)-martingales. The relationships between these probability measures are represented by the following diagram dPn dQ

dPn−1 n

dPn−2 n−1

dP2 3

dP1 2

dP Q −−−−→ Pn −−dP −−→ Pn−1 −− −−→ . . . −−dP −−→ P2 −−dP −−→ P1

where we set (recall that  means that the equality holds up to a normalizing constant) +i−2,i−1 A dPn  dPi−1  ai−1 t : = (1 + ? ai κit )(1 + ai Lit ).  = ηt ,  i−1,i i +t dQ Ft dP Ft ai A In view of (68), it is then not hard to see that n +i−1,i dPi  A ai  κ,L,i t : n−1,n = (1 + ? aj κjt )(1 + aj Ljt ).  := Zt + dPn Ft a n At j=i+1

(69)

We define the equivalent probability measure Pi by postulating that the RadonNikod´ ym density of Pi with respect to P equals Z κ,L,i . In particular, we have that κ,L,n Zt = 1 and thus the equality Pn = P holds. The next result shows that Pi is a martingale measure for the processes κi and Li . Lemma 5.7. For every i = 1, . . . , n, the forward CDS spread κi and the forward LIBOR Li are (Pi , F)-martingales. Proof. To demonstrate that the forward CDS spread κi is a (Pi , F)-martingale, it is enough to use (63) and (69) in order to establish that the product Z κ,L,i(1 + ? ai κi ) is a (P, F)-martingale. Similarly, to prove that the forward LIBOR Li is a (Pi , F)-martingale, it suffices to use (64), (66) and (69) to show that, under the independence assumption, the process ηZ κ,L,i(1 + ai Li ) is a (Q, F)-martingale. It then follows immediately that Z κ,L,i(1 + ai Li ) is a (P, F)-martingale, which in turn implies that 1 + ai Li (and thus also the forward LIBOR Li ) is a (Pi , F)martingale.  Proposition 5.3. Assume that the PRP holds with respect to (P, F) with the spanning (P, F)-martingale M . Then, for every i = 1, . . . , n, the positive (Pi , F)-martingale κi has the following representation under Pi  i i κt = κ0 + κis− σsi · dΨi (M )s , (70) (0,t]

where σ is an R -valued, F-predictable process. Similarly, for every i = 1, . . . , n, the positive (Pi , F)-martingale Li admits the following representation under Pi  i i L t = L0 + Lis− ξsi · dΨi (M )s , (71) i

k

(0,t]

where ξ is an R -valued, F-predictable process. i

k

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The final step in the bottom-up approach is the derivation of the joint dynamics of processes κ1 , . . . , κn and L1 , . . . , Ln under P = Pn . Since the expression for these dynamics is exactly the same as in the top-down approach, we refer to Proposition 5.4 below. 5.2.2. Top-down approach. To construct the market model for forward LIBORs and one-period forward CDS spreads through the top-down approach, we make the following standing assumptions, which are motivated by the analysis of the bottomup approach from the preceding section. It is worth stressing that parts (ii), (iii) and (iv) are motivated by the results obtained within the bottom-up approach, and thus they should be seen as necessary conditions for the arbitrage-free property of the market model that we are going to construct using the top-down approach. Assumptions 5.4. We are given a filtered probability space (Ω, G, F, P) and we postulate that: (i) the initial values of processes κ1 , . . . , κn and L1 , . . . , Ln are given, (ii) for every i = 1, . . . , n, the following process are positive (P, F)-martingales n 

n 

(1 + ? aj κjt ),

j=i+1

(1 + aj Ljt ),

j=i+1

(iii) for every i = 1, . . . , n, the process Z κ,L,i, which is given by (by convention, Z κ,L,n = 1) n  Ztκ,L,i = ci (1 + ? aj κjt )(1 + aj Ljt ), j=i+1

is a positive (P, F)-martingale, where ci is a constant such that Z0κ,L,i = 1, (iv) for every i = 1, . . . , n, the processes κi and Li are (Pi , F)-martingales, where the Radon-Nikod´ ym density of Pi with respect to P equals Z κ,L,i, so that, in n particular, P = P, (v) for every i = 1, . . . , n, the process Li satisfies  i i ? i, Lt = L0 + Lis− ξ?si · dN (72) s (0,t]

? i is an Rli -valued (Pi , F)-martingale and ξ?i is an Rli -valued, where N ? i -integrable process, F-predictable, N (vi) for every i = 1, . . . , n, the process κi satisfies  i i + i, κt = κ0 + κis− σ +si · dN (73) s (0,t]

+ i is an Rki -valued (Pi , F)-martingale and σ where N +i is an Rki -valued, i + F-predictable, N -integrable process.

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The volatility processes ξ?i and σ +i should be considered as model inputs, so that they can be chosen arbitrarily. We only need to ensure that the stochastic integrals in (72) and (73) are well defined, so that the unique solutions to these stochastic differential equations are given by stochastic exponentials. The following definition is also clear, in view of part (iv) in Assumptions 5.4. Definition 5.2. The probability measure Pi is called the martingale measure for the forward LIBOR Li and the one-period forward CDS spread κi . Let k = k1 + · · ·+ kn + l1 + · · ·+ ln . Arguing as in the proof of Lemma 5.3, one can show that there exists an Rk -valued (P, F)-martingale M such that for every i = 1, . . . , n  i i Lt = L 0 + Lis− ξsi · dΨi (M )s (74) (0,t]

and

 κis− σsi · dΨi (M )s ,

κit = κi0 +

(75)

(0,t]

where ξ i = (ξ i,1 , . . . , ξ i,k ) and σ i = (σ i,1 , . . . , σi,k ) are Rk -valued, F-predictable processes, which extend ξ?i and σ +i , respectively. κ,L,n Since Z = 1 so that Ψn (M ) = M , we deduce from (74) and (75) that n under P = P the (Pn , F)-martingales Ln and κn satisfy  Lnt = Ln0 + Lns− ξsn · dMs (0,t]



and

κns− σsn · dMs .

κnt = κn0 + (0,t]

By combining Proposition 4.2 obtained for Setup (B) with formulae (74) and (75) and parts (i)–(iii) in Assumptions 5.4, we obtain the following result, which yields the joint dynamics of forward LIBORs and one-period forward CDS spreads under the martingale measure Pn for Ln and κn . Hence this result uniquely specifies the desired market model up to a choice of the driving (P, F)-martingale M and the volatility processes ξ 1 , . . . , ξ n , σ1 , . . . , σn . Proposition 5.4. Under Assumptions 5.4, the joint dynamics of forward LIBORs L1 , . . . , Ln and forward CDS spreads κ1 , . . . , κn under the martingale measure P = Pn are given by the following 2n-dimensional stochastic differential equation driven by the (P, F)-martingale M : for every i = 1, . . . , n and t ∈ [0, Ti−1 ] dLit =

k 

Lit− ξti,l dMtl −

l=1

−

n 

? aj Lit κjt

n k  aj Lit Ljt  j j=i+1 1 + aj Lt k 

1+? aj κjt l,m=1 j=i+1

ξti,l ξtj,m d[M l,c ,M m,c ]t

l,m=1

ξti,l σtj,m d[M l,c ,M m,c]t −

Lit−

∆Ztκ,L,i κ,L,i

Zt

k  l=1

ξti,l ∆Mtl

Market Models of Forward CDS Spreads

395

and for every i = 1,...,n and t ∈ [0,Ti−1 ] dκit =

k 

κit− σti,l dMtl −

l=1

−

n k  aj κit Ljt 

σti,l ξtj,m d[M l,c ,M m,c ]t j 1 + aj Lt l,m=1 j=i+1

n k  ? aj κit κjt 

1+? aj κjt j=i+1

σti,l σtj,m d[M l,c ,M m,c ]t −

l,m=1

κit−

∆Ztκ,L,i κ,L,i

Zt

k 

σti,l ∆Mtl .

l=1

The uniqueness of the solution to the system of SDEs derived in Proposition 5.4 follows from the classic result on SDEs with locally Lipschitz continuous coefficients. 5.3. Model of one- and two-period CDS spreads In this subsection, we examine the family consisting of all one- and two-period CDS spreads associated with a given tenor structure T = {T0 < T1 < · · · < Tn }. This particular choice of a family of forward CDS spreads originates from Brigo [9, 10], who examined this setup under the assumption that the market model is driven by a multi-dimensional Brownian motion. 5.3.1. Bottom-up approach. In this subsection, we find it convenient to adopt the notation in which the forward CDS spreads are indexed by their maturity date. Specifically, the one-period forward CDS spread is denoted as follows, for every i = 1, . . . , n, P+ i−1,i κit := κi−1,i = ti−1,i , ∀ t ∈ [0, Ti−1 ], t + A t whereas for the two-period CDS spread we write, for every i = 2, . . . , n, κ ?it := κi−2,i = t

P+ti−2,i , +i−2,i A t

∀ t ∈ [0, Ti−2 ].

Our aim is to derive the semimartingale decompositions for forward CDS spreads κ1 , . . . , κn and κ ?2 , . . . , κ ?n under a common probability measure. To achieve this, we will first derive expressions for the ratios of defaultable annuities (annuity deflated swap num´eraires) in terms of one- and two-period forward CDS spreads. Lemma 5.8. The following equalities hold, for every i = 2, . . . , n and every t ∈ [0, Ti−2 ], +i−2,i−1 A κ ?it − κit t = i−1 , i−1,i +t κt − κ ?it A +i−2,i A κi−1 − κit t = ti−1 . i−1,i + κt − κ ?it A t

(76) (77)

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Proof. It is easy to check that the following equalities are valid +i−2,i−1 +i−2,i A A t + 1 = ti−1,i , i−1,i + + A A t t i−2,i−1 + +i−2,i A A κi−1 t i−1,i + κi = κ ?i ti−1,i , + + A A t

t



and this in turn yields (76) and (77).

In view of formula (9) and the assumed positivity of the survival process G, ?n−1,n , which is defined by the formula, for t ∈ [0, Tn ], the process A ?n−1,n := Gt B −1 A +n−1,n = an EQ (B −1 1{τ >T } | Ft ), A t t t n Tn is a positive (Q, F)-martingale. Therefore, this process can be used to define the probability measure Pn on (Ω, FTn ). Specifically, the probability measure Pn equivalent to Q on (Ω, FTn ) is defined by postulating that the Radon-Nikod´ ym density is given by ?n−1,n dPn  A ?n−1,n , (78)  = tn−1,n : A t ? dQ Ft A 0

where, as before, the symbol  denotes the equality up to a normalizing constant. We now observe that formulae (76) and (77) furnish a family of positive processes, which can be used to define a family of equivalent probability measures ? i for i = 2, . . . , n such that κi is a (Pi , F)-martingale and Pi for i = 1, . . . , n and P i i ? κ ? is a (P , F)-martingale. The following commutative diagram summarizes the relationships between these probability measures dPn dQ

dPn−1 n

dPn−2 n−1

dP2 3

d? Pn−1

d? Pn−2

d? P2

dP1 2

dP Q −−−−→ Pn −−dP −−→ Pn−1 −− −−→ · · · −−dP −−→ P2 −−dP −−→ P1         d? Pn−1 d? P2 d? Pn F F F dPn F dP2 dPn−1 n ?3 d? Pn−1 ? n −−d−?P− ? n−1 −− ?2 P → P −−→ · · · −−d−P−→ P

where we set

dPn  ?n−1,n ,  : A t dQ Ft  +i−1,i dPi  A t : =  +i,i+1 dPi+1 Ft A t ? i  +i−2,i dP A t : =  +i−1,i dPi Ft A t

κ ?i+1 − κi+1 t t , κit − κ ?i+1 t κi−1 − κit t . κi−1 −κ ?it t

? i , respectively) is called the martinDefinition 5.3. The probability measure Pi (P gale measure for the one-period forward CDS spread κi (for the two-period forward CDS spread κ ?i , respectively).

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The ultimate goal in the bottom-up approach is the computation of the semimartingale decompositions of processes κ1 , . . . , κn and κ ?2 , . . . , ? κn under the martingale measure Pn . In fact, we are not restricted in the choice of the reference probability measure, so any martingale measure can be chosen to play this role. To find the desired decompositions, we observe that it is easy to compute the fol?i with respect lowing Radon-Nikod´ ym densities of martingale measures Pi and P n to the reference probability P n  +i−1,i dPi  A κ ?jt − κjt t  = ,  +n−1,n dPn Ft κj−1 − ? κjt A t j=i+1 t n ? i  +i−2,i +i−2,i A +i−1,i dP A A κi−1 − κit  κ ?jt − κjt t t t t  = = .  i−1 n−1,n i−1,i n−1,n n i +t +t +t dP Ft κt − κ ?t j=i+1 κj−1 −κ ?jt A A A t Explicit formulae for the joint dynamics under Pn of one- and two-period forward CDS spreads κ1 , . . . , κn and κ ?2 , . . . , κ ?n are now readily available. Since these dynamics are the same as in the top-down approach presented in the foregoing subsection, we refer to Proposition 5.5 below. 5.3.2. Top-down approach. In the top-down approach to the modeling of one- and two-period forward CDS spreads, we will work under the following set of standing assumptions. Note that parts (ii), (iii) and (iv) in Assumptions 5.5 should be seen as counterparts of the relationships between the martingale measures obtained in the previous subsection within the bottom-up approach. Assumptions 5.5. We are given a filtered probability space (Ω, G, F, P) and we postulate that: ?2 , . . . , κ ?n are given, (i) the initial values of processes κ1 , . . . , κn and κ κ,i (ii) for every i = 1, . . . , n, the process Z , which is given by the formula (by the usual convention, Z κ,n = 1) Ztκ,i = ci

n 

κ ?jt − κjt

j=i+1

κj−1 −κ ?jt t

,

(79)

is a positive (P, F)-martingale, where ci is a constant such that Z0κ,i = 1, (iii) for every i = 2, . . . , n, the process Z κ? ,i , which is given by the formula   κi−1 − κi κ,i ci Z κ,i + Z κ,i−1 = ? Z , Z κ? ,i = ? ci i−1 κ −? κi

(80)

is a positive (P, F)-martingale, where ? ci is a constant such that Z0κ? ,i = 1, (iv) for every i = 1, . . . , n, the process κi is a (Pi , F)-martingale, where the RadonNikod´ ym density of Pi with respect to P equals Z κ,i , (v) for every i = 1, . . . , n, the process κi satisfies  +si , κit = κi0 + κis− σ +si · dN (81) (0,t]

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L. Li and M. Rutkowski

+ i is an Rki -valued (Pi , F)-martingale and σ where N +i is an Rki -valued, i + F-predictable, N -integrable process, ? i , F)-martingale, where the Radon(vi) for every i = 2, . . . , n, the process κ ?i is a (P i ? with respect to P equals Z κ? ,i , Nikod´ ym density of P (vii) for every i = 2, . . . , n, the process ? κi satisfies  ? i, κit = κ ? ?i0 + κis− ζ?si · dN ? (82) s (0,t]

? i is an Rli -valued (P ? i , F)-martingale and ζ?i is an Rli -valued, where N i ? F-predictable, N -integrable process. ? i , respectively) is the martingale measure for The probability measure Pi (P the one-period forward CDS spread κi (for the two-period forward CDS spread κ ?i , n κ,n respectively). It is worth noting that the equality P = P holds, since Z = 1, so that the underlying probability measure P is the martingale measure for the process κn . Let k = k1 +· · ·+ kn + l2 +· · ·+ ln. As in preceding subsections, we argue that there exists an Rk -valued (P, F)-martingale M such that for every i = 1, . . . , n the process κi admits the following representation under Pi  κit = κi0 + κis− σsi · dΨi (M )s , (83) (0,t]

and for every i = 2, . . . , n the process κ ?i admits the following representation uni ? der P  i i ? i (M )s , κ ?t = κ ?0 + κ ?is− ζsi · dΨ (84) (0,t]

where σ = (σ , . . . , σ ) and ζ = (ζ i,1 , . . . , ζ i,k ) are Rk -valued, F-predictable processes, which extend σ +i and ζ?i , respectively. The (Pi , F)-martingale Ψi (M l ) is given by  1 ' ( Ψi (M l )t = Mtl − L(Z κ,i )c , M l,c t − ∆Zsκ,i ∆Msl , κ,i Z 0<s≤t s i

i,1

i,k

i

? i , F)-martingale Ψ ? i (M l ) equals whereas the (P  ' ( ? i (M l )t = M l − L(Z κ? ,i )c , M l,c − Ψ t t

1

κ ? ,i 0<s≤t Zs

∆Zsκ? ,i ∆Msl .

We note that since Z κ,n = 1, we have that Ψn (M ) = M and thus (83) yields  n n κt = κ0 + κns− σsn · dMs . (0,t]

Let us observe that the present situation corresponds to Setup (C) introduced in Subsection 4.3. The following result is a rather straightforward consequence of Proposition 4.3 and thus its proof is omitted.

Market Models of Forward CDS Spreads

399

Lemma 5.9. Under Assumptions 5.5, for every i = 1, . . . , n the semimartingale decomposition of the (Pi , F)-martingale Ψi (M ) under the martingale measure P = Pn is given by  j j   n  κ ?s ζs − κjs σsj · d[M c ]s Ψi (M )t = Mt − κjs − κjs ? j=i+1 (0,t]  j−1 j−1   n   1 κs σs − κ ?js ζsj · d[M c ]s + − ∆Zsκ,i ∆Ms , j−1 j κ,i κ − κ ? Z (0,t] s s s j=i+1 0<s≤t ?i , F)-martinand for every i = 2, . . . , n the semimartingale decomposition of the (P ? i (M ) under the martingale measure P = Pn is given by gale Ψ  i−1 i−1   i−1 i−1    κs σs − κis σsi · d[M c ]s κ s σs − κ ?is ζsi · d[M c ]s i ? Ψ (M )t = Mt − + (κi−1 − κis ) κi−1 −κ ?is s s (0,t] (0,t]  j j   n  κs ζs − κjs σsj · d[M c ]s ? − κ ?js − κjs j=i+1 (0,t]  j−1 j−1  n   1  κs σs − κ ?js ζsj · d[M c ]s + − ∆Zsκ? ,i ∆Ms . j−1 j κ ? ,i κ − κ ? Z (0,t] s s s j=i+1 0<s≤t To obtain an explicit expression for the joint dynamics of one-period spreads κ1 , . . . , κn and two-period spreads κ ?2 , . . . , κ ?n under the probability measure P = n P , it suffices to combine Lemma 5.9 with formulae (83) and (84). Note that the system of stochastic differential equations given in Proposition 5.5 is closed, in the sense that as soon as the driving (P, F)-martingale M and the volatility processes σ 1 , . . . , σn and ζ 2 , . . . , ζ n are chosen, the joint dynamics of the processes κ1 , . . . , κ n , ? κ2 , . . . , κ ?n under the martingale measure Pn are uniquely specified. After some standard computations, one can also deduce from this result the joint ?i . dynamics of these processes under any martingale measure Pi or P Proposition 5.5. Under Assumptions 5.5, the joint dynamics of one-period forward CDS spreads κ1 , . . . , κn and two-period forward CDS spreads ? κ2 , . . . , κ ?n under the n martingale measure P = P are given by the following (2n−1)-dimensional stochastic differential equation driven by the (P, F)-martingale M : for every i = 1, . . . , n and t ∈ [0, Ti−1 ] dκit =

k 

κit− σti,l dMtl −

l=1

+

+

n 

κit κ ?jt

k 

j=i+1

κ ?jt − κjt

l,m=1

n 

k κit κjt  i,l j,m σt σt κ ?j − κjt l,m=1 j=i+1 t n 

κit κj−1 t

j−1 j=i+1 κt

−

k 

κjt l,m=1 ?

σti,l ζtj,m d[M l,c , M m,c ]t

d[M l,c , M m,c ]t

σti,l σtj−1,m d[M l,c , M m,c ]t

400

L. Li and M. Rutkowski − −

n 

κj−1 j=i+1 t κit− Ztκ,i

k 

κit κ ?jt

∆Ztκ,i

−

κjt l,m=1 ?

k 

σti,l ζtj,m d[M l,c , M m,c ]t

σti,l ∆Mtl

l=1

and for every i = 2, . . . , n and t ∈ [0, Ti−2 ] d? κit =

k 

κ ?it− ζti,l dMtl −

l=1

k  κ ?it κi−1 t ζti,l σti−1,m d[M l,c , M m,c ]t i κi−1 − κ t t l,m=1

+

k  κ ?it κit ζti,l σti,m d[M l,c , M m,c ]t i κi−1 − κ t t l,m=1

+

k  κ ?it κi−1 t ζti,l σti−1,m d[M l,c , M m,c ]t i κi−1 − κ ? t t l,m=1

k  κ ?it κ ?it − i−1 ζti,l ζti,m d[M l,c , M m,c ]t κt − κ ?it l,m=1

−

+

+

−

n 

κ ?it κ ?jt

k 

j=i+1

κ ?jt − κjt

l,m=1

ζti,l ζtj,m d[M l,c , M m,c ]t

n 

k κ ?it κjt  i,l j,m ζt σt κ ?j − κjt l,m=1 j=i+1 t n 

κ ?it κj−1 t

k 

j=i+1

κj−1 −κ ?jt t

l,m=1

n 

κit κ ? ?jt κj−1 − j=i+1 t

k 

κ ?jt

l,m=1

d[M l,c , M m,c ]t

ζti,l σtj−1,m d[M l,c , M m,c ]t ζti,l ζtj,m d[M l,c , M m,c ]t −

κ ?it− Ztκ? ,i

∆Ztκ? ,i

k 

ζti,l ∆Mtl .

l=1

Unfortunately, we were not able to show the uniqueness of a solution to the system of SDEs derived in Proposition 5.5, since the coefficients are not locally Lipschitz continuous (a similar comment applies to the system of SDEs obtained in Proposition 5.6 below). 5.4. Model of one-period and co-terminal CDS spreads In this subsection, we will study the market model for forward CDS spreads associated with a family of one-period and co-terminal CDSs. Such a model can be seen as a counterpart of the model of co-terminal forward swap rates, which was proposed by Jamshidian [16] as an alternative for the model of forward LIBORs as a tool for valuation of exotic swap derivatives, such as Bermudan swaptions. In the context of CDS spreads, this model was suggested in the paper by Brigo [10]. In this setup,

Market Models of Forward CDS Spreads

401

the goal is to explicitly determine the dynamics of the family of one-period forward CDS spreads and the co-terminal forward CDS spreads κ0,n , κ1,n , . . . , κn−2,n under a common probability measure. As model inputs, we take the driving martingales and a collection of volatility processes for the considered family of forward CDS spreads. The importance of this particular family of forward CDS spreads becomes clear when dealing with the valuation of Bermudan CDS options. 5.4.1. Bottom-up approach. It should be emphasized that throughout this section all CDS spreads are indexed by the starting date, rather than the maturity date (unlike in Section 5.3 where a different convention was adopted). The one-period forward CDS spread is thus denoted as κit := κi,i+1 = t

P+ti,i+1 , +i,i+1 A t

∀ t ∈ [0, Ti ],

for every i = 0, . . . , n − 1. The co-terminal CDS spread with start date Ti and maturity Tn will be denoted by κ ?it := κi,n t =

P+ti,n , +i,n A

∀ t ∈ [0, Ti],

t

for i = 0, . . . , n − 1. It is thus clear that the equality κn−1 = κ ?n−1 holds. The goal of this notation is to simplify the presentation and emphasize some formal similarities between the present model and the model examined in the preceding subsection. We wish to derive the semimartingale decompositions for processes κ0 , . . . , n−2 κ and κ ?0 , . . . , κ ?n−1 under a single probability measure. As in preceding subsections, we need first to express the ratios of defaultable annuities (annuity deflated swap num´eraires) in terms of the underlying family of forward CDS spreads. Lemma 5.10. The following equalities hold, for every i = 0, . . . , n − 2 and every t ∈ [0, Ti ], +i+1,n A κ ?it − κit t = , +i,n κ ?i+1 − κit A t t +i,i+1 A −κ ?it κ ?i+1 t t = . +i,n κ ?i+1 − κit A t t

(85) (86)

Proof. It is easy to check that the following relationships hold true +i,i+1 +i+1,n A A + = 1, +i,n +i,n A A +i,i+1 +i+1,n A A κi +κ ?i+1 =κ ?i . +i,n +i,n A A By solving this system, we obtain (85) and (86).



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L. Li and M. Rutkowski

?0,n , which is given by the formula, It is easily seen from (9) that the process A for every t ∈ [0, T0 ], +0,n = ?0,n A := Gt Bt−1 A t t

n 

aj EQ (BT−1 1{τ >Tj } | Ft ), j

j=1

is a positive (Q, F)-martingale. Therefore, it can be used to define the probability ?0 on (Ω, FT ). Moreover, formulae (85) and (86) furnish a family of measure P 0 positive processes, which can be used to define a family of probability measures Pi ?i for i = 0, . . . , n − 1, such that κi is a (Pi , F)-martingale for i = 0, . . . , n − 2 and P i i ? and κ ? is a (P , F)-martingale for t ∈ [0, T0 ]. The relationships between these martingale measures are summarized in the following commutative diagram d? P0 dQ

d? P1

d? P2

d? Pn−2

? n−1 dP

dP1 0

dP2 1

dPn−2 n−3

dPn−1 n−2

0 1 d? Pn−3 d? Pn−2 ? 0 −−d−?P− ? 1 −−d−?P− ?n−2 −− ?n−1 = Pn−1 Q −−−−→ P → P → · · · −− −−→ P −−→ P         dP0 dP1 dPn−2 F F F F d? P0 d? P1 d? Pn−2 dP dP P0 −−dP −−→ P1 −−dP −−→ · · · −− −−→ Pn−2 −− −−→

Pn−1

where we set ? 0  dP ?0,n  : A t , dQ Ft ? i+1  +i+1,n dP A κ ?it − κit ,  : t i,n = i+1 ?i Ft +t κ ?t − κit dP A +i,i+1 dPi  A κ ?i+1 − κ ?it ,  : t i,n = ti+1 ? i Ft +t κ ?t − κit dP A where, as usual,  denotes equality up to a normalizing constant. Using these equalities, we also obtain the following family of the Radon-Nikod´ ym densities, for every i = 0, . . . , n − 1 and every t ∈ [0, T0 ], i−1  κ ? i  +i,n A ?jt − κjt dP t = ,  : 0,n ? 0 Ft + ?j+1 − κjt A dP t t j=0 κ

and for every i = 0, . . . , n − 2 and every t ∈ [0, T0 ], i−1 +i,n +i,i+1 +i,i+1 A dPi  A A κ ?i+1 − κ ?it  κ ?jt − κjt t = ti+1 .  : t 0,n = t i,n 0,n ? 0 Ft + + + κ ?t − κit j=0 κ ?j+1 − κjt A A A dP t t t t

? 0 of processes κ0 , . . . , It is now not difficult to derive the joint dynamics under P n−2 0 n−1 κ and κ ? , ..., κ ? . However, since they are the same as in the top-down approach presented in the next subsection, we refer the reader to Proposition 5.6 below.

Market Models of Forward CDS Spreads

403

5.4.2. Top-down approach. In this subsection, we present the top-down approach to the modeling of one-period and co-terminal forward CDS spreads. As in the preceding subsections, we start by stating a number of postulates underpinning the top-down approach. They are motivated by the bottom-down approach, meaning that they should be seen as necessary conditions for the consistency of the topdown model. The goal of the top-down approach is to show that Assumptions 5.6 are also sufficient for the unique specification of the joint dynamics for the family of forward CDS spreads considered in this subsection. Assumptions 5.6. We are given a filtered probability space (Ω, G, F, P) and we postulate that: (i) the initial values of processes κ0 , . . . , κn−2 and κ ?0 , . . . , κ ?n−1 are given, κ ? ,i (ii) for every i = 0, . . . , n − 1, the process Z , which is given by the formula (by the usual convention, Z κ? ,0 = 1) Ztκ? ,i = ? ci

i−1 

κ ?jt − κjt

j=0

κ ?j+1 − κjt t

,

∀ t ∈ [0, T0 ],

is a positive (P, F)-martingale and ? ci is a constant such that Z0κ? ,i = 1, (iii) for every i = 0, . . . , n − 2, the process Z κ,i , which is given by the formula Ztκ,i = ci

κ ?i+1 −κ ?it κ? ,i t Zt , κ ?i+1 − κit t

∀ t ∈ [0, T0 ],

is a positive (P, F)-martingale and ci is a constant such that Z0κ,i = 1, (iv) for every i = 0, . . . , n − 2, the process κi is a (Pi , F)-martingale, where the Radon-Nikod´ ym density of Pi with respect to P equals Z κ,i , (v) for every i = 0, . . . , n − 2, the process κi satisfies  + i, κit = κi0 + κis− σ +si · dN (87) s (0,t]

+ i is an Rki -valued (Pi , F)-martingale and σ where N +i is an Rki -valued, i + F-predictable, N -integrable process, ? i , F)-martingale, where the (vi) for every i = 0, . . . , n − 1, the process ? κi is a (P ?i with respect to P equals Z κ? ,i , Radon-Nikod´ ym density of P (vii) for every i = 0, . . . , n − 1, the process ? κi satisfies  ?si , κit = κ ? ?i0 + κis− ζ?si · dN ? (88) (0,t]

? i , F)-martingale and ζ?i is an Rli -valued, ? i is an Rli -valued (P where N i ? F-predictable, N -integrable process. As usual, we argue that the volatilities σ +i and ζ?i can be chosen arbitrarily, provided, of course, that the stochastic differential equations (87) and (88) have ? 0 = P holds, since Z κ? ,0 = 1. unique solutions. Note also that the equality P

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L. Li and M. Rutkowski

Let k = k0 + · · · + kn−2 + l0 + · · · + ln−1 . It is not difficult to show that there exists an Rk -valued (P, F)-martingale M such that for every i = 0, . . . , n − 2, the (Pi , F)-martingale κi admits the following representation under Pi  κit = κi0 + κis− σsi · dΨi (M )s , (89) (0,t]

where σ = (σ , . . . , σ ) is an R -valued, F-predictable process, which extends ? i , F)-martingale κ ?i , σ +i . Similarly, the (P ?i has the following representation under P for every i = 0, . . . , n − 1,  ? i (M )s , κ ?it = κ ?i0 + κ ?is− ζsi · dΨ (90) i

i,1

i,k

k

(0,t]

where ζ i = (ζ i,1 , . . . , ζ i,k ) is an Rk -valued, F-predictable process, which extends ζ?i . Furthermore, the Rk -valued (Pi , F)-martingale Ψi (M ) satisfies, for every l = 1, . . . , k,  1 ( ' Ψi (M l )t = Mtl − L(Z κ,i )c , M l,c t − ∆Zsκ,i ∆Msl , κ,i Z s 0<s≤t ? i , F)-martingale Ψ ? i (M ) satisfies, for every l = 1, . . . , k, whereas the Rk -valued (P  1 ' ( ? i (M l )t = M l − L(Z κ? ,i )c , M l,c − Ψ ∆Zsκ? ,i ∆Msl . t t κ ? ,i Z 0<s≤t s ? 0 and thus Ψ ? 0 (M ) = M . Consequently, the forward CDS In particular, P = P 0 spread ? κ satisfies  κ ?0t = κ ?00 + κ ?0s− ζs0 · dMs . (91) (0,t]

?0 is the martingale measure for ? κ0 . The equality (91) agrees with the fact that P It is not difficult to verify that the current assumptions place us within Setup (D) examined in Subsection 4.4. The following result is a rather straightforward consequence of Proposition 4.4, so that its proof is omitted. Lemma 5.11. Under Assumptions 5.6, for every i = 0, . . . , n−2 the semimartingale decomposition of the (Pi , F)-martingale Ψi (M ) under the martingale measure P = ? 0 is given by P  i+1 i+1   κ ?s ζs − κ ?is ζsi · d[M c ]s i Ψ (M )t = Mt − (? κi+1 −κ ?is ) s (0,t]  i+1 i+1   j j   i−1  κ ?s ζs − κis σsi · d[M c ]s  κs ζs − κjs σsj · d[M c ]s ? + − κ ?i+1 − κis κ ?js − κjs s (0,t] j=0 (0,t]  j+1 j+1  i−1    1 κ ?s σs − κjs σsj · d[M c ]s + − ∆Zsκ,i ∆Ms , j+1 j κ,i κ ? s − κs Z j=0 (0,t] 0<s≤t s

Market Models of Forward CDS Spreads

405

? i , F)and for every i = 0, . . . , n − 1 the semimartingale decomposition of the (P i 0 ? ? martingale Ψ (M ) under the martingale measure P = P is given by  j j  i−1   κ ?s ζs − κjs σsj · d[M c ]s i ? Ψ (M )t = Mt − κjs − κjs ? j=0 (0,t]  j+1 j+1  i−1   κ ?s ζs − κjs σsj · d[M c ]s + κj+1 ? − κjs s j=0 (0,t]  1 − ∆Zsκ? ,i ∆Ms . κ ? ,i 0<s≤t Zs By combining Lemma 5.11 with formulae (89) and (90), we obtain closed-form expressions for semimartingale decompositions of the family of one-period forward CDS spreads κ0 , . . . , κn−2 and the co-terminal forward CDS spreads κ ?0 , . . . , κ ?n−1 0 ? under the probability measure P = P , that is, under the martingale measure associated with the n-period forward CDS spread κ ?0 . We now assume that the driving (P, F)-martingale M and the volatility processes σ 0 , . . . , σn−2 ζ 0 , . . . , ζ n−1 are given. It is also worth recalling that κn−1 = κ ?n−1 . Proposition 5.6. Under Assumptions 5.6, the joint dynamics of one-period forward CDS spreads κ0 , . . . , κn−2 and co-terminal forward CDS spreads κ ?0 , . . . , ? κn−1 un0 ? der the martingale measure P = P are given by the following (2n − 1)-dimensional stochastic differential equation driven by the (P, F)-martingale M : for every i = 0, . . . , n − 2 and t ∈ [0, T0 ] dκit =

k 

κit− σti,l dMtl −

l=1

k  κit κ ?i+1 t σti,l ζti+1,m d[M l,c , M m,c ]t i κ ?i+1 − κ ? t t l,m=1

+

k  κit κ ?it σti,l ζti,m d[M l,c , M m,c ]t i κ ?i+1 − κ ? t t l,m=1

+

k  κit κ ?i+1 t σti,l ζti+1,m d[M l,c , M m,c ]t i κ ?i+1 − κ t t l,m=1

−

k  κit κit σti,l σti,m d[M l,c , M m,c ]t i κ ?i+1 − κ t t l,m=1

−

+

i−1  κit ? κjt

κ ?j j=0 t

−

σti,l ζtj,m j κt l,m=1

i−1  κit κjt j=0

k 

κ ?jt − κjt

k  l,m=1

d[M l,c , M m,c ]t

σti,l σtj,m d[M l,c , M m,c ]t

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+

−

i−1  κit ? κj+1 t

κ ?j+1 j=0 t i−1 

−

k 

κjt l,m=1 k 

κit κjt

κ ?j+1 j=0 t

−

κjt l,m=1

σti,l ζtj+1,m d[M l,c , M m,c ]t σti,l σtj,m d[M l,c , M m,c ]t −

κit− Ztκ? ,i

∆Ztκ? ,i

k 

σti,l ∆Mtl

l=1

and for every i = 0, . . . , n − 1 and t ∈ [0, T0 ] d? κit =

k 

κ ?it− ζti,l dMtl −

j=0

l=1

+

+

− −

i−1  κ ?it κ ?jt

i−1 

−

ζti,l σtj,m j κt l,m=1

i−1  κ ?it ? κj+1 t

κ ?j+1 j=0 t

−

κ ?it κjt

j=0

κ ?j+1 −κ ?jt t ∆Ztκ,i

k 

k 

ζti,l ζtj,m d[M l,c , M m,c ]t

l,m=1

d[M l,c , M m,c ]t

ζti,l ζtj+1,m j κ ?t l,m=1

i−1 

κ ?it− Ztκ,i

κ ?jt − κjt

k 

κ ?it κjt

?jt j=0 κ

k 

d[M l,c , M m,c ]t

ζti,l σtj,m d[M l,c , M m,c ]t

l,m=1

k 

ζti,l ∆Mtl .

l=1

Alternatively, one can derive the joint dynamics of forward CDS spreads un? n (see Rutkowski [27]) or, whenever this is der the martingale measure Pn = P ?i from the family introdesirable, under any other martingale measure Pi or P duced in this subsection. The most convenient choice of the underlying probability measure depends on a particular problem at hand. Our choice of the reference ?0 was motivated by desire to emphasize the symmetrical probability measure P = P features of setups considered in Subsections 5.3 and 5.4, rather than by practical considerations. 5.5. Generic top-down approach We will now make an attempt to describe the main steps in the top-down approach to a generic family of forward spreads. We now consider a generic family of forward swap rates and/or forward CDS spreads, denoted as κ1 , . . . , κn , and we are interested in the top-down approach to the derivation of the joint dynamics of these processes. The generic top-down method hinges on the following set of postulates. Assumptions 5.7. We are given a filtered probability space (Ω, G, F, P). We postulate that the F-adapted processes κ1 , . . . , κn and Z 1 , . . . , Z n satisfy the following conditions, for every i = 1, . . . , n, (i) the process Z i is a positive (P, F)-martingale with EP (Z0i ) = 1, (ii) the process κi Z i is a (P, F)-martingale,

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(iii) the process Z i is given as a function of some subset of the family κ1 , . . . , κn ; specifically, there exists a subset {κn1 , . . . , κnli } of {κ1 , . . . , κn } and a function fi : Rli → R of class C2 such that Z i = fi (κn1 , . . . , κnli ). Parts (i) and (ii) in Assumptions 5.7, combined with Lemma 4.1(i), yield the existence of a family of probability measures P1 , . . . , Pn , equivalent to P on (Ω, FT ) for some fixed T > 0, such that the process κi is a (Pi , F)-martingale for every i = 1, . . . , n. This in turn implies that κi is a (P, F)-semimartingale. We deduce from part (iii) that the continuous martingale part of Z i has the following integral representation Zti,c = Z0i +

li   j=1

t

0

∂fi n1 nl (κ , . . . , κs i ) dκns j ,c , ∂xj s

(92)

where κj,c stands for the continuous martingale part of κj . To establish equality (92), it suffices to apply the Itˆ o formula and use the properties of the stochastic integral. We will argue that, under Assumptions 5.7, the semimartingale decomposition of κj can be uniquely specified under P by the choice of the initial values, the volatility processes and the driving martingale, which is, as usual, denoted by M . For the purpose of an explicit construction of the model for processes κ1 , . . . , κn , we thus select an Rk -valued (P, F)-martingale M = (M 1 , . . . , M k ) and we define the process κi under Pi as follows, for every i = 1, . . . , n,  i i κt = κ0 + κis− σsi · dΨi (M )s , (93) (0,t]

where σ i is the Rk -valued volatility process and the (Pi , F)-martingale Ψi (M ) equals (cf. (24))   1 1 Ψi (M )t = Mt − d[Z i,c , M c ]s − ∆Zsi ∆Ms . (94) i Zsi (0,t] Zs 0<s≤t

Proposition 5.7. Under Assumptions 5.7, if the processes κ1 , . . . , κn satisfy (93)– (94) then for every i = 1, . . . , n the dynamics of κi are dκit =

k 

κit− σti,l dMtl −

l=1

1 nl fi (κnt 1 ,...,κt i )

×

li  ∂fi j=1

−

κit− Zti

∂xj

∆Zti

n

(κnt 1 ,...,κt li )

k  l=1

k 

n

n ,m

κit κt j σti,l σt j

d[M l,c ,M m,c ]t

l,m=1

σti,l ∆Mtl .

(95)

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Proof. In view of (93) and (94), it is clear that the continuous martingale part of κi can be expressed as follows  i κi,c = κ + κis σsi · dMsc , t 0 (0,t] i,c

and thus we infer from (92) that Z admits the following representation li   ∂fi n1 n Zti,c = Z0i + (κs , . . . , κs li )κns j σsnj · dMsc . ∂x j (0,t] j=1

(96)

This allows us to express the process Ψi (M ) in terms of M , the processes κ1 , . . . , κn , their volatilities and the matrix [M c ]. To be more specific, we deduce from (94) and (96) that (note that [M m,c , M c ] is an Rk -valued process)  1 i Ψ (M )t = Mt − n li n1 (0,t] fi (κs , . . . , κs ) × −



li k   ∂fi n1 nl (κs , . . . , κs i ) κns j σsnj ,m d[M m,c , M c ]s ∂x j m=1 j=1

0<s≤t

1 ∆Zsi ∆Ms , Zsi

(97)

n

where Zsi = fi (κns 1 , . . . , κs li ) and nl

nl

1 ∆Zsi = fi (κns 1 , . . . , κs i ) − fi (κns− , . . . , κs−i ).

By combining (93) with (97), we arrive at formula (95). We thus conclude that, under Assumptions 5.7, the choice of the driving (P, F)-martingale M and the volatility processes σ 1 , . . . , σ n in formula (93) completely specify the joint dynamics of κ1 , . . . , κn under P.  At the first glance, it is tempting to argue that the systems of SDEs derived in Propositions 5.1, 5.4, 5.5 and 5.6 can be seen as special cases of the system (95). Although formula (95) is fairly general, it should be stressed that this formula alone cannot be seen as a solution to the problem of modeling of forward swap rates and/or forward CDS spreads. Indeed, the crucial step in the top-down modeling approach is a correct specification of the functions fi such that Z i = fi (κn1 , . . . , κnli ). Obviously, this important step cannot be performed without a thorough analysis of a particular setup. Most likely, this will be done using the bottom-up approach in order to justify explicit formulae for the family of functions fi . In essence, the judicious choice of a considered family of forward swap rates and forward CDS spreads should ensure that the Radon-Nikod´ ym densities of the corresponding martingale measures (that is, the processes Z i appearing in Assumptions 5.7), which are defined as ratios of annuity deflated swap num´eraires, should also admit unique representations in terms of the considered family of

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forward swap rates and/or forward CDS spreads, so that part (iii) in Assumptions 5.7 holds. It is worth to stress that the corresponding functions fi , as derived in Subsections 5.3.1 and 5.4.1 using the bottom-up approach, are not of class C2 and thus the corresponding models require additional assumptions. We have circumvented this additional difficulty by working in Subsections 5.3.2 and 5.4.2 under an a priori assumption that all relevant processes belong to the class S1 . Of course, this assumption should be validated a posteriori by showing that the systems of SDEs derived in these subsections have unique solutions such that the processes of interest are indeed in S1 . These salient properties underpin in fact all particular cases of market models considered in this work. The corresponding general algebraic problem associated with the existence and uniqueness of functions fi will be examined elsewhere. Let us finally observe that, without loss of generality, we may choose one of the processes Z i to be equal identically to 1, so that for the corresponding martingale measure Pi for the process κi we have that Pi = P.

6. Concluding remarks We conclude this work by noting that the derivation of the joint dynamics of CDS spreads does not justify by itself that the model is suitable for the arbitrage pricing of credit derivatives. To show the viability of the model, it would be sufficient to demonstrate that the model of CDS spreads is arbitrage-free in some sense, for instance, that it can be supported by an associated arbitrage-free model for default-free and defaultable zero-coupon bonds. Let us list some further issues that should be addressed in the context of top-down models: (i) the existence and properties of a default time consistent with the dynamics of a given family of forward CDS spreads, (ii) the positivity of processes modeling forward CDS spreads, (iii) the positivity of prices of zero-coupon defaultable bonds implicitly defined by forward CDS spreads, (iv) the monotonicity of prices of zero-coupon defaultable bonds with respect to maturity date, (v) the positivity of other forward spreads that can be computed within a given model. Last, but not least, one has also to analyze the calibration of models to market data and their applications to valuation and hedging of exotic credit derivatives, such as: constant maturity credit default swaps, Bermudan CDS options, etc. All these important issues are beyond the scope of this work and they are currently under study (for some preliminary results, we refer to [21]).

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Acknowledgement We are very grateful to anonymous referees for their insightful reports, which were very helpful in the preparation of the final version of this work. The research of M. Rutkowski was supported under Australian Research Council’s Discovery Projects funding scheme (DP0881460).

References [1] T.R. Bielecki and M. Rutkowski: Credit Risk: Modeling, Valuation and Hedging. Springer-Verlag, Berlin Heidelberg New York, 2002. [2] T.R. Bielecki, M. Jeanblanc and M. Rutkowski: Pricing and trading credit default swaps in a hazard process model. Annals of Applied Probability 18 (2008), 2495–2529. [3] T.R. Bielecki, M. Jeanblanc and M. Rutkowski: Hedging of a credit default swaption in the CIR default intensity model. Forthcoming in Finance and Stochastics. [4] T.R. Bielecki, M. Jeanblanc and M. Rutkowski: Credit Risk Modeling. Osaka University, CSFI Lecture Notes Series 02, Osaka University Press, 2009. [5] A. Brace: Engineering BGM. Chapman and Hall/CRC, Financial Mathematics Series, 2008. [6] A. Brace, D. G¸atarek, and M. Musiela: The market model of interest rate dynamics. Mathematical Finance 7 (1997), 127–154. [7] P. Br´emaud and M. Yor: Changes of filtrations and of probability measures. Z. f¨ ur Wahrscheinlichkeitstheorie verw. Gebiete 45 (1978), 269–295. [8] D. Brigo: Market models for CDS options and callable floaters. Risk Magazine January (2005) (reprinted in: Derivatives Trading and Option Pricing, N. Dunbar, ed., Risk Books, 2005). [9] D. Brigo: Constant maturity CDS valuation with market models. Risk Magazine June (2006). [10] D. Brigo: CDS options through candidate market models and the CDS-calibrated CIR++ stochastic intensity model. In: Credit Risk: Models, Derivatives and Management, N. Wagner, ed., Chapman & Hall/CRC Financial Mathematics Series, 2008, pp. 393–426. [11] D. Brigo and M. Morini: CDS market formulas and models. Working paper, Banca IMI, 2005. [12] D. Brigo and F. Mercurio: Interest Rate Models. Theory and Practice – with Smile, Inflation and Credit. Second Edition. Springer-Verlag, Berlin Heidelberg New York, 2006. [13] T. Choulli, L. Krawczyk, and C. Stricker: E -martingales and their applications in mathematical finance. Annals of Probability 26 (1998), 853–876. [14] S. Galluccio, J.-M. Ly, Z. Huang, and O. Scaillet: Theory and calibration of swap market models. Mathematical Finance 17 (2007), 111–141. [15] J. Jacod and M. Yor: Etude des solutions extr´emales et repr´esentation int´egrale des solutions pour certains probl`emes de martingales. Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und vervandte Gebiete 38 (1977), 83–125.

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[16] F. Jamshidian: LIBOR and swap market models and measures. Finance and Stochastics 1 (1997), 293–330. [17] F. Jamshidian: LIBOR market model with semimartingales. Working paper, NetAnalytic Limited, 1999. [18] F. Jamshidian: Valuation of credit default swaps and swaptions. Finance and Stochastics 8 (2004), 343–371. [19] M. Jeanblanc, M. Yor, and M. Chesney: Mathematical Methods for Financial Markets. Springer-Verlag, Berlin Heidelberg New York, 2009. [20] C. Lotz and L. Schl¨ ogl: Default risk in a market model. Journal of Banking and Finance 24 (2000), 301–327. [21] L. Li and M. Rutkowski: Admissibility of generic market models of forward swap rates. Working paper, University of Sydney, 2009. [22] M. Morini and D. Brigo: No-armageddon arbitrage-free equivalent measure for index options in a credit crisis. Forthcoming in Mathematical Finance. [23] M. Musiela and M. Rutkowski: Continuous-time term structure models: forward measure approach. Finance and Stochastics 1 (1997), 261–291. [24] M. Musiela and M. Rutkowski: Martingale Methods in Financial Modelling. Second Edition. Corrected 2nd printing. Springer-Verlag, Berlin Heidelberg New York, 2007. [25] R. Pietersz and M. van Regenmortel: Generic market models. Finance and Stochastics 10 (2006), 507–528. [26] M. Rutkowski: Models of forward Libor and swap rates. Applied Mathematical Finance 6 (1999), 1–32. [27] Options on credit default swaps and credit default indexes. In: Credit Risk Frontiers: Subprime Crisis, Pricing and Hedging, CVA, MBS, Ratings and Liquidity, T.R. Bielecki, D. Brigo and F. Patras, eds., J. Wiley, 2010, 219–282. [28] M. Rutkowski and A. Armstrong: Valuation of credit default swaptions and credit default index swaptions. International Journal of Theoretical and Applied Finance 12 (2009), 1027–1053. [29] L. Schl¨ ogl: Note on CDS market models. Working paper, 2007. [30] P.J. Sch¨ onbucher: A Libor market model with default risk. Working paper, University of Bonn, 2000. [31] A.N. Shiryaev and A.S. Cherny: Vector stochastic integrals and the fundamental theorems of asset pricing. Proceedings of the Steklov Institute of Mathematics 237 (2002), 6–49. Libo Li and Marek Rutkowski School of Mathematics and Statistics University of Sydney NSW 2006, Australia e-mail: [email protected] [email protected]

Progress in Probability, Vol. 65, 413–429 c 2011 Springer Basel AG


Optimal Threshold Dividend Strategies under the Compound Poisson Model with Regime Switching Jiaqin Wei, Hailiang Yang and Rongming Wang Abstract. In this paper, we consider the optimal dividend strategy for an insurer whose surplus process is modeled by the classical compound Poisson risk model modulated by an observable continuous-time Markov chain. The object of the insurer is to select the dividend strategy that maximizes the expected total discounted dividend payments until ruin. We assume that the company only allows to pay dividend at a small rate. Given some conditions, similar to the results of Sotomayor and Cadenillas (2008) and Jiang and Pistorius (2008), the optimal strategy of our model is also a modulated threshold strategy which depends on the environment state. For the case of two regimes and exponential claim sizes, we obtain an analytical solution. Mathematics Subject Classification (2000). Primary: 93E20; Secondary: 91B70, 60H30. Keywords. Regime switching; dividend strategy; compound Poisson model; stochastic control; HJB equation.

1. Introduction The optimal dividend problem is a classical problem of actuarial mathematics and it has been investigated by many authors in the literature. For earlier work on this problem, see De Finetti (1957), Borch (1967, 1969), B¨ uhlmann, Gerber (1972); Gerber and Buzzi (1974). Recently, with the application of control theory, this problem has been discussed under the diffusion model (see Asmussen and Taksar (1997), Høgaard and Taksar (1999) and Gerber and Shiu (2004), among others.) and the classical compound Poisson model (see Azcue and Muler (2005), Gerber and Shiu (2006), Schmidli (2008) and Kulenko and Schmidli (2008) among others.) Survey articles include Taksar (2000) and Avanzi (2009) among others. Recently, the Markov-modulated insurance risk model becomes popular. This model is proposed by Asmussen (1989) in which the claim inter-arrivals and claim

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sizes are influenced by an external environment process {J(t)}t≥0 . This model can capture the feature that insurance policies may need to change if economical or political environment changes. Under the regime-switching compound Poisson model, Li and Lu (2006) considered the constant barrier strategy, but the paper did not address the optimal dividend strategy. In fact, the optimal strategy is not necessarily a barrier strategy in the regime switching model. Sotomayor and Cadenillas (2008) and Jiang and Pistorius (2008) discussed the optimal dividend strategy under the regime-switching diffusion model. They show that the optimal strategy is not a barrier strategy. We may guess that similar results will hold for the classical insurance risk model with regime switching. This paper is trying to investigate this problem. As we expected, the optimal strategy is dependent on the environment process. In this paper, we assume that the dividend rate is small in our model. This will simplify some mathematics and it makes sense in practice, since the insurance companies seldom pay such high dividends as that given in the barrier strategy. Besides the classical control problem, Sotomayor and Cadenillas (2008) discussed the singular control problem for the case with unbounded dividend rate, and impulse control problem for the case in which there are fixed costs and taxes associated with each dividend payment. Suppose that {J(t)}t≥0 is a homogenous continuous-time Markov chain taking values in a finite set J = {1, 2, . . . , N } with generator Q = (qij )N ×N where −qii = qi , i = 1, 2, . . . , N . We further assume that {J(t)}≥0 is irreducible with stationary distribution π = (π1 , π2 , . . . , πN ). At time t, given J(t) = i, the premium rate is ci , claims arrive according to Poisson process with rate λi , and the size of the claim which arrives at time t follows the distribution Fi with density fi and mean µi . We denote by Xn and Sn , respectively, the size and the arrival time of the nth claim. Given {J(Sn )}n∈N , the claim sizes {X1 , X2 , . . . } are assumed to be mutually independent. Furthermore {X1 , X2 , . . . } are independent of {Sn }n∈N . Define N (t) = max{n ∈ N : Sn ≤ t} as the number of claims up to time t. The counting process {N (t)}t≥0 is called a Markov-modulated Poisson process, which is a special case of the Cox process. Let {R1 (t)}t≥0 , {R2 (t)}t≥0 , . . . , {RN (t)}t≥0 be N independent classical compound Poisson risk processes with premium rate ci , claim arrival rate λi , claim size distribution Fi and zero initial surplus. Suppose the initial surplus is u ≥ 0, the corresponding surplus process {R(t)}t≥0 is given by  t R(t) = u + χ(J(s) = i)dRi (s), t ≥ 0, i∈J

0

where χ(·) is the indicator function. Suppose that the aforementioned random variables and stochastic processes are defined on the filtered probability space (Ω, F, F, P), where F = {Ft , t ≥ 0} is the filtration generated by {R(t)}t≥0 and {J(t)}t≥0 . We also assume that F satisfies the usual conditions. Denote by Pu,i the measure P conditioned on {R(0) = u, J(0) = i}, and write Eu,i for the corresponding expectations.

Optimal Threshold Dividend Strategies

415

A dividend strategy D is described as an F-adapted process {D(t)}t≥0 corresponding to the cumulate amount of dividends paid between time 0 and t. We assume that {D(t)}t≥0 is a non-negative and nondecreasing stochastic process with RCLL paths and D(0−) = 0. Similar to Gerber and Shiu (2006), we study this optimization problem under the constraint that only dividend strategies with dividend rate bounded by a ceiling are admissible and denote by D the set of all admissible strategies. Furthermore, we assume that the ceiling is less than the premium rates. This will lead to the threshold policy provided some conditions. For discussions on threshold dividend strategies, see Lin and Pavlova (2006) which studies the Gerber-Shiu function under a classical insurance risk model with a threshold dividend strategy, and Zhu and Yang (2006) which studies the ruin probability and the deficit at ruin in a Markovian regime switching model under a threshold dividend strategy. In this case, D(t) can be represented as dD(t) = a(t)dt,

0 ≤ a(t) ≤ K,

where a(t) is a F-adapted process that represents the rate at which the dividends are distributed and K < mini∈J {ci }. Schmidli (2008) and Sotomayor and Cadenillas (2008) 1 also discusses the case that K is unbounded. We assume that K is bounded for the reason that it avoids the bankruptcy caused by the payment of dividend, which is not realistic. Furthermore, if K is not bounded, under the compound poisson risk model, an insurer will get into bankruptcy for sure (see Schmidli (2008)). The surplus process in the presence of dividend payments is given by dRD (t) = dR(t) − a(t)dt for all t < τD , where τD = inf{t ≥ 0 : RD (t) < 0} is the time of ruin. Let # τD $ −δt VD (u, i) = Eu,i e a(t)dt

(1.1)

(1.2)

0

be the expected present value of all dividends until ruin, where δ is the force of interest. The objective-function of the insurer is V (u, i) = sup VD (u, i).

(1.3)

D∈D

The problem for the insurer is to identify a dividend strategy D∗ ∈ D that satisfies V (u, i) = VD∗ (u, i) for all u ≥ 0 and i ∈ J. The rest of the paper is organized as follows: In Section 2, we show the Hamilton-Jacobi-Bellman equation and verification theorem. Under some conditions, similar to that in Sotomayor and Cadenillas (2008) and Jiang and Pistorius 1 In

fact, a bounded condition for K is also need in the proof of Lemma 3.1 in Sotomayor and Cadenillas (2008). If K > µi (µi is the same as ci in this paper), then µ ˜i = µi − K < 0 which will lead to φi (+∞) = −∞ + ∞. So, without the bounded condition for K, the systems of equations (3.9), (3.14) and the other systems include the term µi − K can not be solved by Lemma 3.1 in Sotomayor and Cadenillas (2008).

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(2008), we prove that the optimal strategy is also a modulated threshold strategy which depends on the environment state. In Section 3, we construct an analytical solution for a special case in which there are two regimes and the claim sizes are exponentially distributed.

2. Main results 2.1. HJB equation and verification theorem First, we note that the value function is bounded. In fact, for i ∈ J,  ∞ K V (u, i) ≤ e−δt Kdt = . δ 0 Using the same strategy for two different initial surpluses and the same initial state, we know that the value function V (·, i) is nondecreasing. As we all know, HJB equations for regime switching models are extension of one regime HJB equations that presents one extra term in which the generator of the Markov chain is included. This particularity has already been shown for diffusion processes. It is also holds for compound Poisson processes. We omit the proof of the following theorem, since it is standard (see, for example, Schmidli (2008)). Theorem 2.1. For all i ∈ J, if the value function V (u, i) defined by equation (1.3) is continuously differentiable with respect to u ∈ (0, ∞), then it satisfies the Hamilton-Jacobi-Bellman equation %  u max (ci − a)V  (u, i) + λi V (u − y, i)fi (y)dy − (λi + δ)V (u, i) a∈[0,K] 0 &  + qik V (u, k) + a = 0. (2.1) k∈J

Remark 2.1. In fact, using the similar techniques as in Schmidli (2008), we can show that the value function is sufficiently smooth. Here, for simplicity, we assume it holds in the Theorem 2.1. To prove the verification theorem, we give a lemma which severs as an Itˆ o formula with regime switching. We rewrite (1.1) as  ' ( dRD (t) = cJ(t) − a(t) dt − xNJ(t) (dt × dx) (2.2) (0,∞)

where Ni (dt×dx) is a Poisson random measure with intensity measure λi fi (x)dtdx. By a method similar to Bj¨ ork (1980), we have the following lemma. Lemma 2.1. Let f (t, x, i) be a C 1,1 real function, i.e., f is continuously differentiable with respect to the first and the second variable. Then f (t, RD (t), J(t)) is a

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417

finite variation process and the following formula holds: f (t, RD (t), J(t)) − f (0, RD (0), J(0)) (2.3)  t  t = qJ(s)i f (s, RD (s), i)ds + f (s, RD (s−), i)dm(s, i) i∈J i∈J 0 0  t '   ( + ft (s, RD (s), J(s)) + cJ(s) − a(s) fx (s, RD (s), J(s)) ds 0  t + [f (s, RD (s−) − x, J(s−)) − f (s, RD (s−), J(s−))] NJ(s) (ds × dx), 0

(0,∞)

where m(·, i) is a martingale. Proof. Obviously, f (t, RD (t), J(t)) =

 i∈J

f (t, RD (t), i)ξ(t, i)

(2.4)

where ξ(t, i) = χ(J(t) = i). It is shown in Liptser and Shiryaev (1977) that  t ξ(t, i) = ξ(0, i) + qji ξ(s, j)ds + m(t, i), i ∈ J, (2.5) 0

j∈J

where m(t, i) is a square integrable martingale with right-continuous paths. From Itˆ o’s formula and equation (2.2), we know f (t, RD (t), i) is a finite variation process and ' ( df (t, RD (t), i) = ft (t, RD (t), i)dt + cJ(t) − a(t) fx (t, RD (t), i)dt (2.6)  + [f (t, RD (t−) − x, i) − f (t, RD (t−), i)] NJ(t) (dt × dx). (0,∞)

Since ξ(t, i) and f (t, RD (t), i) are finite variation processes, from integration by parts, we obtain  t f (t, RD (t), i)ξ(t, i) = f (0, RD (0), i)ξ(0, i) + f (s, RD (s), i)dξ(s, i) 0  t + ξ(s−, i)df (s, RD (s), i). (2.7) 0

The following equation is derived from equations (2.5), (2.6) and (2.7). f (t, RD (t), i)ξ(t, i) − f (0, RD (0), i)ξ(0, i) (2.8)  t  t = qji ξ(s, j)f (s, RD (s), i)ds + f (s, RD (s), i)dm(s, i) j∈J 0 0  t '   ( + ξ(s, i) ft (s, RD (s), i) + cJ(s) − a(s) fx (s, RD (s), i) ds 0   t + ξ(s−, i) [f (s, RD (s−) − x, i) − f (s, RD (s−), i)] NJ(s) (ds × dx). 0

(0,∞)

Then equation (2.3) follows from equations (2.4) and (2.8).



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Theorem 2.2. Let v(·, i), i ∈ J be a bounded nonnegative function, and let v(·, i) ∈ C 1 ((0, ∞)) satisfy Hamilton-Jacobi-Bellman equation (2.1). (i) Then it holds that v(u, i) ≥ V (u, i) for all u ≥ 0 and i ∈ J; (ii) If, in addition, v = VD for some D ∈ D, then D is the optimal strategy and V ≡ VD . Proof. (i) From equation (2.3), for any D ∈ D, we have e−δ(t∧τD ) v(RD (t ∧ τD ), J(t ∧ τD )) − v(RD (0), J(0))  t∧τD  t∧τD   −δs e qJ(s)i v(RD (s), i)ds + e−δs v(RD (s), i)dm(s, i) = 0



0

i∈J

i∈J

'   ( + e−δs −δv(RD (s), J(s)) + cJ(s) − a(s) v (RD (s), J(s)) ds 0  t∧τD ∞ + e−δs [v(RD (s−) − x, J(s−)) − v(RD (s−), J(s−))] NJ(s) (ds × dx). t∧τD

0

0

Applying conditional expectation to the above equation yields # $ Eu,i e−δ(t∧τD ) v(RD (t ∧ τD ), J (t ∧ τD )) − v(u, i) # t∧τD $  −δs = Eu,i e qJ(s)i v(RD (s), i)ds 0

i∈J

#

t∧τD

+ Eu,i 0

#

0

#

t∧τD

+ Eu,i

$ v(RD (s), i)dm(s, i)

i∈J

t∧τD

+ Eu,i



e−δs

0

3 4 $   e−δs −δv(RD (s), J(s)) + cJ(s) − a(s) v (RD (s), J(s)) ds 

∞

' e−δs v(RD (s−) − x, J(s))

0

$ ( − v(RD (s−), J (s)) λJ(s) fJ(s) (x)dxds $ # t∧τD  −δs = Eu,i e v(RD (s), i)dm(s, i) 0

i∈J

%

t∧τD

+ Eu,i

e−δs

0



#

  cJ(s) − a(s) v  (RD (s), J (s))

RD (s)

+ λJ(s)

v(RD (s) − x, J(s))fJ(s) (x)dx

0

− (λJ(s) + δ)v(RD (s), J(s)) +

 i∈J

$ & qJ(s)i v(RD (s), i) ds .

(2.9)

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Since v(·, i) is bounded and m(t, i) is a martingale, it is sufficient to show  Eu,i 



t∧τD

e−δs

0



 v(RD (s), i)dm(s, i) = 0.

i∈J

Then from the Hamilton-Jacobi-Bellman equation (2.1) and equation (2.9), we obtain 3 4 v(u, i) ≥ Eu,i e−δ(t∧τD ) v(RD (t ∧ τD ), J(t ∧ τD )) # t∧τD $ + Eu,i e−δs a(s)ds . 0

Taking t → ∞ yields v(u, i) ≥ VD (u, i). Then (i) follows from the arbitrariness of D. (ii) The equality follows from the definition of V and (i).



2.2. Modulated threshold strategy In the setting of regime-switching, Jiang and Pistorius (2008) considers an unbounded dividend strategy and defines the modulated barrier strategy. Since the bounded dividend strategy will lead to the threshold policy (see Gerber and Shiu (2006)), we will generalize the threshold strategy by allowing the threshold to depend on the current regime. Definition 2.1. A modulated threshold strategy at level b = (b1 , b2 , . . . , bN ), where bi ≥ 0, i ∈ J, is a dividend strategy D ∈ D satisfying   a(t) = χ RD (t) ≥ bJ(t) K. The corresponding expected value of all dividends until ruin is denoted by Vb (u, i). Under the modulated threshold strategy at b, by the standard method (see Dickson and Waters (2004), Gerber and Shiu (2006)), we can show that Vb (u, i) satisfies following system of equations:
u  0 = ci Vb (u, i) + λi 0 Vb (u − y, i)fi (y)dy    N    − (λi + δ)Vb (u, i) + k=1 qik Vb (u, k), 0 ≤ u < bi ; (2.10)
u   0 = (ci − K)Vb (u, i) + λi 0 Vb (u − y, i)fi (y)dy    N  − (λi + δ)Vb (u, i) + k=1 qik Vb (u, k) + K, u ≥ bi ,

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for i ∈ J. In fact, by the iterative rule for expectation, we have for i ∈ J and u ≥ bi , Vb (u, i)  ∞  t = qi e−qi t λi e−λi s 0 0  s

×

e

−δx

Kdx + e

0

 +

∞

λi e−λi s

=





0



−δs

s

qi e−qi t 

u+(ci −K)s



0

K λi ∆i (u) + + Λi ci − K

Vb (u + (ci − K)s − y, i)fi (y)dy dsdt

0 t

e−δx Kdx + e−δt

0



∞

e−Λi t

0

 qik k=i



qi

 Vb (u + ci t, k) dtds

qik Vb (u + ci t, k)dt,

(2.11)

k=i


∞ −Λ z
z u Λ where Λi = λi + qi + δ and ∆i (u) = e i ci −K u e i ci −K 0 Vb (z − y, i)fi(y)dydz. Note that the time of ruin τD → ∞ when u → ∞. Thus, we have  ∞ Vb (u, i) → e−δt Kdt = K/δ, u → ∞. (2.12) 0

Consequently, we have  ∞  1  K qi K e−Λi t qik Vb (u + ci t, k)dt → qik = , Λi δ Λi δ 0 k=i

u → ∞.

k=i

Furthermore, using using de l’Hˆ opital’s rule, we have  ci − K u lim ∆i (u) = lim Vb (u − y, i)fi (y)dy. u→∞ u→∞ Λi 0 Letting u → ∞ in (2.11) yields that  u K lim Vb (u − y, i)fi (y)dy = . u→∞ 0 δ

(2.13)

Therefore, letting u → ∞ in the second equation of system (2.10) yields lim Vb (u, i) = 0 for i ∈ J.

u→∞

(2.14)

The following lemma can be regarded as a net-profit condition. Lemma 2.2. Let Rb (t) be the surplus process under a modulated threshold strategy at b with Rb (0) = u. If N  i=1

then limt→∞ Rb (t) = ∞, a.s.

πi [(ci − K) − λi µi ] > 0,

(2.15)

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Proof. If we replace ci by ci − K in the definition of R(t) and denote by R (t) the corresponding surplus process, then R (t) ≤ Rb (t). In fact, (2.15) is the netprofit condition of R (t) (see Ng and Yang (2006)), thus, we have limt→∞ Rb (t) ≥ limt→∞ R (t) = ∞, a.s.  Lemma 2.3. Assume (2.15) holds, and the density fi (u) is decreasing, differentiable and Λi fi (u) + fi (u) ≤ 0, (2.16) ci − K for u ∈ [0, ∞) and i ∈ J. If, for i ∈ J, Vb (u, i) ∈ C 1 ((0, ∞)), then Vb (u, i) is concave on [0, ∞), i ∈ J, and as a consequence, Vb (u, i) ≥ 1 on [0, bi ), and Vb (u, i) ≤ 1 on [bi , ∞). Proof. Letting u → bi in the system (2.10), we have ci Vb (bi −, i) = (ci − K)Vb (bi +, i) + K. Since Vb (·, i) is continuously differentiable, the above equation yields that Vb (bi −, i) = Vb (bi +, i) = 1.

(2.17)

Differentiating the system (2.10) yields that 
u 0 = ci Vb (u, i) + λi 0 Vb (u − y, i)fi (y)dy − (λi + δ)Vb (u, i)     N   + k=1 qik Vb (u, k) + λi Vb (0, i)fi (u), 0 ≤ u < bi ;
u  0 = (ci − K)Vb (u, i) + λi 0 Vb (u − y, i)fi (y)dy − (λi + δ)Vb (u, i)     N  + k=1 qik Vb (u, k) + λi Vb (0, i)fi (u), u ≥ bi ,

(2.18)

for i ∈ J. Note that the second equation of above system can be rewritten as 0 = (ci − K)Vb (u, i) + λi gi (u) + λi Vb (u, i)fi (0) − (λi + δ)Vb (u, i) +

N 

qik Vb (u, k),

(2.19)

k=1


u where gi (u) = 0 Vb (u − y, i)fi (y)dy. Noting that f  (y) ≤ 0 and Vb (·, i) is increasing, it is easy to see gi (u) is decreasing and  u gi (u) ≥ Vb (u, i) fi (y)dy = Vb (u, i)(fi (u) − fi (0)) (2.20) 0

≥−

K fi (0), δ

which implies that limu→∞ gi (u) exists. Therefore, from equations (2.12), (2.14) and (2.19), we know limu→∞ Vb (u, i) exists and, lim Vb (u, i) = 0.

u→∞

(2.21)

422

J. Wei, H. Yang and R. Wang Let w(u, i) = Vb (u, i). Differentiating the system (2.18) yields that 
u  0 = ci w (u, i) + λi 0 w(u − y, i)fi(y)dy − (λi + δ)w(u, i)      + N 0 ≤ u < bi ; k=1 qik w(u, k) + λi zi (u),
u   0 = (ci − K)w (u, i) + λi 0 w(u − y, i)fi (y)dy − (λi + δ)w(u, i)     N  + k=1 qik w(u, k) + λi zi (u), u ≥ bi ,

(2.22)

where zi (u) = Vb (0+, i)fi (u) + Vb (0+, i)fi(u). If bi > 0 for i ∈ J, letting u → 0 in the first equation of system (2.10), we get  ci Vb (0+, i) = (λi + qi + δ)Vb (0+, i) − qik Vb (0+, k) k=i

≤ (λi + qi + δ)Vb (0+, i). From the above inequality and condition (2.16), we have zi (u) ≤ 0, for u ∈ [0, ∞) and i ∈ J. Similarly, if there exists i0 ∈ J such that bi0 = 0, (2.16) also assures that zi (u) ≤ 0, for u ∈ [0, ∞) and i ∈ J. It is worth to mention that w(u, i) may be not continuous only at bi and the Itˆo formula is still applicable to it. Consider M large (u < M ) and τM = inf {t ≥ 0 : Rb (t) = M }. Note that Lemma 2.2 implies that τM < ∞ a.s. Using the Lemma 2.1 and taking conditional expectation (similar to equation (2.9)), from the system (2.22) and the second result of Lemma 2.2, we have, for 0 ≤ u < ∞, 3 4 w(u, i) = Eu,i e−δ(τM ∧τD ) w(Rb (τM ∧ τD ), J(τM ∧ τD ))  (τM ∧τD ) + Eu,i e−δt λJ(t) zJ(t) (Rb (t))dt , i ∈ J. 0

If τM < τD , then w(u, i) ≤ maxj∈J {w(M, j)}. By letting M → ∞ and equation (2.21), we have that w(u, i) ≤ 0. If τM ≥ τD , the surplus process is stopped at the time of ruin and there is no dividend after ruin, thus, we have w(Rb (τD ), J (τD )) = 0 and w(u, i) ≤ 0. Thus, we know that Vb (u, i) is concave and, from (2.17), we get  that Vb (u, i) ≥ 1 on [0, bi ), and Vb (u, i) ≤ 1 on [bi , ∞). The following theorem is a direct consequence of the Lemma 2.3 and the Theorem 2.2. Theorem 2.3. Assume that the conditions (2.15) and (2.16) hold. If there exists b = (bi ≥ 0, i ∈ J) such that Vb (u, i), i ∈ J are continuously differentiable with respect to u on (0, ∞), then the optimal strategy D∗ is the modulated threshold strategy at b and the value function V (u, i) = Vb (u, i). Remark 2.2. Does condition (2.15) contradict with (2.16)? The answer is negative at least when the claim size is exponentially distributed with mean µi . In fact, in this special case (2.16) simplifies to ci − K ≥ (λi + qi + δ)µi which implies (2.15).

Optimal Threshold Dividend Strategies

423

3. Case study: Two regimes and exponential claim density In this section we assume that fi (x) = γi e−γi x , i = 1, 2. In the case of one regime, exponential claim density is considered in Example 2.35 of Schmidli (2008) which also analyses the case with unbounded K. The techniques used in this section are similar to Schmidli (2008) and Section 3 of Sotomayor and Cadenillas (2008). From the Theorem 2.3 and the system (2.10), we only need to find bi ≥ 0, and v(u, i) which is continuously differentiable, nonnegative, nondecreasing and bounded solve the system
u  0 = ci v  (u, i) + λi γi 0 v(u − y, i)e−γi y dy    2   − (λi + δ)v(u, i) + k=1 qik v(u, k), 0 ≤ u < bi ;
u (3.1)  0 = (ci − K)v (u, i) + λi γi 0 v(u − y, i)e−γi y dy    2  − (λi + δ)v(u, i) + k=1 qik v(u, k) + K, u ≥ bi , for i = 1, 2. In this section, we consider b1 ≤ b2 ; the case b1 > b2 may be treated similarly. Here we have to consider three cases: u ∈ [0, b1 ), u ∈ [b1 , b2 ) and u ∈ [b2 , ∞). If u ∈ [0, b1 ), system (3.1) gives the following system of integro-differential equations: .
u −c1 v  (u, 1) = λ1 γ1 e−γ1 u 0 v(y, 1)eγ1 y dy − (λ1 + q1 + δ)v(u, 1) + q1 v(u, 2),
u −c2 v  (u, 2) = λ2 γ2 e−γ2 u 0 v(y, 2)eγ2 y dy − (λ2 + q2 + δ)v(u, 2) + q2 v(u, 1). The right-hand sides of the both equations are differentiable. Therefore, 
u c1 v (u, 1) = λ1 γ12 e−γ1 u 0 v(y, 1)eγ1 y dy − λ1 γ1 v(u, 1)      + (λ1 + q1 + δ)v (u, 1) − q1 v  (u, 2),
u  c2 v (u, 2) = λ2 γ22 e−γ2 u 0 v(y, 2)eγ2 y dy − λ2 γ2 v(u, 2)     + (λ2 + q2 + δ)v (u, 1) − q2 v  (u, 2). Plugging in the original equations yields  0 = c1 v  (u, 1) − (λ1 + q1 + δ − c1 γ1 )v  (u, 1)     − γ1 (q1 + δ)v(u, 1) + γ1 q1 v(u, 2) + q1 v (u, 2),  0 = c2 v  (u, 2) − (λ2 + q2 + δ − c2 γ2 )v  (u, 2)    − γ2 (q2 + δ)v(u, 2) + γ2 q2 v(u, 1) + q2 v (u, 1). We conjecture that the solution is given by . v(u, 1) = A11 eα1 (u−b1 ) + A12 eα2 (u−b1 ) + A13 eα3 (u−b1 ) + A14 eα4 (u−b1 ) , v(u, 2) = A21 eα1 (u−b1 ) + A22 eα2 (u−b1 ) + A23 eα3 (u−b1 ) + A24 eα4 (u−b1 ) .

(3.2)

(3.3)

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By plugging (3.3) in (3.2) and comparing the coefficients, we know that if we can show the system of equations . c1 α2 − (λ1 + q1 + δ − c1 γ1 )α − γ1 (q1 + δ) + γ1 q1 β + q1 βα = 0, (3.4) c2 α2 − (λ2 + q2 + δ − c2 γ2 )α − γ2 (q2 + δ) + γ2 q2 /β + q2 α/β = 0 has four roots (α, β), then we can determine (αj , A2j /A1j ), j = 1, 2, 3, 4. Lemma 3.1. The system of equations (3.4) has four real roots (αj , βj ), j = 1, 2, 3, 4, and α1 < α2 < 0 < α3 < α4 . Proof. Let gi (α) = ci α2 − (λi + qi + δ − ci γi )α − γi (qi + δ),

i = 1, 2

and G(α) = g1 (α)g2 (α) − q1 q2 (γ1 + α)(γ2 + α). Obviously, for i = 1, 2, gi (α) = 0 has two roots θi1 and θi2 , such that θi1 < 0 < θi2 . Since G(∞) = G(−∞) = ∞ and G(0) > 0, the proof is then completed by showing that there are θ1 and θ2 , such that θ1 < 0 < θ2 and G(θi ) < 0, i = 1, 2. Firstly, we have θ12 > 0 and G(θ12 ) < 0. Secondly, for i = 1, 2, gi (−γi ) > 0, then −γi < θi1 < 0. Thus, we have θ1 = max{θ11 , θ21 }. In fact, G(θ1 ) = −q1 q2 (γ1 + θ1 )(γ2 + θ1 ) < 0.  Hence, we identify (αj , A2j /A1j ), j = 1, 2, 3, 4 as the four roots of (3.4). Hence for j = 1, 2, 3, 4, we have A2j = −

g1 (αj ) γ 2 q 2 + q 2 αj A1j = − A1j . γ1 q1 + q1 αj g2 (αj )

(3.5)

If u ∈ [b1 , b2 ), system (3.1) gives the following system of integro-differential equations: 
u −(c1 − K)v (u, 1) = λ1 γ1 e−γ1 u 0 v(y, 1)eγ1 y dy      − (λ1 + q1 + δ)v(u, 1) + q1 v(u, 2) + K,
 −γ2 u u  −c v (u, 2) = λ γ e v(y, 2)eγ2 y dy  2 2 2 0    − (λ2 + q2 + δ)v(u, 2) + q2 v(u, 1). Similar to (3.2), we have  0 = (c1 − K)v (u, 1) − [λ1 + q1 + δ − γ1 (c1 − K)] v  (u, 1)     − γ1 (q1 + δ)v(u, 1) + γ1 q1 v(u, 2) + q1 v (u, 2) + γ1 K,  0 = c2 v  (u, 2) − (λ2 + q2 + δ − c2 γ2 )v  (u, 2)    − γ2 (q2 + δ)v(u, 2) + γ2 q2 v(u, 1) + q2 v (u, 1).

(3.6)

Optimal Threshold Dividend Strategies We conjecture that the solution is given by  v(u, 1) = B11 eα˜ 1 (u−b2 ) + B12 eα˜ 2 (u−b2 )      + B13 eα˜ 3 (u−b2 ) + B14 eα˜ 4 (u−b2 ) + η1 ,  v(u, 2) = B21 eα˜ 1 (u−b2 ) + B22 eα˜ 2 (u−b2 )     + B23 eα˜ 3 (u−b2 ) + B24 eα˜ 4 (u−b2 ) + η2 .

425

(3.7)

Plugging (3.7) in (3.6) and comparing the coefficients show that ηi =

q2 + (2 − i)δ K, δ(q1 + q2 + δ)

i = 1, 2.

Similarly, to obtain (α˜j , B2j /B1j ), j = 1, 2, 3, 4, we need to show  (c1 − K)˜ α2 − [λ1 + q1 + δ − γ1 (c1 − K)]˜ α      −γ1 (q1 + δ) + γ1 q1 β˜ + q1 β˜α ˜ = 0,  c2 α ˜ 2 − (λ2 + q2 + δ − c2 γ2 )˜ α − γ2 (q2 + δ)     +γ2 q2 /β˜ + q2 α/ ˜ β˜ = 0

(3.8)

(3.9)

˜ which directly follows from Lemma 3.1 (note that K < has four roots (˜ α, β), c1 ). Hence, we identify (α˜j , B2j /B1j ), j = 1, 2, 3, 4 as the four roots of (3.9), and without loss of generality, let α˜1 < α˜2 < 0 < α˜3 < α˜4 . Hence for j = 1, 2, 3, 4, we have g˜1 (˜ αj ) γ2 q2 + q2 α ˜j B2j = − B1j = − B1j , (3.10) γ1 q1 + q1 α ˜j g˜2 (˜ αj ) where g˜1 (˜ α) = (c1 − K)˜ α2 − [λ1 + q1 + δ − γ1 (c1 − K)]˜ α − γ1 (q1 + δ), g˜2 (˜ α) = c2 α ˜ 2 − (λ2 + q2 + δ − c2 γ2 )˜ α − γ2 (q2 + δ). Finally, if u ∈ [b2 , ∞), system (3.1) gives the following system of integrodifferential equations: 
u −(c1 − K)v  (u, 1) = λ1 γ1 e−γ1 u 0 v(y, 1)eγ1 y dy      − (λ1 + q1 + δ)v(u, 1) + q1 v(u, 2) + K, (3.11)
u  −(c2 − K)v  (u, 2) = λ2 γ2 e−γ2 u 0 v(y, 2)eγ2 y dy     − (λ2 + q2 + δ)v(u, 2) + q2 v(u, 1) + K. Similar to (3.2), we have the following system of differential equations  0 = (c1 − K)v  (u, 1) − [λ1 + q1 + δ − γ1 (c1 − K)] v  (u, 1)     − γ1 (q1 + δ)v(u, 1) + γ1 q1 v(u, 2) + q1 v  (u, 2) + γ1 K,  0 = (c2 − K)v  (u, 2) − [λ2 + q2 + δ − γ2 (c2 − K)] v  (u, 2)    − γ2 (q2 + δ)v(u, 2) + γ2 q2 v(u, 1) + q2 v  (u, 1) + γ2 K.

(3.12)

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We conjecture that the solution is given by . v(u, 1) = C11 eαˆ 1 u + C12 eαˆ 2 u + C13 eαˆ 3 u + C14 eαˆ 4 u + ηˆ1 , v(u, 2) = C21 eαˆ 1 u + C22 eαˆ 2 u + C23 eαˆ 3 u + C24 eαˆ 4 u + ηˆ2 .

(3.13)

Plugging (3.13) in (3.12) and comparing coefficients show that ηˆ1 = ηˆ2 =

K . δ

Similarly, to determine (ˆ αj , C2j /C1j ), j = 1, 2, 3, 4, we need to show  (c1 − K)ˆ α2 − [λ1 + q1 + δ − γ1 (c1 − K)]ˆ α      ˆ ˆ −γ1 (q1 + δ) + γ1 q1 β + q1 β α ˆ = 0,  (c2 − K)ˆ α2 − [λ2 + q2 + δ − γ2 (c2 − K)]ˆ α     −γ2 (q2 + δ) + γ2 q2 /βˆ + q2 α/ ˆ βˆ = 0

(3.14)

ˆ Since K < min{c1 , c2 }, this is assured by Lemma 3.1. Hence, has four roots (ˆ α, β). we identify (ˆ αj , C2j /C1j ), j = 1, 2, 3, 4 as the four roots of (3.14). Hence for j = 1, 2, 3, 4, we have C2j = −

gˆ1 (ˆ αj ) γ2 q2 + q2 α ˆj C1j = − C1j , γ1 q1 + q1 α ˆj gˆ2 (ˆ αj )

(3.15)

where gˆ1 (ˆ α) = (c1 − K)ˆ α2 − [λ1 + q1 + δ − γ1 (c1 − K)]ˆ α − γ1 (q1 + δ), gˆ2 (ˆ α) = (c2 − K)ˆ α2 − [λ2 + q2 + δ − γ2 (c2 − K)]ˆ α − γ2 (q2 + δ). Without loss of generality, let α ˆ1 < α ˆ2 < 0 < α ˆ3 < α ˆ 4 . Because v(u, 1) and v(u, 2) are bounded, we have C13 = C14 = C23 = C24 = 0. Therefore, . v(u, 1) = C11 eαˆ 1 u + C12 eαˆ 2 u + K , δ (3.16) K α ˆ1 u α ˆ2u v(u, 2) = C21 e + C22 e + δ is the solution of the system (3.12), where (3.15) is satisfied for j = 1, 2. Plugging (3.16) into (3.11), we have .

C11 α ˆ 1 +γ1 C21 α ˆ 1 +γ2

+ +

C12 α ˆ 2 +γ1 C22 α ˆ 2 +γ2

+ +

K δγ1 K δγ2

= 0, = 0,

from which and (3.15) we can find Ci,j for i, j = 1, 2.

(3.17)

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Now to find the thresholds b1 and b2 , and the coefficients in (3.3) and (3.7), we use the following smooth-fit condition:  v(bi −, i) = v(bi +, i),      v(b3−i −, i) = v(b3−i +, i),   v  (bi −, i) = 1, (3.18)     v (bi +, i) = 1,      v (b3−i −, i) = v  (b3−i +, i), for each i = 1, 2. The system (3.18) assures v is smooth enough to satisfy the condition of Lemma 2.3. The solution of the system of equations (3.18) will give us the values for b1 and b2 , and also the values for A1j and B1j , j = 1, 2, 3, 4. The values for the corresponding A2j , B2j can be found from equations (3.5), (3.10). Example 3.1. Consider the case where c1 = 1.6, c2 = 2, λ1 = 0.2, λ2 = 0.4, q1 = 0.25, q2 = 0.75, γ1 = 2, γ2 = 1, K = 0.7 and δ = 0.08. Note that conditions (2.15) and (2.16) are satisfied. By the using of Matlab, we solve the systems (3.4), (3.9), (3.14), (3.17), and numerically solve the system of nonlinear equations (3.18). We obtain the thresholds b1 = 0.3545, b2 = 0.5309, and the value functions v(u, 1) and v(u,2) are given by   −0.2241e−1.8862(u−0.3545) − 0.1314e−0.8585(u−0.3545)      +8.3104e0.0522(u−0.3545) + 0.0480e0.6387(u−0.3545),     0 ≤ u < 0.3545     −0.2876e−1.8118(u−0.5309) − 0.2000e−0.8540(u−0.5309) v(u, 1) =  +2.0261e0.0784(u−0.5309) − 0.1053e0.7914(u−0.5309) + 6.7245,      0.3545 ≤ u < 0.5309      −0.7539e−1.8108u − 0.4665e−0.8073u + 8.7500,    u ≥ 0.5309 and

  −0.0308e−1.8862(u−0.3545) − 0.8162e−0.8585(u−0.3545)      +8.3615e0.0522(u−0.3545) − 0.1235e0.6387(u−0.3545),     0 ≤ u < 0.3545     −0.0403e−1.8118(u−0.5309) − 0.7597e−0.8540(u−0.5309) v(u, 2) =  +2.1638e0.0784(u−0.5309) + 0.1371e0.7914(u−0.5309) + 6.0764,      0.3545 ≤ u < 0.5309     −0.1387e−1.8108u − 1.7189e−0.8073u + 8.7500,     u ≥ 0.5309

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Acknowledgment We would like to thank the referee for careful reading the paper and the helpful comments and suggestions. J. Wei would like to acknowledge the PHD Program Scholarship Fund of ECNU (No. 2010050). H. Yang would like to acknowledge the Research Grants Council of the Hong Kong Special Administrative Region, China (project No. HKU 754008H). R. Wang would like to acknowledge the National Natural Science Foundation of China (10971068), National Basic Research Program of China (973 Pro gram) under grant number 2007CB814904 and Program for New Century Excellent Talents in University(NCET-09-0356).

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