Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering

THEORY OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS AND APPLICATIONS

MATHEMATICAL AND ANALYTICAL TECHNIQUES WITH APPLICATIONS TO ENGINEERING

MATHEMATICAL AND ANALYTICAL TECHNIQUES WITH APPLICATIONS TO ENGINEERING Alan Jeffrey, Consulting Editor

Published: Inverse Problems

A. G . Ramm Singular Perturbation Theory

R. S. Johnson Methods for Constructing Exact Solutions of Partial Differential Equations with Applications

S. V . Meleshko Stochastic Differential Equations with Applications

R. Situ

Forthcoming: The Fast Solution of Bounda~yIntegral Equations

S. Rjasanow and 0. Steinbach

THEORY OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS AND APPLICATIONS MATHEMATICAL AND ANALYTICAL TECHNIQUES WITH APPLICATIONS TO ENGINEERING

RONG SITU

a - Springer

Library of Congress Cataloging-in-Publication Data Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering By Rong Situ ISBN-10: 0-387-25083-2 e-ISBN 10: 0-387-25 175-8 ISBN-13: 978-0387-25083-0 e-ISBN-13: 978-0387-25175-2

Printed on acid-free paper.

O 2005 Springer Science+Business Media, Inc.

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. U se in connection w ith any form o f in formation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereatler developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1

(BSIDH)

SPIN 1 1399278

Contents

Preface

xi

Acknowledgement

xvii

Abbreviations and Some Explanations

xix

I Stochastic Differential Equations with Jumps in Rd 1 1 Martingale Theory and the Stochastic Integral for Point Processes 1.1 Concept of a Martingale 1.2 Stopping Times. Predictable Process 1.3 Martingales with Discrete Time 1.4 Uniform Integrability and Martingales 1.5 Martingales with Continuous Time 1.6 Doob-Meyer Decomposition Theorem 1.7 Poisson Random Measure and Its Existence 1.8 Poisson Point Process and Its Existence 1.9 Stochastic Integral for Point Process. Square Integrable Martingales 2 Brownian Motion, Stochastic Integral and Ito's Formula

3 3 5 8 12 17 19 26 28 32 39

Contents 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Brownian Motion and Its Nowhere Differentiability Spaces C° and £ 2 Ito's Integrals on C2 Ito's Integrals on £ 2 -' oc Stochastic Integrals with respect to Martingales Ito's Formula for Continuous Semi-Martingales Ito's Formula for Semi-Martingales with Jumps Ito's Formula for d-dimensional Semi-Martingales. Integration by Parts Independence of BM and Poisson Point Processes Some Examples Strong Markov Property of BM and Poisson Point Processes Martingale Representation Theorem

39 43 44 47 49 54 58

Stochastic Differential Equations 3.1 Strong Solutions to SDE with Jumps 3.1.1 Notation 3.1.2 A Priori Estimate and Uniqueness of Solutions . . . 3.1.3 Existence of Solutions for the Lipschitzian Case . . . 3.2 Exponential Solutions to Linear SDE with Jumps 3.3 Girsanov Transformation and Weak Solutions of SDE with Jumps 3.4 Examples of Weak Solutions

75 75 75 76 79 84

Some Useful Tools in Stochastic Differential Equations 4.1 Yamada-Watanabe Type Theorem 4.2 Tanaka Type Formula and Some Applications 4.2.1 Localization Technique 4.2.2 Tanaka Type Formula in d—Dimensional Space . . . 4.2.3 Applications to Pathwise Uniqueness and Convergence of Solutions 4.2.4 Tanaka Type Formual in 1-Dimensional Space . . . 4.2.5 Tanaka Type Formula in The Component Form . . . 4.2.6 Pathwise Uniqueness of solutions 4.3 Local Time and Occupation Density Formula 4.4 Krylov Estimation 4.4.1 The case for 1—dimensional space 4.4.2 The Case for d—dimensional space 4.4.3 Applications to Convergence of Solutions to SDE with Jumps

103 103 109 109 110

2.9 2.10 2.11 2.12

62 64 65 67 68

86 99

112 116 121 122 124 129 129 130 133

Stochastic Differential Equations with Non-Lipschitzian Coefficients 139 5.1 Strong Solutions. Continuous Coefficients with p— Conditionsl40 5.2 The Skorohod Weak Convergence Technique 145

Contents

vii

5.3 Weak Solutions. Continuous Coefficients 147 5.4 Existence of Strong Solutions and Applications to ODE . . 153 5.5 Weak Solutions. Measurable Coefficient Case 153

II

Applications

161

6 How to Use the Stochastic Calculus to Solve SDE 163 6.1 The Foundation of Applications: Ito's Formula and Girsanov's Theorem 163 6.2 More Useful Examples 167 7 Linear and Non-linear Filtering 169 7.1 Solutions of SDE with Functional Coefficients and Girsanov Theorems 169 7.2 Martingale Representation Theorems (Functional Coefficient Case) 177 7.3 Non-linear Filtering Equation 180 7.4 Optimal Linear Filtering 191 7.5 Continuous Linear Filtering. Kalman-Bucy Equation . . . . 194 7.6 Kalman-Bucy Equation in Multi-Dimensional Case 196 7.7 More General Continuous Linear Filtering 197 7.8 Zakai Equation 201 7.9 Examples on Linear Filtering 203 8 Option Pricing in a Financial Market and BSDE 8.1 Introduction 8.2 A More Detailed Derivation of the BSDE for Option Pricing 8.3 Existence of Solutions with Bounded Stopping Times . . . . 8.3.1 The General Model and its Explanation 8.3.2 A Priori Estimate and Uniqueness of a Solution . . . 8.3.3 Existence of Solutions for the Lipschitzian Case . . . 8.4 Explanation of the Solution of BSDE to Option Pricing . . 8.4.1 Continuous Case 8.4.2 Discontinuous Case 8.5 Black-Scholes Formula for Option Pricing. Two Approaches 8.6 Black-Scholes Formula for Markets with Jumps 8.7 More General Wealth Processes and BSDEs 8.8 Existence of Solutions for Non-Lipschitzian Case 8.9 Convergence of Solutions 8.10 Explanation of Solutions of BSDEs to Financial Markets . . 8.11 Comparison Theorem for BSDE with Jumps 8.12 Explanation of Comparison Theorem. Arbitrage-Free Market 8.13 Solutions for Unbounded (Terminal) Stopping Times . . . . 8.14 Minimal Solution for BSDE with Discontinuous Drift . . . .

205 205 208 209 209 213 215 219 219 220 223 229 234 236 239 241 243 250 254 258

viii

Contents 8.15 Existence of Non-Lipschitzian Optimal Control. BSDE Case 262 8.16 Existence of Discontinuous Optimal Control. BSDEs in R1 . 267 8.17 Application to PDE. Feynman-Kac Formula 271

9 Optimal Consumption by H-J-B Equation and Lagrange Method 277 9.1 Optimal Consumption 277 9.2 Optimization for a Financial Market with Jumps by the Lagrange Method 279 9.2.1 Introduction 280 9.2.2 Models 280 9.2.3 Main Theorem and Proof 282 9.2.4 Applications 286 9.2.5 Concluding Remarks 290 10 Comparison Theorem and Stochastic Pathwise Control 291 10.1 Comparison for Solutions of Stochastic Differential Equations 292 10.1.1 1—Dimensional Space Case 292 10.1.2 Component Comparison in d—Dimensional Space . . 293 10.1.3 Applications to Existence of Strong Solutions. Weaker Conditions 294 10.2 Weak and Pathwise Uniqueness for 1-Dimensional SDE with Jumps 298 10.3 Strong Solutions for 1-Dimensional SDE with Jumps . . . . 300 10.3.1 Non-Degenerate Case 300 10.3.2 Degenerate and Partially-Degenerate Case 303 10.4 Stochastic Pathwise Bang-Bang Control for a Non-linear System 312 10.4.1 Non-Degenerate Case 312 10.4.2 Partially-Degenerate Case 316 10.5 Bang-Bang Control for d—Dimensional Non-linear Systems 319 10.5.1 Non-Degenerate Case 319 10.5.2 Partially-Degenerate Case 322 11 Stochastic Population Control and Reflecting SDE 11.1 Introduction 11.2 Notation 11.3 Skorohod's Problem and its Solutions 11.4 Moment Estimates and Uniqueness of Solutions to RSDE . 11.5 Solutions for RSDE with Jumps and with Continuous Coefficients 11.6 Solutions for RSDE with Jumps and with Discontinuous Coefficients 11.7 Solutions to Population SDE and Their Properties 11.8 Comparison of Solutions and Stochastic Population Control

329 330 332 335 342 345 349 352 363

Contents 11.9 Caculation of Solutions to Population RSDE

ix 372

12 Maximum Principle for Stochastic Systems with Jumps 12.1 Introduction 12.2 Basic Assumption and Notation 12.3 Maximum Principle and Adjoint Equation as BSDE with Jumps 12.4 A Simple Example 12.5 Intuitive thinking on the Maximum Principle 12.6 Some Lemmas . 12.7 Proof of Theorem 354

377 377 378

A A Short Review on Basic Probability Theory A.I Probability Space, Random Variable and Mathematical Expectation A.2 Gaussian Vectors and Poisson Random Variables A.3 Conditional Mathematical Expectation and its Properties . A.4 Random Processes and the Kolmogorov Theorem

389

B Space D and Skorohod's Metric

401

379 380 381 383 386

389 392 395 397

C Monotone Class Theorems. Convergence of Random Processes407 C.I Monotone Class Theorems C.2 Convergence of Random Variables . . . . . . . . . . . . . . C.3 Convergence of Random Processes and Stochastic Integrals

407 409 411

References

415

Index

431

Preface

Stochastic differential equations (SDEs) were first initiated and developed by K. Ito (1942). Today they have become a very powerful tool applied to Mathematics, Physics, Chemistry, Biology, Medical science, and almost all sciences. Let us explain why we need SDEs, and how the contents in this book have been arranged. In nature, physics, society, engineering and so on we always meet two kinds of functions with respect to time: one is determinstic, and another is random. For example, in a financial market we deposite money rt in a bank. This can be seen as our having bought some units r)! of a bond, where the bond's price Pf satisfies the following ordinary differential equation dPf = Pfrtdt, P : = 1, t E [O,T], where rt is the rate of the bond, and the money that we deposite in the t bank is r t = r)!Pf = r)! exp[Jl r,ds]. Obviously, usually, Pf = exp[& r,ds] is non-random, since the rate rt is usually deterministic. However, if we want to buy some stocks from the market, each stock's price is random. For simplicity let us assume that in the financial market there is only one stock, and its price is P i . Obviously, it will satisfy a differential equation as follows: d P i = Pi(btdt d(a stochastic perturbation)), Po' = Po', t E [O,T], where all of the above processes are 1-dimensional. Here the stochastic perturbation is very important, because it influences the price of the stock, which will cause us to earn or lose money if we buy the stock. One important' problem arises naturally. How can we model this stochastic perturbation? Can we make calculations to get the solution of the stock's price Pi, as we do in the case of the bond's price P f ? The answer is positive, usually a

+

continuous stochastic perturbation will be modeled by a stochastic integral / 0 asdws, where Wt,t > 0 is the so-called Brownian Motion process (BM), or the Wiener process. The 1—dimensional BM wt, t > 0 has the following nice properties: 1) (Independent increment property). It has an independent increment property, that is, for any 0 < t\ < • • • < tn the system {wo, w^ — wo, w42 — Wt,, • • • , Wtn — tvtn^i, } is an independent system. Or say, the increments, which happen in disjoint time intervals, occured independently. 2) (Normal distribution property). Each increment is Normally distributed. That is, for any 0 < s < t the increment Wj — ws on this time interval is a normal random variable with mean m, and variance a2(t — s). We write this as wt — ws ~ N(m,a2(t — s)). 3) (Stationary distribution property). The probability distribution of each increment only depends on the length of the time interval, and it does not depend on the starting point of the time interval. That is, the m and a2 appearing in property 3) are constants. 4) (Continuous trajectory property). Its trajectory is continuous. That is BM wt, t > 0 is continuous in t. Since the simplest or say, the most basic continuous stochastic perturbation, intuitively will have the above four properties, the modeling of the general continuous stochastic perturbation by a stochastic integral with respect to this basic BM Wt,t > 0 is quite natural. However, the 1—dimensional BM also has some strange property: Even though it is continuous in t, it is nowhere differentiate in t. So we cannot define the stochastic integral / 0 as(u})dws{(jj) for each given w. That is why K. Ito (1942) invented a completely new way to define this stochastic integral. Our first task in this book is to introduce the Ito stochastic integral and discuss its properties for later applications. After we have understood the stochastic integral jQ as(uj)dws(uj) we can study the following general stochastic differential equation (SDE): xt = xo + / 0 * Z ( s , x s ) d s + Jo a(s,xs)dws,

t>0,

or equivalently, we write dxt — b(t,xt)dt + 0.

(1)

Returning to the stock's price equation, we naturally consider it as the following SDE: dPl = P}(btdt + ctdwt),Pj

= P o \* € [0,T\.

(2)

Comparing this to the solution of Pt°, one naturally asks could the solution of this SDE be P / = PQ exp[J0 bsds + JQ crsdws}'! To check this guess, obviously if we can have a differential rule to perform differentiation on PQ expxt, where Xt = f0 bsds + JQ crsdws, then we can make the check. Or more generally, if we have an f(x) € C2(R) and dxt = btdt + atdwt, can we have df(xt) = f'(xt)dxt = f'(xt) (btdt + crtdwt)f

If as in the deterministic case, this differential rule holds true, then we immediately see that P / = PQ exp[/ 0 bsds + JQ asdws] satisfies (2). Unforturnately, such a differential rule is not true. K. Ito (1942) has found that it should obey another differential rule - the so-called Ito's formula:

df'(xt) = f'(xt)dxt + |/"(x t ) Wtf dt. By this rule one easily checks that P / = PQ1 exp[/0* bsds +jl asdws - \ /„* | 0; then solving this ODE we immediately get this quantity xt- However, if the dynamics are disturbed by some continuous random noise, say such that dxt = b(t,Xt)dt + a(t,xt)dwt,xo = XQ,t > 0, then for the amount Xt, or say, the solution of this SDE, two situations can arise. The first one is, if we think that the random noise - BM ws,s < t, is an input, then after disturbing the dynamics we get an output xt. This means that the solution xt is a functional of the given noise - BM ws,s < t for each t. We will call such a solution a strong solution. Another situation is that for a given noise we cannot immediately find the solution. However, we can find a random process xt, t > 0, and maybe another random noise that is also a BM wt,t > 0, such that (xt,Wt),t > 0 satisfy the SDE dxt = b(t,xt)dt + a(t,xt)dwt,xo = xo,t > 0. In this case we will call (xt,wt),t > 0 or simply, Xt,t > 0, a weak solution of the original SDE. Obviously, from the engineering point of view the strong solutions is more realistic and useful. However, since, if the strong solution xt, t > 0 exists, then all finite dimensional probability ditributions of (xt,wt)t>0 are the same as that of (S t , w t ) t > 0 . So, if we do not know the existence of a strong solution, but we do know the existence of a weak solution (xt,Wt),t > 0, then from the probability point of view the weak solution can still help us in some sense. Therefore, for solutions of SDEs there are two kinds that need to be considered: strong solutions and weak solutions. Our third task in this book is to introduce the concepts of solutions and to discuss their existence and uniquess and the related important theory. (For example, Girsanov's theorem and the matingale representation theorem, the first of which can help us find weak solutions, while the second

one is necessary for finding the solutions of backward SDE and the filtering problem considered later). Since, actually, in the realistic world we will always meet some jump type stochastic perturbation, in this book we also consider stochastic integrals with respect to a Poisson counting measure (which is generated by a Poisson point process), the Ito formula and SDE for this case. (To find the reason why we consider the Poisson point process and its related integral as a jump type stochastic perturbation see the subsection "The General Model and its Explanation" in Chapter 8 - "Option Pricing in a Financial Market and BSDE"). The first three Chapters are intended to solve the above three tasks. They are the basic foudation of the SDE theory and its applications. However, interesting and important things for SDE do not only come from the above mentioned three chapters, where they exhibit the following facts: the definition of Ito's stochastic integrals and Ito's differential rule are completely different from the deterministic case, etc. The interesting and important things also come from the following facts: __ 1) For an ordinary differential equation (ODE) dxt — b(t,Xt)dt,xo — XQ, t > 0 if b(t, x) is only bounded and jointly continuous, then even though the solution exists, is not necessary unique. However, for the SDE (1) in 1—dimensional case if 6(£, a;) and a(t, x) are only bounded and jointly Borelmeasurable, and |5(i,a;)|~ is also bounded and a(t, x) is Lipschitz continuous in x, then (1) will have a unique strong solution. (Here "strong" means that Xt is #2"—measurable). This means that adding a non-degenerate stochastic perturbation term into the differential equation, can even improve the nice property of the solution. 2) The stochastic perturbation term has an importaqnt practical meaning in some cases and it cannot be discarded. For example, in the investment problem and the option pricing problem from a Financial Market, the investment portfolio actually appears as the coefficient of the stochastic integral in an SDE, where the stochastic integral acts like a stochatic perturbation term. 3) The solutions of SDEs and backward SDEs can help us to explain the solutions of some deterministic partial differential equations (PDEs) with integral terms (the Feynman-Kac formula) and even to guess and find the solution of a PDE, for example, the soluition of the PDE for the price of a option can be solved by a solution of a BSDE - the Black-Scholes formula. 4) More and more. So we have many reasons to study the SDE theory and its applications more deeply and carefully. That is why we have a Chapter that discusses useful tools for SDE, and a Chapter for the solutions of an SDE with nonLipschitzian coefiicients. These are Chapter 4 and 5. The above concerns the first part of our book, which represents the theory and general background of the SDE.

The second part of our book is about the Applications. We first provide a short Chapter to help the reader to take a quick look at how to use Stochastic Analysis (the theory in the first part), to solve an SDE. Then we discuss the estimation problem for a signal process : the so-called filtering problem, where the general linear and non-linear filtering problem for continous SDE systems and SDE systems with jumps, the KalmanBucy filtering equation for continuous systems, and the Zakai equation for non-linear filtering, etc. are also considered. Since, now, research on mathematical finance, and in particular on the option pricing problem for the financial market has become very popular, we also provide a Chapter that discusses the option pricing problem and backward SDE, where the famous Black-Scholes formulas for a market with or without jumps are derived using a probability and a PDE approach, respectively; and the arbitrage-free market is also discussed. The interesting thing here is that we deal with the mathematical financial problem by using the backward stochastic differential equation (BSDE) technique, which now becomes very powerful when treating many problems in the finacial market, in mathematics and in other sciences. Since deterministic population control has proved to be important and efficient, and the stochastic population control is more realistic, we also provide a Chapter that develops the stochastic population control problem by using the reflecting SDE approach, where the existence, the comparison and the calculation of the population solution and the optimal stochastic population control are established. Besides, the stochastic Lagrange method for the stochastic optimal control, the non-linear pathwise stochastic optimal control, and the Maximum Principle (that is, the necessary conditions for a stochastic optimal control) are also formulated and developed in specific Chapters, respectively. For the convenience of the readers three Appendixes are also provided: giving a short review on basic probability theory, space D and Skorohod's metric, and monotone class theorems and the convergence of random processes. We suggest that the reader studies the book as follows: For readers who are mainly interested in Applications, the following approach may be considered: Appendix A —> Chapter 1 —+ Chapter 2 —> Chapter 3 — • > Chapter 6 —» Any Chapter in The second part "Applications" except Chapter 10, and at any time return to read the related sections in Chapters 4 and 5, or Appendixes B and C, when necessary. However, to read Chapter 10, knowledge of Chapters 4 and 5 and Appendixes B and C are necessary.

Acknowledgement

The author would like to express his sincere thanks to Professor Alan Jeffrey for kindly recommending publication of this book, and for his interest in the book from the very beginning to the very end, and for offering many valuable and important suggestions. Without his help this book would not have been possible

Abbreviations and Some Explanations

All important statements and results, like, Definitions, Lemmas, Propositions, Theorems, Corollaries, Remarks and Examples are numbered in sequential order throughout the whole book. So, it is easy to find where they are located. For example, Lemma 22 follows Definition 21, and Theorem 394 is just after Remark 393, etc. However, the numbers of equations are arranged independently in each Chapter and each Appendix. For example, (3.25) means equation 25 in Chapter 3, and (C.4) means the equation 4 in Appendix C. The following abbreviations are frequently used in this book. a.e. almost everywhere. a.s. almost sure. BM Brownian Motion. BSDE backward stochastic differential equation. FSDE forward stochastic differential equation. H-J-B equation Hamilton-Jacobi-Bellman equation. IDE integral-differential equation. ODE ordinary differential equation. PDE partial differential equation. RCLL right continuous with left limit. SDE stochastic differential equation. P - a.s almost sure in probability P. a+ max {a, 0) . amax {-a, 0). a v b max{a,b). a A b min{a,b).

fj. « v measure /i is absolutely continuous with respect to v\ that is, for any measurable set A, v(A) = 0 implies that n(A) = 0. £ n —> £, a.s. £ n converges to £, almost surely; that is, £n(w) —> £(w) for all UJ except at the points w € A, where P(A) = 0. £ n —> ^, in P £n converges to ^ in probability; that is, Ve > 0, lim^oo P(w : |C(w) - e(w)| > E) = 0. # {•} the numbers of • counted in the set {•} . cr(xs, s < t) the smallest cr—field, which makes all a;s, s < t measurable. S[^|?7] E[£\(T(T})]. It means the conditional expectation of ^ given cr(r)). The following notations can be found on the corresponding pages. For example, #, 4,387 means that notation J can be found in page 4 and page 387. 0,4,387 S, 4,387 34 34 34 oc 34 , p Rdl C , 43 43 43 c\ 43 48 ) 51 Rd{

T

h

11

(

212

212 ), 212 )(/ 12 f,P)Zd®d2

^, 35

O, 7 T5, 7 P, 4,387 , 212

Part I

Stochastic Differential .. Jumps in Equations with

Martingale Theory and the Stochastic Integral for Point Processes

A stochastic integral is a kind of integral quite different from the usual deterministic integral. However, its theory has broad and important applications in Science, Mathematics itself, Economic, Finance, and elsewhere. A stochastic integral can be completely charaterized by martingale theory. In this chapter we will discuss the elementary martingale theory, which forms the foundation of stochastic analysis and stochastic integral. As a first step we also introduce the stochastic integral with respect to a Point process.

1.1 Concept of a Martingale. In some sense the martingale conception can be explained by a fair game. Let us interprete it as follows: In a game suppose that a person a t the present time s has wealth x, for the game, and at the future time t he will have the wealth xt. The expected money for this person a t the future time t is naturally expressed as E[xt15,], where E[.]means the expectation value of ., 8, means the information up to time s , which is known by the gambler, and E[.lSs] is the conditional expectation value of - under given 5,. Obviously, if the game is fair, then it should be E[xt15sl = xs,W 2 s. This is exactly the definition of a martingale for a random process xt, t 0. Let us make it more explicit for later developement.

>

4

1. Martingale Theory and the Stochastic Integral for Point Processes

Let (R, 5 , P ) be a probability space, {St)t,o be an information family (in Mathematics, we call it a a-algebra f a d y or a a-field family, see Appendix A ) , which satisfies the so-called "Usual Conditions": ( 2 ) 5, c St, as 0 5 s 5 t ; (ii)5t+ =: nh>05t+h. Here condition (i)means that the information increases with time, and 1 gt, condition (ii)that the information is right continuous, or say, as h 10. In this case we call - a a-field filtration. Definition 1 A real random process { x ~ ) , , is ~ called a martingale (supermartingale, submartingale) with respect 15{5t}t,o - , or { x t , &}t20 is a martingale (supermartingale, submartingale), i f (2) xt is integrable for each t 1 0; that is, E lxtl < w , V t 1 0; (ii) xt is gt-adapted; that is, for each t 2 0 , xt is &-measurable; (iii) E(xtlTs] = x,, (respectively, <, 2),a.s. VO 5 s 5 t. For the random process {xt)tE~O,Tl and the random process { x , ) ~ =with ~ discrete time similar definitions can be given.

Example 2 If { ~ t ) , ,is~ a random process with independent increments; that is, VO < tl < t2 7 . . . < t,, the family of random variables {xo, X t l - 2 0 , xtz - X t l , . . . ,xt, is independent, and the increment xt - x,, W > s , is integrable and with non-negative expectation, moreover, xo is also integrable, then {xt)t>o is a submartingale with respect to {$;),Lo, where 5; = a(x,, s 5 t ) , whizh is a a-field generated by {x,, s 5 t ) (that is, the smallest a-field which makes all x,, s 5 t measurable) and makes a completion. In fact, by independent and non-negative increments, 0 5 E ( x t - x S ) = E[(xt - ~ ~ ) 1 5 : ] , V2t S . Hence the conclusion is reached. is a submartingale, let yt := xt V 0 = max(xt,O), Example 3 If then {yt)t20 is still a submartingale. In fact, since f ( x ) = x V 0 is a convex function, hence by Jensen's inequality for the conditional expectation E [ x t V OISs] 1 E[xtlSs]V E[Ol5s] 2 2 s V O,Vt 2 S . So the conclusion is true.

Example 4 If { x ~ } , ,is~ a martingale, then { I ~ t -l ) i~s ,a~submartingale. In fact, by Jensen's inequality E[Ixtl 15,] > IE[xtlSsll = Ix.4 ,W 1 S. Thus the conclusion is deduced. Martingales, submartingales and supermartingales have many important and useful properties, which make them become powerful tools in dealing with many theoretical and practical problems in Science, Finance and elsewhere. Among them the martingale inequalites, the limit theorems, and the

1.2 Stopping Times. Predictable Process

5

Doob-Meyer decomposition theorem for submartingales and supermartingales are most helpful and are frequently encountered in Stochastic Analysis and its Applications, and in this book. So we will discuss them in this chapter. However, to show them clearly we need to introduce the concept called a stopping time, which will be important for us later. We proceed to the next section.

1.2 Stopping Times. Predict able Process Definition 5 A random variable ~ ( w E) [O, co] is called a &-stopping time, or simply, a stoping time, if for any (co >)t > 0, { ~ ( w5) t ) E St. The intuitive interpletation of a stopping time is as follows: If a gambler has a right to stop his gamble at any time ~ ( w )he , would of course like to choose the best time to stop. Suppose he stops his game before time t , i.e. he likes to make ~ ( wI) t , then the maximum information he can get about his decision is only the information up to t , i.e { T ( w ) t ) E The trivial example for a stopping time is ~ ( w=) t ,Vw E 0. That is to say, any constant time t actually is a stopping time. For a discrete random variable T ( W ) E {0,1,2,. . . ,CO) the definition can be reduced to that ~ ( wis) a stopping time, if for any n E {0,1,2,. - . ) , { T ( w )= n ) E since { ~ ( w=) n ) = { ~ ( w ) n ) - { ~ ( w ) n - I ) , and { T ( w ) 5 n ) = UZzl { ~ ( w=) k) . The following examples of stopping time are useful later.

<

zn,

<

zt.

<

Example 6 Let B be a Bore1 set i n R' and {xn)z="=le a sequence of real gt-adapted random variables. Define the first hitting time T B ( W ) to the set B (i.e. the first time that { x , ) ~ = hits ~ B) by T B ( w ) = inf { n : xn(w) E B ) . Then T B ( w ) is a discrete stopping time. In fact, {TB(W)= n ) = n r:; {xk E B C )n { x , E B ) E 8,. For a general random process with continuous time parameter t we have the following similar example.

Example 7 Let xt be a d- dimensional right continuous zt -adapted process and let A be an open set i n Rd. Denote the first hitting time O A ( W ) to A by a A ( w ) = inf { t > O : xt(w) E A ) . Then a A ( w ) k a stopping time. In fact, by the open set property and the right continuity of xt one has that { ~ A ( wI) t ) = n ? = l { ~ A ( w )< t + - 00 u T ~ ~ , r < t +{~ ~/ n( wE)A ) E gt+o = zt,

i)

6

1. Martingale Theory and the Stochastic Integral for Point Processes

where Q is the set of all rational numbers. The following properties of general stopping times will be useful later. ) t ) E 5t, W. L e m m a 8 T(W)is a stopping time, if and only if { ~ ( w<

<

i)

Proof. 3: { ~ ( w< ) t) = UFZ1 { ~ ( w ) t E 5t. +==:{T(w) t ) = flF=l {T(w)< t E $t+o = %t. H

+ i)

<

L e m m a 9 Let a, T, a,, n = 1,2, . . be stopping times. Then (i) a A ~ , a V r , (ii) a = limn,, a,, when a, T or a, 1, are all stopping times. +

<

< <

<

Proof. (i): {a A T t) = {a t) U {T t ) E &, { a v ~ t) ={a t ) n { ~ t) E S t . (ii): If a, f a , then {a t) = flFZl {a, t) E 3t. If a, 1a, then {a < t) = UFZl {a, < t ) E 5 t . By Lemma 8 a is a stopping time. H Now let us introduce a a- field which describes the information obtained up to stopping time T. Set 5, ={A E 5,: Vt E [O,oo),Afl{~(w)5 t) E 5t), where we naturally define that 5, = Vtlo3t, i.e. the smallest a-field ) t, including all Zt, t E [0,m). Obviously, 3, is a a- algebra, and if ~ ( w = then 8, = &.

<

<

<

Proposition 10 Let a,T, a,, n = 1,2, . . - be stopping times. (1) If 4 ~ T(w), ) Vw, then 5, c &. (2) If an(^) 14 w ) , vw, then nF=15,, = 5,. (3) a E 5,. (We use f E 5, to mean that f is 5,-measurable).

<

<

<

Proof. (1): A n {T I t ) = (A n {a t)) fl { r t) E 5t. (2): By (1) 5, c n,"!15,,. Conversely, if A E n~=15,,, then Afl{o, < t ) = ~ ~ ~ ( A n { a , < t - ~ ) ) ~ S ~ , V t 2 0 , V n . Hence A n {a < t) = u,"!~(A n {a, < t)) E &, and A n { a t) = fl&(An {a < t + E &+o = 5t, i.e. A E 5,. (3): For any constant 0 c < oo one has that {a c)n{a t) E StAc c &,so { a < c ) E&. It is natural to ask that if { x ~ ) , , ~ is &-adapted, and a is a stopping time, is it true that z, E go? Generally speaking, it is not true. However, if {xt)tZo is a progressive measurable process, then it is correct. Let us introduce such a related concept.

<

.

<

i))

<

<

Definition 11 An valued process { ~ t ) ~is ,called ~ measurable (respectively, progressive measurable), if the mapping (t, w ) E (0,oo) x Q -+ xt(w) E Rd

1.2 Stopping Times. Predictable Process

7

(respectively, for each t > 0 , ( s , w ) E [0,t] x R -+ xt(w) E Rd) is %([O,m ) )x 5 / % ( R d ) -measurable (respectively, %([0,t ] )x gt / % ( R d ) -measurable); that is, {(t,w) : xt(w) E B ) E %([0,m))x S , V B E % ( R d ) , E B ) E %([0,t ] )x & , V B E % ( R d ) ) . (respectively, { ( s ,w ) : s E [O, t ] x,(w) Let us introduce two useful a-algebras as follows: Denote by P (respectively, 0 )as the smallest a-algebra on [O, oa) x R such that all leftcontinuous (respectively, right-continuous) $t-adapted processes yt(w) E Rd vt(w) : [ o , ~x ) are measurable. P (respectively, 8 ) is called the predictable (respectively, optional) a-algebra. Thus, the following definition is natural. +

Definition 12 A process { x ~ ) , , is ~ called predictable (optional), i f the mapping ( t , ~E )[o,W ) x n -+ x ~ ( wE) R~ is P / % ( R d ) -measurable (respectively 0 / % ( R d ) -measurable). Let us use the notation f E P t o mean that f is P - measurable; etc. I t is easily seen that the following relations hold: f E P + f E 0 + f is progressive measurable + f is measurable and St -adapted. W e only need t o show the first two implications. T h e last one is obvious. Assume that {xt)t>o - is left-continuous, let x: = x $ , as t E [ kF , 2n ), k = 0,1,. - . ; n = 1,2,. . . . T h e n obviously, x: is right-continuous, and by the left-continuity o f x t , x:(w) xt(w), as n co,Vt,Vw. So {xt)tlo E 8. From this one sees that PC 0. Let us show that {xt)tyo E 8 implies that 0 we show {xt)t20 is progressive measurable. For this for each given t that {x,),>~ restricted on ( s ,w ) E [0,t ] x R is %([0,t ] )x 5t- measurable. In fact, without loss o f generality we may assume that {xt)t>o is rightcontinuous. Now for each given t > 0 , let x> x+ as s E [&, k= 0,1, - .. ,2" - 1;n = 1,2,. . . . Then obviously, { x ~ ) , ~is[ %([0, ~ , ~t ]~) x 3t-measurable, so is { x ~ ) , , ~ ~since , ~ ~by , the right continuity o f xt we have that as n -+ m , x y ( w ) -+ x,(w) ,Vs E [O,t],Vw. Let us show the following -+

-+

>

w),

Theorem 13 If { x ~ ) , , is ~ a Rd-valued progressive measurable process, then for each stopping f me a, Z,I,<, is &-measurable. W e will use the compositon o f measurable maps t o show this theorem. For this we need the following lemma. Lemma 14 If fi is a measurable mapping from ( R ,5) to ( R ; ,g:), i = 1,2,. . - ; then f(w) = (f1(w),f2(w), is a measurable mapping from (a,5) to (0;x R i x . . . ,zi x 5; x - . - ).

1. Martingale Theory and the Stochastic Integral for Point Processes

8

In fact, for any Bi

E $,i = 1,2, ...

,f-l(B1 x B 2 x . - . ) = ng1fz:l(~i)

E

5. So f-l(k-7; x x ; x ...) c5. Now let us prove Theorem 13. Proof. Let 52, = {a < co). We need to show that x, is a measurable mapping from ( a u ,3,) to (Rd,123(Rd)). For any given t 2 0 by Proposition 10 a E 5,. So by the definition of 5,, a is a measurable mapping from ({a t) ,St) to ([O,tl,B([O,tl)). Hence by Lemma 14 gl(w) = (a(w),w) is a measurable mapping from ({a t) , S t ) to ([0, t] x 52, 123([0,t]) x s t ) . Note that by the progressive measurability of { x ~ ) ~ g2(s, , ~ , W)= xs(w) is a measurable mapping from ([O, t] x a,123([0,t]) x St) to (Rd,123(Rd)).Hence = g2 o gl(w) is a measurable mapping from ({a t) , z t ) x,(,)(w)I,<, to ( R ~ , % ( R ~ This ) ) shows that for any B E 123(Rd), {x,I,<, E B) n {a t) E ?jt.Since t 2 0 is arbitrary by definition {x,I,<, E B) E 3,.

<

<

<

<

1.3 Martingales with Discrete Time First we will show the Doob's stopping theorem (or called Doob's optional sampling theorem) for bounded stoping times.

Theorem 15 Let { x ~ ) ~ = ~ ,be ~ a, ~martingle ,.. (supermartingle, submartinis, a. .strong . gale), a < T be two bounded stopping times. Then { x ~ ) , = ~ , ~ , ~ martingle (respectively, strong supermartingle, strong submartingle), i.e. E[x71&] = xu (respectivly, 5 , >), a.s. Proof. We only prove the conclusion for the case of submartingle. By assumption there exists a natural number 0 no such that T no. So lxT1<max{1xnl,n=0,1,2,... ,no) < x z o l x n l . So E lxT1< o . By the same manner E Ix,~ < co. Note that by the definition of a stopping time and for A E 5, and 0 n 5 no A~{a=n)fl{~>n}E~~. 1 in addition. Then by the definition of a submartinNow suppose T -a gale SA(xu - xT)dP = C 2 o SAn{o=n)n{T>n)(~n - xn+l)dP I 0. In the general case set Tn = T A ( a n), n = 1,2, . - . ,no. Then all T, are stopping times, and a
<

<

<

<

+

<

zT,,,.

< <

Theorem 16 Let { x ~ ) ~ = ~ , be ~ , a. .submartingale. . Then for every X > 0 and natural number N

1.3 Martingales with Discrete Time

9

Proof. Let us use the first hitting time technique and strong submartingale property to show this theorem. Set u = min{n N : x, 2 A); u = N , if {.) = 4. Then a is a bounded stopping time. By Theorem 15 E X NL Exu = E ~ u I r n a x ~ ~x ,,?, ~< + ~ Ex~lXnaXo-X -A~(mino<~-A). ~,<~ Thus the second inequality is derived.

<

+

<

<

<

<

<

+

+

Corollary 17 1) Assume that {x,),,~,~,... is a real submartingale such that ~ ( ( x : ) ' ) < m,n = 0,1,. . . , for some p > 1. Then for every N, and A > 0, + P AP P(ma% 1,

<

P

E(m=O
(5)

Proof. 1): By Example 3 {x:),=~,~ ,... is a non-negative submartingale. Using Jensen's inequality again one has that { ( x ~ ) ~ ) ~ =is~still , ~ a, .non,, negative submartingale. Hence the first inequality is obtained from Theorem 16. Now if p > 1, set [ = maxo<,=x ) d ~ p-1-+ zN] < [E(X$)P]~/~[E~~](~~)/P. Now if E (tP) = 0, then the second inequality is trivial. I f E (tp) > 0, dividing both sides by [ E [ ~ ] ( P - ~the )IP second , inequality is also obtained. 2): I f {x,),,~,~,... is a real martingale, then by Jensen's inequality we have that { ~ x ~ ~ ) ~ = is a~ submartingale, ,~,... and lxn(+= lx,l. So by 1 ) the conclusions are derived in this case.

6

1. Martingale Theory and the Stochastic Integral for Point Processes

10

In the following we will show the upcrossing inequality for a submartingale, which is the basis for proving the important limit property of a submartingale. First we introduce some notations. , ~ ,an . . interval [a, b], where For a real &- adapted process { x ~ ) ~ = ~and b > a , let 71 =min{n 0 : x, < a ) , 7 2 = min{n > 71 : x, b),

>

......

= min{n 72n+2 = min{n

72,+l

...... .

>

> 72, : x, < a), > 72n+l : xn > b),

I

where we recall that min 4 = +co.Then (7,) is an increasing sequence of stopping times. In fact, Vk 2 0, {TI = k) = {XO > a, x1 > a , . . . ,xk-1 > a, xk a ) E Sk; ( 7 2 = k) = U =: {TI = j, 7 2 = k)

<

=::u ;

(71 = j, x j

{73 = k)

=

:;u

< a, X j + l

(72

,xk-1 < b, xk 2 b)

< b,

€

8k;

= j, 73 = k)

= U : Z : { T ~ = ~ , X ~ >bb,xj+l > a , . . - , x ~ - I> U , X I , < U ) E ~ ~ . Hence r 1 , ~ 2 and , 73 are stopping times. The proofs for the rest are similar. Now set ~,b[x(.), N](w) = max {k 2 1 : T ~ ~ ( WN) ) , D: [x(.),N](w) = max {k 2 1 : T ~ (w)~ -N ) ~. Obviously the first one is the number connected to the upcrossing of {x,),=,N for the interval [a,b], and the second one is the number connected to the ~ the interval [a, b]. downcrossing of { x , ) ~ = for

<

<

Proof. For a submartingale { x , ) ~ = by ~ Example 3 one sees that { Y n ) ~ == o {(x, - a)+):=o is a non-negative submartingale. Clearly ~;-"[y(.), N](w) = U,b[x(.),N](w). Again define 7 1 , 7 2 , . . as above, but with x, a, and b replaced by y, 0, and b - a respectively. Then if 2k > N k E(YN-YO)= EC?=~(Y~,,AN-YT~-IAN) = ECn=1(~7znAN-~Tzn-1AN) k-1 + C n = oE ( ~ ~ z -nY+T ~~ ~~A2N~( b) - a ) ~ @ - ~ [ y ( . ) , N ] , where we have used the fact that {yn}F=o is a submartingale, and hence -~y T Z n A ~ 0; ) and a strong submartingale (Theorem 15), so E ( y 7 2 n + l A y, 2 0, Vn. The first inequality is proved. Now observe that 0 2 E(YTZ,.AN- Y ~ ~ + ~ A N ) - E [ ( ~ -~~72n+lh~)(I72n1N
>

1.3 Martingales with Discrete Time

11

= E[(b - a - Y N ) ~ T Z , , < N
{~,)r=~

+

+

+

+

Proof. Let yn = -x,. Then {y,)r==, is a submartingale. Hence u M . > ,Nl = DIba[Y(.), Nl, and D m . ) , Nl = UZb"[Y(.>,Nl. Applying Theorem 18 we arrive a t the results. w Theorem 18 and Corollary 19 are the classical crossing theorems on martingale. We can derive some other useful crossing results which are very useful in the mathematical f i n a n ~ e . [ ~ ~ ] ~Here [ ~ ~ we ] ~ [apply ~ * ~ some ] of them to derive the important limit theorem on martingales.

1. Martingale Theory and the Stochastic Integral for Point Processes

12

Theorem 20 If {x,)~'~ is a submartingale such that there exists a subsequence of {n) , denote it by {nk) , such that

then x, = limn,,xn exists a s . , and x, is integrable. In particular, i f x, O,Vn, then condition (1.1) is obviously satisfied, and i n this case Qn E[~ml5n2 ] xn, a.8.

<

Proof. First, clearly condition (1.1) u supnEx: < oo ++ sup,Elxn) < oo. In fact, by the properties of submartingales one has that EX: 5 Ex&, Vlc. and E lxnl = 2Ex: - EX, 2Ex: - EX^. Hence the equivalent relations hold. Now let U,b(x(.)) = lim~,, U,b(x(.), N). Then by Theorem 18 EU,~(X(.))I SUPNE(XN- a)+ < m. Hence U,b(x(.)) < oo, a.s. Let W = Ua,b€Q,a
<

&

<

>

1.4 Uniform Integrability and Martingales It is well known in the theory of real analysis that if a sequence of measurable functions is dominated by an integrable function, then one can take the limit under the integral sign for the function sequence. That is the famous Lebesgue's dominated convergence theorem. However, sometimes it is difficult to find such a dominated function. In this case the uniform integrability of that function sequence can be a great help. Actually, in many cases it is a powerful tool .

Definition 21 A family of functions A c L1(R,5, P ) is called uniformly integrable, if limx,, s u p f , ~E(f Il = 0, where L1(R, 5,P ) is the totality of random variables <, (that is, all J are 5-measurable) such that E IJI < m.

1.4 Uniform Integrability and Martingales

13

Lemma 22 Suppose that {x,)r==, c L1(R, 5,P ) is uniformly integrable, and as n + oo, x, -+ x, i n probability, i.e. VE > 0, limn,, P(lx, -XI > E) = 0, then limn,, E 12, - X I = 0. (i.e. x, 4 x, i n ~ l ( R , 5P)). , In particular, limn,, Ex, = E x Proof. In fact, VE > 0, E Ixn - 21 E(1xn - XI ~ l x n - x l +E(lxn > ~ ) - X ( I l x , - x l < ~= ) I:" +I;". Hence one can take a X large enough such that I:" < e/2, since clearly {x, - x)rZl is uniformly integrable. Then for this fixed X by using Lebesgue's dominated convergence theorem one can have a sufficiently large N such that as n 2 N, I:" < ~ / 2 . For the sufficient conditions of uniform integrability of a family A we have

<

Lemma 23 Suppose that A c L1(S1,5, P ) . Any one of the following conditions makes A uniformly integrable: 1) There exists an integrable g E L1 (R, 5,P ) such that 1x1 1 9 , Vx E A. 2) There mists a p > 1 such that SUPxE~ E lx(w)IP < oo. Proof. 1): Since as X + oo > A) 5 f SUPXEAE~X(W)I 5 ;EI~(w)I SUP,EAP(~X(W)I So by the integrability of g one has that as X -, oo E lx(w)l Ilx(w)l>x 5 E ~ ( W ) I ~ , (-+~ 0, ) ~uniformly >~ w.r.t. x E A. E Ix(w)I --,0, as x 4 oo. SO 2): Since SUPxE~ P ( ~ X ( W>) ~A) 5 SUPXE~ asX4oo E ~x(w)IIIX(W)I>X SUPXEA 0. w 1S U P , ~ ~ Ix(w)I~)~IP (E S U P , ~ ~ [ P ( ~ X> (W ~)](p-')/p )I Now we know that the uniform integrability condition is weaker than the domination condition. Actually, it is also the neccessary condition for the L1 -convergence of the sequence of integrable random variables or, say, integrable functions. Theorem 24 Suppose that {x,):=~ c L1(S1,5,P). Then the following two statements are equivalent: 00 1) {x,),=~ is uniformly integrable. ,... E lx,( < oo; and VE > 0,36 > 0 such that VB E 5, as 2) P(B) < 6 SUP,,^,^, ... E 1xnI IB< E. Furthermore, if there exists an x E L1(S1,5, P ) such that as n -+ oo, x, --, x, i n probability; then the following statement is also equivalent to 1): 3) xn + x , i n L1(S1,5,P).

1. Martingale Theory and the Stochastic Integral for Point Processes

14

Proof. Since 1) ==+ 3) is already proved in Lemma 22, we will show that 2) ==+ 1) *2). 1) ==+ 2): Take a Xo large enough such that sup,,,,,,... E lx, 1 Ilx,l>Xo< 1. Then SUP,,^,^, ... E 1xnI I XO 1. On the other hand, for any B E 5 since E 1xnl IB= E 1xn1I { ~ X , ~ > X+) E ~ B1xn1I{1x , I~) XP(B) = I: QB. Hence VE > 0 one can take a A, > 0 large enough such that I? < 5 , then let 6, = &. For this 6, > 0 one has that VB E S,P(B) < 6, j SUP,,^,^, ... E IxnI IB< E. 2) =+ 1): VE > 0 Take 6 > 0 such that 2) holds. Since , 2E , IxnI. P(IxnI > A) I S ~ ~ n = 1... Hence one can take an N large enough such that as X > N, P(lx,l > A) < 6, Vn = 1,2,.. . . Thus by 2) as X > N, E Ixn(Ilxnl>X < E, Vn = 1,2,. . . . 3) & 2): Take an No large enough such that as n > No, EIx-x,l < 1. Thus ,... E IxnI I maxi1 E 1x1, E 1x11,.. . ,E 1 ~ ~ <~ co. 1 ) On the other hand, observe that E lxn(IB E 1% - XI E 1x1IB. Hence VE > 0, one can take an N, large enough such that as n > N,, E I x - x n l < f. Then take a 6 > 0 small enough such that VB E 5, as P ( B ) < 6, maxn=l,...,N={E lxnl IB)< E, and E 1x1 IB< ,512. Thus as P ( B ) < 6, E lx,l IB< E, Vn = 1,2, - - . . rn Now let us use uniform integrability as a tool to study the martingales.

3)

*

+

+

<

+

+

+

<

Theorem 25 If {x~):=~ is a submartingale such that { ~ , f ) z =is ~ uniformly integrable, then x, = limn,, x, exists, a.s., and E[xwl5n] 2 X n , Vn, i.e. { x , ) , ~ ~ , ~,,., , ,,w is also a submartingale, and we call it a right-closed submartingale.

This theorem actually tells us that a uniformly integrable submartingale is a right-closed submartingale. Proof. By uniform integrability one has that supn=o,l,2,... Ex; < 00. Hence applying Theorem 20 one has that x, = limn,, x, exists, a.s. Now by the submartingale property { ~ , f ) ~is =also ~ a submartingale (Example 3). Hence for any a > 0, and B E 5,, as m 2 n, SB[(-a) xn]dP SB[(-a) x,]dP. Letting m -+ co by the uniform integrability of {x,+):=~ one has that

<

1.4 UniformIntegrability and Martingales

15

JB[(-a) v xn]dP < JB[(-a)v xm]dP. Now letting a t ca by Fatou's lemma one obtains that JB xndP 5 JB x,dP = JB E[xml5n]dP,VB E 5,. The conclusion is established. We also have the following inverse theorem. Theorem 26 If { x , ) , = ~ , ~ ,.,., ,~ is a submartingale, where x , = limn,, x, exists, a.s., ,,,., is integrable. then { x ~ ) , = ~ ~ uniformly ,~ Proof. By Jensen's inequality { x $ ) , = ~ ,...,, , ~ , ~ is also a submartingale. Now VX > 0, denote By = { x i > A), then by the submartingale definition asX-+ca P(B;I) < *EX$ < *Ex& 4 0, uniformly w.r.t. n. Therefore as X 4 ca JBj: x$dP JB j: x&dP 4 0, uniformly w.r. t. n.

<

Corollary 27 1) If {x,),=~,~,~,... ,, is a right-closed martingale, then one has that { x , ) , = ~ , ~, ,. ~is uniformly integrable. 2) For a sequence of random variables {yn)n,o,,,2,,.,if there exists a z E L1(R,5,P ) such that lynl 5 E ( ( z (I5,), where 5, c Sn+l c 5 are a-fields, then {yn)n,o,l,2,... is uniformly integrable. Proof. 1): I t can be derived from Theorem 26, since {xn)n=o,l,2 ,...,, is both a submartingale and a supermartingale. 2): In fact, let z, = E(zl3,). Then, obviously {zn)n=0,1,2,...is a martingale, and SUP,,^,^,^ ,... E I z , ~ < E lzl < ca. Hence by Theorem 20 z, = lim,,,z, exists, a.s., z, E L1(R,5,P), and z, = E(z,15,),Vn. Therefore {zn)n=0,1,2,.., ,, is a right-closed martingale. Now by 1) it is uniformly integrable, so is { ~ n ) , = ~ , ~ ., ~fl, . . . Condition in 2) for the uniform integrability is weaker than the usual Lebesgue's dominated condition. Moreover, from the proof of 2 ) in Corollary 27 one also can obtain the following thoerem Theorem 28 (Levi's theorem) If z E L1(R,5,P ) , and 3, C c 5 are a-fields, then as n t co,E[zlzn]-+ E[zl5,], where 5, = Vn=l,2,...5,, i.e. 5, is the smallest a-field including all gn, n = 1,2, . . . . Proof. By the proof of 2) in Corollary 27 one already has that z, = limn+, z, = limn,, E[zlS,] exists, a.s., z, E L1(R,5,P),and z, = E(zWl5,),Vn. Let us show that z, = E[zl5,], a.s. In fact, by limit one has z, E 5,, i.e. it is 5,-measurable. Moreover, Vn,VB E 5,, Ez,IB = EznIB = E ( E [ z ( ~ ~ = ] IE(E[~IBIS,]) B) = EZIB =E E [ ~ I ~ ~ ] I B . From this one also has that VB E 5, = Vn=l,z,...Sn, Ez,IB = E E [ Z ~ ~ , ] I B .

16

1. Martingale Theory and the Stochastic Integral for Point Processes

Since z, and E [z15,] both are 8, -measurable. Hence z, = E[zl&,], rn

a.s.

+

Now let us consider the discrete time {. . . , -5, -5 1,. - . , -2, -1,O) with right-end point 0 but without the initial left-starting-time. We still call {x,),~{... ,-k,-k+l ,,.,,-2,-1,0) a martingale (supermartingale, submartingale), if (i) x, is integrable for each n = 0, -1, -2,. . . ; (ii) x, is &-adapted, i.e. for each n = 0, -1, -2,. . . , x, is $,-measurable, where &, n = 0, -1, -2, - .. , are a-fields still with an increasing property, i.e. 8, C ~ m , a s n I m ; V n , m {... ~ ,- 5 , - 5 + 1 , . . . ,-2,-1,O); a.s. W I n I m;Vn,m E (iii) E[x,lzs] = x,, (respectively, 5 , {... ,- 5 , - 5 + 1 , . . . ,-2,-1,O). We have the following limit thoerem.

z),

Theorem 29 If {x,),,{... ,-k,-k+l ,...,-2,-1,0) is a submartingale such that infn,0,-l,-2 ,... Ex, > -oo, then {x,),,{ ... ,-k,-k+l ,.. ,-2,-1,0) is unifomly integrable, x-, = limn,-, x, exzsts a.s., and as n --+ -00, x, --+ x-,, in L1(fl,5,p). Proof. For each N consider the finite sequence of random variables {x,),=-~ -N+l ,...,-1,0 . Denot by U,b[x(.),-N] the number of upcrossing of { x , ) , = - ~ , - ~ + ~ ,... for the interval [a,b]. Then by Theorem 18 EU,b[x(.),-N] 5 &E(XO - a)+, and EU,~[X(.)] 5 &E(XO - a ) + < oo, where U,b[x(.)] = lim~,, ~,b[x(.), -N]. By the proof of Theorem 18 one has that x-, = limn+-,x, exists a.s. However, it still remains to be proved that x-, is finite, a.s.. Let us show that {x,),,(... -,,,,., ,-2,-l,o) is uniformly integrable. If this can be done, then all concfusions will be derived immediately by the property of uniform integrability. Observe that Exn 1, as n 1, since { ~ n ) , ~ { .,-k,-k+l .. ,... ,-2,-1,0) is a submartingale. Hence by assumption a finite limit exists: infn,o,-l,-2 ,... Ex, > -oo. limn,-, Ex, Note that {E[~ol3n]),~{... ,-k,-k+l,... ,-2,-1,0) is a martingale and uniformly integrable. Hence {x, - E[x015,]),,{... ,-k,-k+l ,... ,-2,-1,0) is a non-positive submartingale, and the uniform integrability of it is the same as that of { ~ n ) , ~ { . .,-k,-k+l . ,.. ,-2,-l,o) . So we may assume that xn I 0. Now VE > 0, Ex, < E . Then by the take a -5 large enough such that Exk - limn,-, property of submartingales and the property that Ex, 1as n 1, P(xn < -A) 5 +Elx,l = i ( 2 ~ x R- Ex,) 5 - limn,-, Ex,) -+ 0, uniformly w.r.t. n, as X --+ oo. and if n k 0, 0 I E[(-xn)Ixn<-~] I -Exn - E[(-xn)Ix,,>-~] I -limn+-, Ex, E[xkIXn2-~] 5 EX^ E[X~IZ,,~-A] E = E[(-xlc)IX,<-~] E .

,+,

>

i EX: < <

+

+

+

+

1.5 Martingales with Continuous Time

From this one easily derives that {x,),~(... is uniformly integrable.

,-k,-k+l

,...,-2,-1,0)

17

with x, I 0

1.5 Martingales with Continuous Time Now let us consider the martingale (submartingale, and supermartingale) {xt)tE[o,oo)with continuous time. First, we will still introduce the upcrossing numbers of a random process for an interval. Let { x ~ ) ~ ~ = >{xt)tEIO,m) be an adapted process, and U = {tl,t 2 , .. . ,t,) be a finite subset of R+ = [0, 00).Denote its rearrangement to the natural order by Isl, s2,. - - ,s,) , i.e. sl < s 2 < .. . < s,. Let ~,b[x(.), U] be the number of upcrossings of {x,,);=~ for interval [a,b], and we also call it the number of upcrossings of {xt)tEU for interval [a,b]. For any subset D of R+, define ~,b[x(.), D] = sup {~,b[x(.), U] : U is a finite subset of D) . In case D = {tl, t2, . - . ,t,, . - . ) , obviously U,b[x(.),Un], U,b[x(.),D] = limn--+m where U, = {tl, t 2 , . - . ,t,) . By using the results on discrete time we have the following theorem.

Theorem 30 If {xt)t>o is a submartingale, D = {tl, t2,. . . ,t,, for a n y 0 5 r < s , a < B and X > 0 one has that +2E(~z), Xp(suptGD,[,,] lxt 1 > -A) 5 EU,b[x(.),D n [r,s]] &(E[(x6 - a)+ - (xr - a)+).

. . . ) , then

<

Proof. Set 6 = {r,s,t l , t2,. . . ,t,, . . . ) Notice that un = ({r, S, t l , t2, . . ,tn) n j , 4 ) T (6n [r,s]). So the conlusions for the set D U [r,s] are derived by applying Theorem 16 and 18 and taking the limit for n -+ co.However, ( D n [r, s]) c ( 6 n [r,s]). So the two conclusions for ( D n [r, s]) hold true. Now we can generalize the limit theorem to submartingales with continuous time. Theorem 31 Let be a submartingale. Then ?& = limrlt,rEQxr exists and finite a.s. ana {2t)t>o is still a submartingale such that Zt is right continuous with left-handlimits a s . Furthermore, xt I st a.s. for W 2 0. (Recall that Q = the totality of real rational numbers). To establish the above theorem we divide it into two steps. The first step can be written as the following lemma.

Lemma 32 If {xt)t>o is a submartingale, then Vt 2 0 l i m , ~ t 2, , ~ and ~ ~l h l t , r E Q xr exist and are finite a s . (Here, we define XO- = xo, and 80-= 80).

18

1. Martingale Theory and the Stochastic Integral for Point Processes

Proof. Denote Q = {tl ,t2, . . . ,t,, - . . ) . Notice that for any given T > 0 if w E 0 is such a point that there exists a t E [0,T] such that one of the limits: limrpTEQxV(w)and limrLttrEQ x,(w), does not exist, then there will exist a < b such that U,b[x(.),Qn [O,Tll(w) = On the other hand, if one of the above limits even exists but it is not finite, then S U P t ~ ~ n [I ox t, (~~]) I= However, by Theorem 30 one has that VT > 0, and for all a, b with a < b EHT,a,b = 01 where HT,,.~= {@[x(.), Q n [O, TI] = m ) U { s u ~ t , ~ n p1% ,~ 1 =~ m ) . Hence the conclusion can be established. The second step to show Theorem 31 is as follows: Proof. By the above lemma Zt = lim,lt,,EQxr exists and are finite a s . Obviously, it is right continuous with left-hand limit. Since Zt is 3t = &+o-adapted and {xt)t>o is a submartingale. Hence Yen 4 0, { x ~ + , , ) ~ ~ ~ is uniformly integrable by Theorem 29, and EPtI~ = limn,, IB limn+, Exsfan IB= E ~ J B , for s > t and B E S t . This shows that {2t)t20is asubmartingale. Moreover, since EZtIB = limn,, Ex~+E,IB 2 E x ~ I B , V BE St. SO xt ??t a s . for Vt20.

<

<

Corollary 33 Under the assumption of Theorem 31 one has the following conclusions: E(xt) is right-continuous, Vt 0. 1) P(xt = &) = 1, Vt 2 0 2) If {xt}tlo is a martingale, then P(xt = Zt) = 1,W 2 0. So {2t}t>o - may be called a right-continuous version of { x ~ )-, > ~ .

>

Proof. 1): +==: By the uniform integrability of {~t+~,,):=~ ,where en 10, and the right-continuity of E(xt) one has E$ = limn,, Ext+En = E(xt). Since xt 5 Pt, a s . Hence xt = Pt, a s . ===+: By assumption and the uniform integrability one has that Ye, 1 0 , Ext+€,. E(xt) = E& = E limn+ooxt+€, = limn,, So E(xt) is right-continuous. 2): Since in this case E(xt) = E(xo),Vt 0. The conclusion is established. w F'rom now on we consider right-continuous martingales (supermartingales, submartingales) only. The martingale inequalities, Doob's stopping time theorem (Doob's optional sampling theorem) now are naturally generalized to the continuous time case.

>

T h e o r e m 34 1) If { x ~ ) , , ~ is a real right-continuous submartingale such that ~ ( ( x : ) ~ ) < m , t 2 ,for some p 2 1, then for every T > 0, and X > 0,

1.6 Doob-Meyer Decomposition Theorem

19

P(suP~E[o,T] x f 2 A) 5 E( ( x $ ) ~ ) I x ~ , and if p > 1, then P

(5)

E ( s u ~ t c a ~(x:)'r) 5 E((X$)~). 2) If {xt)t20 is a real right-continuous martingale such that E((xtlP) < oo,t 2 0, then the conclusions in 1) hold true for x t and x$ replaced by lxtl and lxTl, respectively. Theorem 35 (Doob's stopping time theorem). Let {xt),,, be a real rightcontinuous martingle (supermartingle, submartingale) w i a respect to {St), and be a family of bounded stopping times such that P ( a t 5 a,) = 1, if t < s. Then {xt)tlo is a strong martingle (respectively, strong supermartingle, strong submartingle), 2.e. as t < s, E[X,~~S,~] = x U t (respectivly, 5, 3,a.s.

In Theorem 34, 1) can be proved by using the same technique as the one in Theorem 30, and 2) is derived from 1). To show Theorem 35 let us use the standard discretization technique. Proof. We show the conclusion for martingales only. For any bounded stopping time at in the given family let ar=&,asat~[~,&),n,k=1,2,..-. Then for each n, ~7 is a discrete stopping time. In fact, {a? = = k: 5 at < 2;;) E S*. Moreover, as n t oo,ar 1at. By Theorem 15 one has that 'dB E Satc So;, as t < s, EX,: IB= EX,; IB. Moreover, since a? 1 and is bounded, if we let y-n = xu;, S-n = 8,; then {yn)n=-l -z ... with the a-field family {Sn}n=-1,-2 ,... obviously satisfies all conditio& in Theorem 29. In fact, clearly it is a submartingale by Theorem 15, and inf, Eyn 2: Exa > -CO. So {yn)n,-l,-q,,. is uniformly integrable. Hence by the right-continuity of Ex,: IB= xt one has that limn,, Ex,; IB= Ex,, IB.Similarly, limn,, Exv8IB. Therefore, V B E Sot, as t < s, Ex,.I,q = ExatIB. ~ martingale. rn So { x ~ ) is, a~ strong

(v

&)

1.6 Doob-Meyer Decomposition Theorem In the incomplete financial market to price some option will involve the problem connected to the Doob-Meyer decomposition of some submartingales or supermartingales. Besides, this decomposition theorem is also a fundamental tool in stochastic analysis and its applications. First we will look at the discrete time case.

20

1. Martingale Theory and the Stochastic Integral for Point Processes

Theorem 36 Let {x,),=~,~,~,... be a submartingale. The there exists a unique decomposition such that x, = M n + A n , n = 0 , 1 , 2 , . . . , where {Mn)n,o,l,2 ,.., is a martingale, and {An)n,o,l,z ,... is an increasing process, both are {5n)n=o,l,2 ,... -adapted, and {An)n=o,l,2,...is predictable, where predictable means that A, E Zn-l, Vn = 1,2,. . . , and A. = 0. Proof. Uniqueness. If x, has the above decomposition, where A, E then it is easily checked that it will satisfy A. = 0, A, = A n - l + E [ x n - x , - l l ~ n - l ] , n = 1 , 2 , - . . , and M, = X, - A,. So the decomposition must be unique. Existence. Now let us define {An)n=0,1,2, . and {Mn)n,o,l ,,,... as above. Then obviously {An)n,o,l,2,... is increasing in n and predictable. Moreover, {Mn)n=o,l,2,,..is a martingale. In fact, E[Mnl3n-l] = E[xn-AnISn-l] = E[~n-(An-1+E[~n-~n-115n-1])15n-1] - ~ ~ - An-1 1 - = M,-l. Hence we have a decomposition for x,. m In the continuous time case the situation is much more difficult. Let us introduce some neccesary related concepts first. In the following we fix a probability space (fl,5,P ) with an increasing a-field family {St)t>o , C 3,Vt 2 0. (Such a {&}tlo is called a reference family). All martinggles, adaptness, stopping times, etc. are relative to this reference family. Now let us introduce the first concept needed for DoobMeyer decomposition theorem. For 0 < a < co let Sa be the set of all stopping times a such that u < a, a s .

st

Definition 37 A submartingale {xt),,, is said to be of class (DL) (class (D)), if for every 0 < a < co (for a =-co), the family of random variables {xu : a E Sa) is uniformly integrable. Clearly, (D) c (DL). Recall that we only consider the right continuous martingales (supermartingales, submartingales).

Theorem 38 1) Any martingale {xt)t>o is of class (DL). 2) Any uniformly integrable m a r t i n g a l e ~ ~ t ) ,is, ~of class (D). 3) Any non-negative submartingale { ~ t ) -, , ~iB of class (DL). Proof. 1): By the optional sampling theorem (Theorem 35), for any O A) S U P ~ ~ SE, lxul /A 5 E lxal /A -+ 0, E Ixol Ilx,l>X E Ixal Ilx,l>X-' 0, uniformly w.r.t. u E S,, where we have applied Jensen's inequality to show that { I X ~-/ ) is ~> a sub~ martingale.

<

<

1.6 Doob-Meyer Decomposition Theorem

21

2): Since { x ~ )-, >is~a uniformly integrable martingale, so by Theorem 25 one can derive that x, = limtEQ,tT, xt exists, x, E L1(R,5,P), and xt = E [ x , I ~ ~ ] , V ~0. Hence by using the standard discretization technique as in the proof of Theorem 35 one also has that Xu~jv= E[x,lSu~~],b'uE S,,VN = 1,2,. . . . Letting N f m and applying Levi's Theorem one obtains x, = E [ X , I ~ ~ ] , V E~ S,. By this the uniform integrability of {xu : u E S,) is easily derived. (See Corollary 27). 3): Since lxtl = xt 2 0, the proof in 1) goes through. The second important concept we need for the Doob-Meyer decomposion theorem is the natural increasing process.

>

.

is called an Definition 39 1) A right continuous random process is {&}t20 -asdaited, (ii) A0 = integrable increasing process, if (i) O,A, < A t , a . s . , a s s < t, (222) EAt < mTW 2 0 . 2) An integrable increasing process {At),,,, - is called natural, if for every bounded martingale {Mt),,,E J , ~ M , ~ A=EJ,~M,-~A,, , W E [o,oo).

Lemma 40 The following statements are equivalent: 1) The integrable increasing process - is natural. 2) For every bounded martingale {Mt)t>o EMtA, = E Ma-dA,, W E [0, w).

~,t

Proof. In fact, let 0 = to < tl < tz < - . . < t, = t. Then E M A t = EL=,E[Mt(Atk -At,-, )I = EL=,E[(At, - At,-, )E(Mt15tk)l = C;=l'EIMtk(Atk - Atk-l)] ES;Msd~8> as maxllk2n(tk - tk-1) 0The following are examples of natural processes. -f

.

Example 41 1) Any continuous integrable increasing process - is natural. is natural if and only if it is 2) An integrable increasing process predictable. Proof. 1): Any martingale has a right continuous version (actually, as we said before, we need only consider the right continuous martingales, supermartingales, and submartingales). However, for a right continuous function it can only have at most countable discontinuous points. Now At MsdA, = is continuous. So we have that I M ~ ~ M , - ~=A0., Hence M,-~A,, a.s. 2): This proof needs more information, so we refer to Theorem 5.33 in [431 However, here we can give some intuitive view to see why it is possible that this is true. From 1) it is known that if At is left continuous, then for any

sot

$

1. Martingale Theory and the Stochastic Integral for Point Processes

22

bounded martingale {Mt)t10

,

In particular for any right continuous martingale {Mt)t,o which is a simple process, i.e. there exists a sequence of times 0 = to 7 tl < t2 < - .. < t, < - . . and a sequence of random variables {[n)r=l ,(, E st,, such that Mt = En, as t E [t,, t,+l),n = 0,1, - .. ; one has that (1.2) is true. Fix such an Mt temporarily. Notice that the predictable u-field is generated by left continuous st-adapted process, and we have already known that (1.2) is true, if At is left continuous. So we can derive that (1.2) is still true, if At is st-predictable. Finally approximating the bounded right continuous martingale {Mt)t,o by the simple processes we obtain that (1.2) is true when At is st-pr6dictable. Conversely, suppose that At is st-predictable. We notice that At E ~t-,becauseit is true when At is left continuous, hence this is so when At is st-predictable. From this we intuitively have E J M,dA, ~ = E J E[M,dA,IS,-] ~ = E J M,-dA,. ~ Beware of the fact that the last equality is not rigorous. Now suppose that a submartingale { ~ t ) ~-, , , can be expressed as

.

where {Mt)t.,o is a martingale, and is an integrable increasing process. ~ h e n { x ~ ) , is, ~of class (DL). In faTt, by Theorem 38 {Mt)t,o - is of class (DL), and Gbviously by 0 5 A, 5 A,, as a E S,, {At),,, is of class (DL) too, so is { x ~ ) , , ~ .Conversely, we have the following so-called Doob-Meyer decomposition theorem: is a submartingale of class (DL), then there exists Theorem 42 If a unique expression (1.3) for { x ~ ) ~ where , ~ , {Mt),Lo is a martingale, and {At)tlo is a natural integrable increasing process. We call this decomposition a D-M decomposition.

+

+

Proof. Uniqueness. If xt = Mt At = M,' A: are two such decompositions, then At - A: = Mt - M,' is a martingale. Let 0 = to < t l < tz < - .. < t, = t, and IAl = maXk,l, ..., {tk - tk-1). Suppose that E is a bounded random variable. Then mt = E[(lst]is a bounded martingale. So E J: m,-d(A, - A',) =

l i m l ~ l + ~mtk[(Atk+l ~ [ ~ F-~A;k+l) i - (Atk -

= limlal-+o E C~I; mtk[E(Atk+l- A;k+,[st,) - (Atk - A&)] = 0, since At - A: is a martingale. By the naturality property one has that EmtAt = EmtA:. So EEAt = EEA:. Hence for each t, At = A:,a.s. Now by the right continuity of At and A: one derives that At = A:, for all t 2 0,a.s.

23

1.6 Doob-Meyer Decomposition Theorem

Existence. By uniqueness we only need to show the result holds on interval [0,a], for every a > 0. Set yt = xt - E[xa13t],t € [0, a]. Then y, = 0, a.s. and {yt)tE[o,alis a non-positive submartingale, since {xt)tE[O,alis a submartingale and hence xt E[x,I$~],W E [O,a]. Obviously, xt has a D-M decomposition ++-yt has a D-M decomposition. However, yt has nicer properties. Now it is natural to discretize yt and to use the D-M decomposition (Theorem 36) for the submartingale with discrete time, then to take the limit and get the general result. Let A, be the partition 0 = tg < ty < , .. < ?t& = a of [0,a] by ty = ja/2". Then by Theorem 36 for each n, {yt;) j=o,l ,...,2n has a D-M decomposition yt? = M e +A??, j = 0 , 1 , . . . ,2,, where

<

In what follows we are going to show that as n --+ CQ, will have a limit A,. Noice that if yt = Mt At holds, then by 0 = Ma A, one should have At = yt EIAalgt]. This means that for given yt from A, one can define At, and then define Mt. So, possibly, the problem is solved.

+

+

+

Lemma 43 {A~),=,,,,... is uniformly integrable. Proof. Suppose that t = t F l E A,. By (1.4) A? = A?7-1 + E[ytjn- Yt?-,13t?-ll, 2" take the summation ClcZj AT; then take the conditional expectation E[-lSt] on both sides and notice that y, = ytyn = 0 as a result one obtains that vt E A, E[A:I3tl = A: - yt. (1.5) Now let A > 0 be fixed and set a: = inf tz-l : AYE > X ; a! = a, if {.) = 4. Then a: E S,. In fact, {a;=t?-,) = {A?? < x , ~ = o , I , . . ., ~ - I ; A ? ; > A ) ES~;-,, where we have used the result that by (1.4) A?; E St;-,. By Doob's stopping time theorem ;: - YU; . E[At13u;] = A (1.6)

{

}

Noting that {A: > A) C {a: < a ) . On the other hand, by the increasing property of A? one also has that {A: > A) > {a: < a ) . So {A: > A) = {a! < a ) . Hence EAu;Iu;<, I -Ey,; I,;., XP($ < a). EAEIA;>A = -EY,;L;<~ Notice also that by (1.6) -E~u;/,.To;,,
+

+

1. Martingale Theory and the Stochastic Integral for Point Processes

24

where we have used the result that A; > A, as ol < a; and A,;/= I $ Therefore, EAEIA;>A I -Eyu;Iul A) EAEIX -Eyo/X + 0, uniformly w.r.t. n, where we have used the result that by (1.5) EAE = -Eye. Similarly, one also has that as X -+ oo, P(al12< a ) + 0, uniformly w.r.t. n. So applying the fact that {yO)~ES,is uniformly integrable, one finds that as X --+ oo, EA;IA;>A --+ 0. That is, {AE)n,l,2,.,, is uniformly integrable. rn However, for the uniform integrability family we can quote the following lemma.

<

<

Lemma 44 12& p292 A subset K of L1(O,z,P) is uniformly integrable if and only if it is weakly sequentially compact, i.e. for any sequence {f,) c K, there exists a subsequence {f,,) C { f,) and a f E L1 (R,5,P) such that as k W, E(gfnk) E(gf),vg 'g LW(%S,P). +

-+

Now let us return t o the proof of Theorem 42. Proof. By the above two lemmas there exists a subsequence {A~k}kk=1,2,... of {AE}n=1,2,... and a A, E L ' ( R , ~ , P )such that as k -+ oo, E ( g A 2 ) + E(gAa),Vg E Lw(O, 5 7 ) . Let At = yt E[A,l&], t E [0, a], where we always take the right continuous version of martingale EIAalzt]. Recall that yt = xt - E[x,l$t], t E [0, a]. Then xt = At E[x, - A,[&]. So we have a decomposition for xt. Obviously, At is integrable, since yt and A, are integrable. However, to be a D-M decomposition we still need to prove: 1) At is increasing in t; 2) At is natural. 1): Since by (1.5) one has that A: = yt E[AElzt], t E A, c A,, as n 2 m. Hence one can show that as k -+ oo, E(gAyk) --+ E(gAt),Vg E Lw(O,z,P),t E A,,Vm. In fact, a s k -too, E(gE[A? 13t1) = E(E[gl3tlE[A? l3tl) = E(E[9l3t1AEkl) E(E[gIiTt]Aa]) = E(gE[AaIStI). Moreover, by (1.4) one sees that A? I A,! as t I s, and t, s E A,, n 3 m. Therefore, one can easily show that At I A,, as t I s, and t, s E A,,Vm. By the right continuity of At it follows that At is increasing in t. 2): For each t E [0, a] if mt is a bounded martingale, then E m , - d ~ . = limn,, E [ m t ; ~ t ( A t ; + ~~ tAt;~t)]

+

+

+

-+

x:lil

xilil~ [ m t ; ~ t ( ~ k +-~YcA~)] At = limn.-+, x:lil E[mt:~t(A?;+~,,~ - A?;At)l = limn-, xiLilE[mt(Ar;+lAt - A?;,,)] = limn,, = limn-w

Therefore At is natural.

EmrA: = EmtAt.

1.6 Doob-Meyer Decomposition Theorem

25

Corollary 45 If {x~)~.,~ is a submartingale of class (D), then {Mt)t,o - in the D-M decomposition-is uniformly integrable. Proof. In fact, in this case s ~ p t 2 lxtl ~ E = ko < m , since is uniformly integrable. So we also have suptloE lytl 5 2ko < m , since yt = xt - E[x,l&], t E [0, a], Va > 0. Furthermore, by (1.5) one has that 0 5 EA: = -Eyo 5 2ko < CQ, Va > 0. Therefore 0 5 EA, 5 2ko < m , Va > 0. This shows that suptlo E I Mt 1 3ko < m , since Mt = xt - At, W 2 0. Thus by Theorem 25 and the proof of Lemma 32 M, = limtEQ,tt, Mt E L1(n, 5 , P ) exists, and Mt = E[Mm15t],W 2 0. {Mt)t20 is a uniformly integrable martingale. rn Now let us discuss when At can be continuous.

<

Definition 46 A submartingale is called regular, if forVa > 0 and a, E S, such that a, f a , one has Ex,, -+ Ex,. Theorem 47 For a submartingale {xt)t20 of class (DL) the natural increasing process in the D-M decomposition (1.3) is continuous, if and only if { x ~ )-, , ~is regular. Proof. d: Notice that any martingale {Mt),,, is always regular, since EM, = EMo,Va E S,. Moreover, if {At)t,O is c&tinuous, then as a, f a, 0 5 A,,- 1 A,. Hence by Fatou's lemma EA,, 1 EA,. So {xt)t20 is regular, if is continuous. +==: Suppose That {st),,, is regular. Then At = at - Mt is also regular. Moreover, as a, f a, a, E S,, one has that 0 5 A,,

f A,,a.s.

(1.7)

In fact, by regularity VB E 5, 0 5 E(A, - A o n ) I ~5 E(A, - A,,,) 1 0, as n f m . So if P ( B ) = P(lim,,,A,, < A,) > 0, this will be a contradiction. Now define the sequence A, of the partition of [0, a] as in the proof of Theorem 42. Let, for c > 0,

as t E (tz, t;+l]. Then 2~is a bounded martingale on t E (t;, t;+l], and by the naturality of At one has E J;: A y d ~ ,= E J; ,:?:-d~,, Vt E (t;, t;+,],

J;?'

$5'

X y d ~ ,= E Xy-d~,. Taking the summation for j from 0 and E 3 to k - 1 one finds that Vt E [0, a], E Jot X y d ~ .= E]~ $ 1 ; AydA, J;, X y d ~ , ]= E X y - d ~ ~ , i.e.

J;p1-

E

I'

+

Ji

(X$i:-)d~,=o.

From this one sees that if one can prove the following statement:

(1.9)

26

1. Martingale Theory and the Stochastic Integral for Point Processes

Statement (A): There exists a subsequence (

IA-F

2 ~such ) that

I

lim., supt,p,l - At c = 0, a s . Then At A c is continuous in t, and so is At, since c can be taken arbitrarily. In fact, if Statement (A) is true, then by (1.9) one has that t O = EJ,(A,Ac-A,-Ac)dA, E{C,,,[(A,AC)-(A,Ac)12). So At c is continuous in t. Now we are going to establish Statement (A). - At A cl --+ 0, For this we only need to show that as n -, m ,

>

1 2

in probability; i.e. for any E > 0, as n --+ m , P [ ~ u p , , ~ ~ ,-~ At ~ A cl > E] --+ 0. Now define a! = inf{t E [0, a] : AT - At A c > E); a! = a, if {.) = 4. Then a! is a stopping time by Example 7. Moreover, it has the following properties 1) for every n, a! = a implies that - At A c &,'it E [0,a]. 2) a! f , as n f . In fact, A, is increasing in s, hence by definition is decreasing in n. So a! f , as n f . Thus a, = limn,, a! 5 a exists. Notice that is a martingale on t E (t;, t;+J. Hence by (1.8) one has that

-

x~

2

<

3

-

EIt€(t;,tC+l]AF= EIt€(t;,t;+l](At;+l A 4. Now let us generalize this equality for t to that for the stopping time a! by means of Doob's stopping time theorem. For this we introduce a function cpn(t) by defining cpn (t) = t;+l, as t E (t; ,tE+l]. Then cpn (a!) is obviously a discrete stopping time, since {cpn(a~)= t ~ + = ~ }{t; < a! t;+,) E Ti:+, . Moreover, cpn(a:) a, since a! a. Furthermore, limn,, cpn(a;) = a,, since limn,, a! = a,. Now applying Doobls stopping time theorem one finds that EI,F,(t;,t;+llA!: = EI,:,(t;,tgll(A,n(,:) A c). Take the

<

-

<

<

-

summation for t = 0 up to t = 2n - 1. One finds that EAE: = E(A,,,.qo:) A c). Hence by this and by the property 1) of a!, and after applying (1.7) one finds that 0 = limn,, E[APn(,:) A c - A,: A c] = limn,, E[AZ: - A:, A c]

-

> limn,, &P(a!< a) 2 limn,, The proof is complete.

I-

~ P ( s u p ~ , [ ~A,T, ~- At A c

> E).

1.7 Poisson Random Measure and Its Existence A dynamical system will always encounter some jump stochastic purterbations. The simplest type comes from a stochastic point process. To understand it properly requires some preparation. Let (2,B z ) be a measurable space. Definition 48 A map p(B,w) : Bz x i2 --+ R+ U {co) is called a random measure on ?Bz x a, if 1) for any fixed B E Bz, p(B,.) is a random variable but with values in R+ U {m); 2) for any fixed w E 0, p(., w) is a

1.7 Poisson Random Measure and Its Existence

a-finite measure. (Here, a-finite means that there exists {Un)z=l such that Z = Ur=lUn and p(Un, w ) < co, V n ) .

27 C

BZ

Here the definition of a random variable taking values in R+ U {OD),is the same as that taking values in R. Let us introduce the Poisson random measure as follows:

Definition 49 A random measure p ( B , w ) is called a Poisson random measure o n Bz x 0, i f it is non-negative integer valued (possibly oo) such that 1) for each B E B2, p ( B , .) is Poisson distributed; i.e. = 0,1,. . . ; p({w : p ( B , w ) = n ) ) = e- WW,, n! where X(B) = E p ( B , w ) , V B E B Z , is usually called the mean measure, or the intensity measure of p; 2) if Bz 2 { B j ) Z l are disjoint, then { p ( B j ,. ) ) Z l are independent. Here as in the real analysis we still define 0 . oo = 0. Thus if X(B) = oo, then all p({w : p ( B , w ) = n ) ) = 0 , n = 0 , 1 , . . - , hence p ( B , w ) = co, P as. The existence of a Poisson random measure is given by the following theorem.

Theorem 50 For any a-finite measure X on (2,B z ) there exists a Poisson random measure p with X(B) = E p ( B ) , V B e B z . Proof. Since X is a a-finite measure on (2,B z ) ,there exists a {Un)z=l c Bz such that all U,, n = 1, - ,. , are disjoint, i.e. Ui n U j = 4, i # j, V i ,j; and Z = Ur=lUn and 0 < X(Un) < co,Vn. Let us construct a probability space and construct random variables on it as follows: (i) for each n = 1 , 2 , . . . , and i = 1,2,... , construct a Un- valued random variable J r such that P(.$ E du) = X(du)/X(Un); (ii) p,, n = 1,2, . . - , is an integral-valued random variable with P(pn = k ) = X(Uk)k exp[-X(Uk)]/k!,k = 0,1,2, . - . ; (iii) J r , p n , n = 1,2, - . - ,i = 1,2,. - . are multually independent. , Ipn21, V B E B z . Let us show that Now set p ( B ) = C 2 = l C::l I B " ~ (J;) for every disjoint B1,. . - ,Bn E B z , V a i > 0 , i = 1,2,. . . ,m

If this is true, then p is a Poisson random measure with the intensity measure A. Let us simply show the case for m = 2. Note that by the independence E exp(a j p ( B j ) ) = flr=l E e x p ( - a i CfZ1IU,B~(ET)Ip,,ti -2 C:z1Iu,,~~(J;)Ip,tl)= Jn. However, by the complete probability formula one can derive that k Jn = 1 . P(pn = 0 ) + C;=i E [ e x ~ ( - a lCi=1 IU,BI (J1)

Zz1

28

1. Martingale Theory and the Stochastic Integral for Point Processes

1.8 Poisson Point Process and Its Existence Now let us introduce the concept of random point processes. Assume that (2,B z ) is a measurable space. Suppose that Dp c (0,cm)is a countable set, then a mapping p : Dp 2, is called a point function (valued) on 2. Endow (0,oo) x Z with the product a-field %((o,oo))x B z , and define a counting measure through p as follows: Np((O,t]x U ) = # { s E D p : s 5 t,p(s) E U ) , Vt > 0,U E % z , where # means the numbers of . counting in the set 1.). Now let us consider a function of two variables p(t, w) such that for each w E 52, P(.,w) is a point function on Z , i.e. P(.,w) : DP(.,,) -+ Z , where Dp(.,,) c (0,oo) is a countable set. Naturally, its counting measure is defined by Np((0,tl x U,w) = Np(w)((O,tl x U ) =#{~ED~:S~~,~(S,W)EU),~~>O,UE%Z, and we introduce the definition as follows: -+

Definition 51 1) If Np((O,t]x U,w) is a random measure on (%((o,oo))x B z ) x R, then p is called a (random) point process. 2) If Np((O,t] x U,w) is a Poisson random measure on (%((o,oo))x % z ) x R, then p is called a Poisson point process. 3) For a Poisson point process p if its intesity measure np(dtdx) = E(Np((dtdx))satisfies that np(dtdx)= w(dx)dt,

1.8 Poisson Point Process and Its Existence

29

where ~ ( d x )is some measure on (Z,%z), then p is called a stationary Poisson point process. ~ ( d x is ) called the characteristic measure of p. Because for a Poisson random measure on % z x R (1.10) is its sufficient and necessary condition. Now we consider the Poisson random measure defined on (%((o, oo)) x B z ) xR. Then it is easily seen that p is a stationary Poisson point process with the characteristic measure d t ~ ( d x )if, and only if W > s 2 0, disjoint {Ui)z1 c %z and Xi > 0, P - a.s.

g,

where = a[Np((O,s'] x U); s' 5 s, U E %z]. Now let us use this fact to show the existence of a stationary Poisson point process. Firstly, we will show a lemma on Poisson random process.

Lemma 52 If {Nt),,,, - is a Poisson process with intensity tp, i.e. {Nt)t20 is a random process such that P(N(t) = Ic) = e - p t w , EN(t) = tp, and it has stationary independent incremenrs; then 1) E [ ~ - X ( N ~ - N * ) I ~ ,= ] e(t-s)k'(eix-l), Vt > > , A > 0; 2) ~ [ ~ - X ( N t + u - N u ) l g , ] = e t ~ ( e i h - l ) ,t(t > O,A > 0,

-

and Vu-bounded zt-stopping time. Proof. 1): Since {Nt),,,- has independent increments, E[~-x(N~-N.)~= ~ ~E[~-x(N~-N.) I

I = e(t-~)~(eiA-l).

The last equality follows from the elementary probability theory. 2): Let us make a standard approximation of a, i.e for any bounded stopping time u with 0 5 u 5 T,where T is a constant, let an = a s u [~~ T , & T ) n, , I c = 1 , 2 , . . . . Then for each n , an is a bounded discrete stopping time. Moreover, as ntoo,anLu.By ~ ) V B E & , -*(Nt+on-No,) -*(Nt+&& - N ~ T S { W , , < ~ } ~ B ee dP = .f{r.=q~}nB ee T=d p = et"("*-l)d~. So --*(Nt+a,- N u n ) d p = J~et~(e'A-l)dp. SB ee C In particular, VB E -NNt+an -Nan) d P = JB etP(ei*-l)d~. SB ee Letting n + oo one obtains that VB E -h(Nt+a-Na) d P = SBetk'(eix-')d~. JB ee Thus 2) is proved. rn By means of this lemma one immediately obtains the following lemma.

g,

S{un=qF},,B

go gun,

g,.

30

1. Martingale Theory and the Stochastic Integral for Point Processes

Lemma 53 If {N(t))t,o is a Poisson process with an intensity function pt, denote 7-1 = inf { t 5 0 : N ( t ) = I ) , . . . , and r k = inf { t - r k - 1 > 0 : N ( t ) - N(7k-1) = I ) , then 1) { r k ) L l is an independent family of random variables, and P ( r k > t ) = e-xt,Vk = 1,2,. . . ;

2) P(c:~:

7 j

< t < c:=~

ri) = e- pt

&, (P

k-l.

Proof. 1): In fact, P ( r l > t ) = P ( N ( t ) - N ( 0 ) = 0 ) = e-pt. Hence by Lemma 52 P(7-k > t ) = P ( N ( t 7 k - 1 ) - N(7k-1) = 0 ) = P ( N ( t ) - N ( 0 ) = 0 ) = e-pt,Vk = 1,2,... . Since {N(t))t,o has independent increments, V t l ,Vtz,. . - ,Vt, P ( n C 1 {rk-> t ) ) = P(nT=, { N ( t 71-1) - N(rk-1) = 0 ) ) = P ( N ( t r k - I ) - N(Q-1) = 0 ) = n r = l P ( n > t). Hence is an independent family. 2): p(rl 7 2 > t 2 r l )= P ( N ( ~=) 1 ) = e - p t y .

+

nrZl

+

+

P(C:~: 7 j < t < c:=~

+

&

pt ~t

k-l.

.

r j )= P ( N ( t ) = k - 1) = eNow we are in a position to show the existence of a Poisson point process.

Theorem 54 Given a n-finite measure n on (2,B z ) there exists a sationary Poisson point process on Z with the charteristic measure n. Proof. Since .rr is a a-finite measure on (2,B z ) , there exists a disjoint { U k ) E 1 c !BZ such that r ( U k ) < m, and Z = U g l U k . Let us construct probability spaces and the random variables defined on them as follows: 50,Po), V k ,i = 1,2, . . . ;[$ is a Uk-valued (i) On a probability space (no, E d x ) = n(dx)/n(Uk) defined on it such that random variable with P ( J ~ {[$,V k ,i = 1,2,. - . is an indepent random variable system. (ii) For each k = ;, 2, . on a probability space ( R k ,5 k , Pk), { N ! ) , ~ ,is a Poisson random process with intensity t.rr(Uk) defined on it, set V i = 1,2,... , 7: = inf { t - ~ i - 1> 0 : ~ ~ ( N t~ ( )T ~ - 1 = ) 1); then by Lemma 53 { r b ) z l is an independent variable system such that P(T: > t ) = exp[-t.rr(Uk)],for t 2 0. S Now let R = X ~ = ~ R=~ X, *k=OSk, P = x g 0 P k . Then [$,N: are naturally extended to be defined on probability space (R,$,P), i.e. for w = (wO,wl,... ,wk,...) E R let [$(w) = [$(wo), N ~ ( w = ) N $ ( w ~ ) Then . r t ,V k ,i = 1,2, - . is an indepent random variable system on ( R ,5 , P ) .

{I!,

Moreover, { N I , J $ , V k ,i = 1,2,. maps. Now set D, = U g l ( T ; , T ~+T!,-.. ,T: and

- I is a independent system of random +T! + . - . + r k , . . - ) ,

1.8 Poisson Point Process and Its Existence

31

p ( r ~ + ~ ~ + . . . + 7 ~ ) = ~ ~ , ~ k , .m = 1 , 2 , . . . Introduce a counting measure by p as follows: Np((s, t] x (Uk n B)) = # {r E Dp : r E (s, t],p(s) E Uk n B ) Then we have that

Note that if w E 0 is such that s < CG1T;(w) 5 t, then there exists an F E (s, t] (actually, Z = CGl T?(w)) such that Nk(F,w) = m, and N ~ ( uw) , = m - 1,Vu < F. Conversely, if there exists an F E (s, t] such that Nk(F,w) = m, and N"U,W) = m - 1,Vu < ;F; then F E (s,t] (and -s = CEl r?(w)), because {T;(W))~=~ w is just the set of all jump times which have happened for the Poisson process Nk(u,w). Thus by (1.12) and by the independence one has that

zEl

~u~ns(~k)~(~.tl(~~Nk~~~l. ~ ~ ( (tl sx ,B) = Now the proof of (1.11) for this Np((s,t] x B) can be completed by the complete probability formula, as in the proof of Theorem 50. 4 A special case is the following corollary.

Corollary 55 Given a finite measure x on (2, B z ) there exists a finite sationary Poisson point process on Z with the charteristic measure x. (Here, a finite measure x means that x(Z) < co; and, a finite point process p means that the counting measure Np((O,t], Z) generated by p, is always finite, VO t < co).

<

In the case of Corollary 55 by construction the domain of a finite Poisson point process p is Dp = {71,71+72,..+ , T I + T ~ + . . . + T , , . . . ) , and Np((O, t], Z), t 2 0 is a Poisson process, where its first jump happened at the random time 71, and its m-th jump happened at the random time 71 7 2 . . . 7,. However, if one concerns with the counting measure Np((O,t], U), where U E Bz, then even Np((O, t], U), t 2 0 is still a Poisson process, but 71 is not necessary its first jump time, because now p(r1) E U is not necessary true. Actually, Poisson processes Np((O,t],Z) and Np((O, t], U) have different intensity functions tx(Z) and tx(U), respectively, if x(U) < x(Z). Therefore, one sees that the concept of a Poisson point process is finer than a Poisson process, because it also considers where jumps occur, as well as the jumps themselves. Sometimes to such situations more attention should be paid. For example, in many cases to count how many times the degree of an earthquake exceeds some level (that is, the point process drops in some area), where the earthquake happened in some area, is more important than counting all of the times it has happened. Actually, the forecast of an earthquake is only that its power is stronger than some degree. When the earthquake is very very small, usually, it is not necessary to forecast it. So the point process is more realistic. For the reason

+ + +

32

1. Martingale Theory and the Stochastic Integral for Point Processes

why we consider the Poisson point process as a basic jump type stochastic perturbation see the subsection "The General Model and its Explanation" in Chapter 8 - "Option Pricing in a Financial Market and BSDE".

1.9 Stochastic Integral for Point Process. Square Integrable Martingales In a dynamical system the stochastic jump perturbation usually can be modeled as a stochastic integral with respect to some point process (i.e. its counting measure), or its martingale measure. In this section we will discuss how to define such stochastic integral. The idea is first to define it in the simple case by Lebesgue-Stieltjes integral for each or almost all w E 0 (is said to define it pathwise). Then consider it in the general case through some limits. For this now let us consider a probability space ( 0 , S , P ) with an increasing a- field family {3t)t,o, which satisfies the usual condition explained in the first section of tliis chapter. From now on all random variables and random processes are defined on it if without further explanation.

Definition 56 Suppose that p is a point process on Z, and Np(t, U) = CsED Iu(p(s)) is its counting measure. called gt-adapted, if its counting measure is ~t-measurable for 1) p each t 2 0 and each U E Qz. 2) p is called a- finite, if 3{U,)z=1 C Cz, such that ENp(t,Un) < oo,W > O,Vn, and Z = U,"=lUn. From now on we only discuss the gt-adapted and a- finite point process p. Denote rp = {U E CZ : ENp(t, U) < cm,W > 0). Obviously, for any U E rP,Np(t, U) is a submartingale and is of class (DL), since it is nonnegative and increasing in t. Hence by Doob-Meyer's decompositio~theorem (Theorem 42) there exists a unique 3t-adapted martingale Np(t, U) and a unique &-adapted natural increasing process fip(t, U) such that Np(t, U) = K ( t , U)

+ fip(t, U).

(1.13)

Notice that the equality only holds true P - a.s. for the given U. Hence fip(t, U)may not be a measure for U E Cz, a.s. Moreover, it also may not be continuous in t. However, in most practical case we need N,(t, U) to have such properties. A

Definition 57 A point process p is said to be of class (QL) (meaning Quasi Left-continuous) if in the D-M decomposition expression (1.13) (i) fip(t, U) is continuous in t for any U E r,; (ii) fip(t, U) is a a- finite measure on (2,CZ)for any given t 2 0, P-as.

1.9 Stochastic Integral for Point Process. Square Integrable Martingales

33

We will call Gp(t, U) the compensator of Np(t, U) (or p). We now introduce the following definition for the Poisson point process.

st-

st-

Definition 58 A point process p is called an Poisson point process, zf it is a Poisson point process, st-adapted, and a- finite, such that Np(t h, U) - Np(t, U) is independent of for each h > 0 and each U E F,.

+

st

st,

gt

Notice that = a[Np((O,s] x U); s 5 t, U E Bz] c and in general these may not equal to each other. This is why we have to assume that for a Poisson point process Np(t h, U) - Np(t, U) is independent of From now on we only discuss point processes which belong to class (QL). By definition one can consider that the following proposition holds true.

+

st.

(st-)

Proposition 59 A point processes p is a stationary St- Poisson point process of class (QL), if and only if its compensator has the form: S P ( t , u ) = tr(U), W > 0, U E I?,, where r(.) is a a-finite measure on Bz. In fact, the "only if part" of the Proposition can be seen from the dePoisson point process, then its counting finition: If p is a stationary measure Np(t, U) is a Poisson random measure with the intensity measure ENp(t, U) = t?r(U), where r ( . ) is a 0-finite measure on B z . From this, one sees that Vt 2 0,Vh > 0, E[(Np(t + h, U) - (t + h)r(U)) - (Np(t, U) - t.n(U))l&] = E[Np(h, U) - hr(U)] = 0. , ~a~st-martingale, i.e. Hence {Np(t, U) - ~ T ( U ) ) is Np(t, U) - tr(U) = Mt, where Mt is st-martingale. However, by the uniqueness of decomposition of the submartingale N,(t, U) (Theorem 42) one should have Mt = fip(t,U), and fip(t,U) = t?r(U). The "if part" of the above Proposition will be proved in the next chapter by using Ito's formula. Now let us discuss the integral with respect to the point process. In the simple case it can be defined by the Lebesgue-Stieltjse integral. First, we have the following Lemma.

st-

Proof. Assume that f (t, w) is a left-continuous bounded st-predictable (s) process. Let f,(t) = f (O)I{,=O~ f ( & ) I ( + ,(s). ~ ~Then by definition one easily sees that J,"fn(s)dNp(s, U ) = f (&)[fiP(% At, U) - fip(& At, U)].

+

CEO

1. Martingale Theory and the Stochastic Integral for Point Processes

34

So the left hand side is obviously an &-martingale. Now since f (t) is left-continuous and bounded, applying Lebesgue's dominated convergence theorem and u ~ i n g E [ J fn(s)dNp(s, ~ U)15s] = J," fn(s)dfip(s, u), vs I t. one obtains tkat as n -+ co f (s)dNp(s, u)l&l = J," f (s)dfip(s, U), vs I t. Now by the Monotone-classTheorem (Theorem 392) it is easily seen that it also holds true for all bounded st-predictable process. H The integral defined in the above lemma motivates us to define the stochastic integrals with respect to the counting measure and martingale measure generated by a point process of the class (QL) for some class of stochastic processes as the integrands thru LebesgueStieltjes integral. First, let us generalize the notion of predictable processes to functions f (t, z, w) with three variables.

m,t

Definition 61 1) By P we denote the smallest a-field on [0,GO)x Z x Q such that it makes all g having the following properties P/%(R1)- measurable: (i) for eacht > 0, Z X R 3 (z,w) +g(t,z,w) E R1 is%zx&-measurable; (ii) for each (z, w), g(t, z, w) is left-continuous in t. 2) If a real function g(t, z,w) is p/%(R1)- measurable, then we call it gt- predictable, and denote g E P. Now for any given Zt-point process of the class (QL) let us introduce four classes of random processes as follows: Fp= {f (t,x,w) : f is Tt- predictable such that W > 0, SZ If (3, x,w)I NP(d%dz) < 00, a.s.1, = { f (t, x, W): f is St-- predictable such that Vt > 0,

J,t+

Fi

q,t+ Jz l f ( ~ ' ~ , ~ ) l f i P ( d<~w), ,d~) Fi = {f(t, x, w) : f is gt-- predictable such that 'v't > 0, EJ,t+ Jz lf ( s , x , 4 l 2fip(ds,dz) < GO), is St- predictable such that 30, f co, a s . , an is a stopping time, and IIo,o,l (t) f (t, x, w) E F:, Vn = 1,2, . ). It is natural that we define the stochastic integral for f E Fpwith respect to the counting measure by $ + J Z f (~7.7 w)NP(ds, dz) = P(s)?W) = Cslt,sEDp(,) f (s,P(s, w), w), since the last series absolutely converges for a s . w E 0. pote that E Jz If (s, x,w)l Np(ds, dz) = E S,t+ Jz If (s,x,w)l Np(ds, dz). Actually, the above equality holds for f being an 3t-simple process. Applying the monotone class theorem (Theorem 391) one easily sees that it is also true for f being an Et- predictable process. So 3; c Fp,and for f E 3; we can define the stochastic integral for f E Fj with respect to the martingale measure by = {f(t, x,w) : f

el1oc

+

J,t+

+

1.9 Stochastic Integral for Point Process. Square Integrable Martingales

35

As the proof of Lemma 60 one can show that it is a martingale. In fact, it is true for f being an 5t-simple process. Applying the monotone class theorem (Theorem 391) again one easily sees that it also holds true for f being an 5,- predictable process. Thus for f E Fj the stochastic integral with respect to s p ( d s ,dz) can be defined by (1.14), and it is a 5,- martingale. However, for f E 3; we cannot define the stochastic integral by (1.14), since each term on the right hand side of (1.14) may not have meaning in this case. So we have to define it through limit. Let us introduce the following notion: M2 = {{mt)t20 : {mt)t20 is a square integrable martingale, i.e. {m,),,, -

~ 2 , 1

is a martingale, and for each t > 0, E lmt12 < oo. Moreover, mo = 01, is a locally square integrable martingale, 0 C-{{mt)t20 : {mt)

i.e.

30, f co,each a, is a stopping time, such that for each n, {mt/\un) t > O E M2 1,

M$ = {{mt)t€lo.TI : {mt}t20 E M2)

7

M>~OC

={{mt)t€[O,*] : {mt)t20 E M2>loch For each {mt),,o- E M2 by Jensen's inequality

{ lmtl') t>O is a non-

negative submartingale. So it is of class (DL). In fact, for ;iyconstant a > 0 one has that sup,,ss E lm,12 5 E lm,12. So {m,),,s, is uniformly integrable. Now by the D-M decomposition theorem lrntl2 has a unique decomposition, and we denote it by (m), , lmt12 = a martingale i.e. (m), is the natural increasing process for the decomposition of submartingale lmt12. Usually, (m), is called the characteristic process of mt. Let us show the following lemma.

+

Lemma 62 Iff E Fi n F;, then

{J:+

Sz f (s, . z , u ) ~ ~ ( d s , d ntzo )) E M2;

and

1. Martingale Theory and the Stochastic Integral for Point Processes

36

Proof. Let us consider a special case first. Assume that f(s, z, w) = Iu(z), U E I?. We are going to show that u)) = g p ( t , U). In fact, let Vm= 1 , 2 , . . . , om = inf {t 2 0 : l&(.,U)l > m, or GP(t,U) > m ) . Then o m is a stopping time, since Vt > 0, {om > t) = {lGP(.,U)l m, and fip(t, U) m) E k.

(&(.,

<

<

Y

I

J

Denote g(t) = N;(t, U) = Gp(t A urn,U), and fi;(t, U) = fip(t A om,U). Then both of them are bounded in t, and as n -+ co, k t 2 - 2 t 9(t)2 = C ; = ~ [ S ( ~ ) 9( 2" I = E L g ( V ) [ g ( % - g ( k-1 W)l E;=l s ( g ) [ g ( g ) - d V ) l S,t g(s-)dg(s) + ds)dg(s) = G p - , u)iS,"(d~, u ) J : G;(s, u)G;(d~, u) = 2 N,"(s-, U)G;(ds, U ) G[N;(S, U) - fip"(s-, U)]fi;(ds, U) = a martingale Ji [N,"(s, U) - N T (s- ,U)]N r ( d s , U) = a martingale + N r ( t , U) = a martingale fim(t, u), where we have applied Lemma 60 to obtain the result that the_integral $F(s-, U ) & ~ ( d sU) , is a martingale, and used the fact that NF(t, U) is continuous. Hence

+

C Sot

+ +

+

Sot

-+

+

Sot

So bv Fatou's lemma

2

Elimm-+mlfip(t A om,U)l A

= limm,,ENP(t

A om,U) = ENp(t,U)

-

< lim- , m INp(t ~ A om,U)l

< m.

That is to say, G ( t , U) is a square integrable martingale, so (fip(-, u)) exists and has meaning. However, (Gp(-, u))

tAo,

= (fip(. A

t

om,u ) ) ~ .So

by (1.15) letting m + m one obtains that u)) = fip(t, U). t Now let us return the proof of Lemma 62. However, the conclusions are true for f being a simple process (just proved). So applying the monotone class theorem (Theorem 391) one sees that the conclusions also hold true for f being a predictable process belonging to Fj nF;. The proof is complete. (&(el

Now assume that f E F i . We are going to define the stochstic integral for f with respect to the martingale measure Gp(ds,dz). Since we only consider the point process being &-adapted, a-finite and of class (QL), we have 3U, E Bz,Un f 2, such that ENp(t, U,) < w,Vt > 0, Vn. Let fn(s, Z,W)= I[-n,n] (f (S,~,W))IU, (z)f (s, 2 , ~ ) . Then obviously, fn is bounded and f, E F j n e , V n . By (1.14) and Lemma ': Jz fn(s, z, w)fip(ds,dz) is defined and be62 the integral I(fn)(t+) = J longs to M 2 . Moreover, by Lemma 62 for each fixed T > 0, as 0 t T ,

< <

2

1.9 Stochastic Integral for Point Process. Square Integrable Martingales

E[II(fn)(t+) - I(fm)(t+)l2] = E

< ES,T

37

J JZ~ I(fn - f m ) ( ~ , z , w ) l ~ @ p ( d ~ , d z )

I(fn - fm)(s,z,w)12@p(ds, dz). So {I(fn)(t+))tEIO,Tlis a Cauchy sequence in M$ with the norm llmllT =

JE[lmTlal

= d m . By the completeness of M$ with such norm (Lemma 63 below) there exists a unique limit denoted by

{JiCJz

{I(f )(t+)}tE[o,T] = f ( ~ , z , w ) f i p ( d ~ ~ d~E[O,TI z)) E MF. Again by the uniqueness of the limit one can define VT > 0 Jz f (s, z,w)Np(ds, dz) = JZf (s, z,w)NP(ds,dz), as t E [(),TI. E M2. The integral Then obviously, {J:' JZ f (s, z, w)Gp(ds,dz)) t>o J:+ JZf (s, z, w)@p(ds,dz) is usually called a "compensated sum". Finally, for f E by assumption 30, f co,a.s., a, is a stopping time, and I[o,m,l(t) f(t,x,w) E G , V n = 1 , 2 , . . - . Let {xt)t20 E M2110c such that JZ f (s, 2, w)~[O,Onl ( ~ > ~ p dz). (d~, x t ~ o ,= Then the integral is uniquely defined.

~0t+

~i+

~ i * ' ~ ~

Ji+

Lemma 63 1) For {mt}tE[o.TlE M$ let llmllT = \ / E [ I ~ T=I ~ dI m then (M$, II.IIT) is a Banach space.

,

M2 : {mt)t10 is continuow in {mt}tto E M2 : {mt)t,o is right continuous

2) Let M21C= {{mt)t,o

E

with left Zmit in t

~ 2 = '{{mt)tE[o,Tl

{mt)t20 E M21c)1

and

M$""

= {{mt)tElo,Tl: {mt)t20 E M ~ J ' ~ ~ ) .

Then M$" C M$'"" both are closed subspaces of M$. 3) For {mt)t>o E M2 let llmll = C,"'=l (llmlin A I), and d(m,n) = Ilm -rill, then ( M 2 , d ) is a complete metric space. Moreover, M2*" c both are closed subspaces of M 2 .

f

M2711rc

Proof. 1): By functional analysis all others are true except that the completeness under this norm still needs to prove. Now suppose that {m;)tEIO.Tl E M$ is a Cauchy sequence under this norm, i.e. as n, k 4 oo ~ [ l mm&12 ~ -t 0. By L2- theory there exists an mT E &- such that ~ [ l r n ~
38

1. Martingale Theory and the Stochastic Integral for Point Processes

limn--rcu, limn - milT = limn,, E[lmF - mTI 2 ] = 0. By the martingale inequality (Theorem 34) one has that as n -+m , E[suptGP.q Im? - mt12]5 4E[lmF - ~ T I ~0. I So 3 {nk)T=fi{n),"=l such that as k -+ m , suptE[o.T] Imrk - mtl 2 -+ 0,P - a.s. Since the convergence of {m~k)tE[O,Tl to {mt)tEIO,Tl is uniform in t , we T lcontinuous in t (right continuous have that if for each k , { m ~ ) t , [ o , is (respectively, so also is {mt)tEIO,Tl). with left limit in t ) ,so is {mt)tElo,Tl 3): We only need to note that Ilm - rill = 0 Ilm - nlln = 0,Vn; and that Ilmn - mil -+ 0 Ilm- nil, -+ 0,Vn;provided that {m;)tLo, {mt)tloE M 2 . Then all of the conclusions are easily derived by means of 1) and 2). +

++

Brownian Motion, Stochastic Integral and Ito's Formula

For a dynamic system the simplest continuous stochastic perturbation is naturally considered to be a Brownian motion (BM),since it is a Normal process (or say, a Guassian process) with independent increments which are also normally distributed. In general, a continuous stochastic perturbation will be modeled as some stochastic integral with respect to the BM. However, the BM has the strange property that even though its trajectory is continuous in t, it is not differentiablefor all t. So for a stochastic integral with respect to BM one has to use a different approach - the martingale approach is used t o define it. In this chapter we will discuss such problems.

2.1 Brownian Motion and Its Nowhere Differentiability Definition 64 A d-dimensional random process { x ~ -) , ,is~called a Brownian Motion (BM) or a Weiner process, if 1) its initial probability law is given by some probability measure p, i.e. vr E % ( R ~ P) (,X O E r) = p ( r ) ; 2) it has independent increments, i.e. VO = to < tl < . - . < t,, the increments xto,xtl - xto,xtz - xi1,.. . ,xt, - xtmTl are independent; 3) VO 5 s < t , x f - x i N ( O , ( t - s ) ) , i = 1 , 2 , . - . , d , ie. each real component increment xi - xi is Normally distributed with the mean E(xt x,) = 0 and the variance V ( x t - x,) = ( t - s); where xt = (x:,xf,. . ,x f ) , d and is an independent random variable family for each t > 0;

40

2. Brownian Motion, Stochastic Integral and Ito's Formula

4 ) it is continuous in t , a.s., that is, for almost all w E 0 the trajectory xt(w) is continuous in t. NOW for t > 0 , x E Rd let p(t, x ) = ( 2 ~ t ) -exp[~ / ~ 1x12121. Then we have the following proposition.

Proposition 65 If {xt)t20 is a continuous d-dimensional random process,

the following statements are equivalent: (i) {xt)t20 is a BM with some initial law p. (ii)VO=to< tl < . . . < t,, a n d I ' i ~ % ( ~ d ) , i = 0 , 1 , 2 , .,m, -.

(iii) E[exp(i(X,xt - x,))l$T] = exp[-(t - s) / X I 2 121, as., VXER~,O<S<~, where 5: = n r = l ~ ( :~s , 2 t + and a(x,; s 5 t is the smallest a-field such that it makes all x,, for s 5 t measurable.

i), + k,

+ i)

Proof. ( i ) Consider the probability law o f P(xto E ro,xt, (i): xto E rl,.- . ,xtm - xtm-, E r m )I f we let yo = x , yl = xl - x , y2 = $2 - X I , - . . ,ym = xm - xm-1, then one can easily finds that the absolute = 1. Thus by (ii) one has value of the Jacobi determinant that

which is equivalent to saying that { x ~ ) is~a,BM ~ with the initial law p. (i) ==. (ii) : Similarly, now considgr the probability law o f P(xt, E r o , x t , E rl,...,xt, E I?,). I f w e l e t x = y 0 , x l - x = y1,x2-x1 = y2,. . . ,x , - x,-~ = ym, then one still finds that the absolute value o f the Jacobi determinant = 1. Thus by (2.2) we find that (ii) holds true. (iii) +( i ) : By (iii) VO = to < tl < . . . < t,, Xk E R d ,k = 0,1,2, . , m, E[ex~{i(Xoxo CyZl( h , x t k - xt,-,)>)l = E[eiXoX0E(exp(i C:=l (Ah,xt, - xtk-,))15~)]

+

- ~ [ ~E ( e iiX l ( x~t l - x ~~) E [ e xi X 2 ( x ~t 2 - " t l ) . .. ~ ( ~ ~ ' " ' ( " t -m"tm-l'~$t,-I) .. . f8tl]180)I

2.1 Brownian Motion and Its Nowhere Differentiability

- ( E ~ ~ X O X )O . = (EeiXoxo) .

nYzlexp[-(tj n m

3=1

41

1Ajl2 /2]

tj-1)

Eeib ( x t j-=tj+1

Thus the increments xto,xtl - xto,xtz - xtl , - . . ,xt, - ~ t , - ~are independent, and the increment xtj - xtj_, is Normally distributed with mean 0 and with the variance matrix such that all elements on the diagonal are equal to t j - tj-1, and all other elements are equal 0. Thus {xt)t>o - is a (d-dimensional) BM with some initial law. (i)==+ (iii) : I f { x ~ ) , ,is~ a BM, then by the independent increment property, and because i t s increments are Normally distributed, one finds that E[exp(i(A,xt - xs))13;] = E[exp(i(A,xt - x s ) ) ]= exp[-(t - s ) !At2121. Now let us set W d = the set o f all continuous d-dimensional functions w(t) defined for t 2 0. % ( W d )= the smallest a-field including all Borel cylinder sets in W d , where a Borel cylinder set means a set B c wdof the following form B = {W : ( ~ ( t l. ). -,~ ( t , ) )E A ) , for some finite sequence 0 tl < t2 < -.. < tn and A E %(Rnd). From above one sees that given a Brownian motion { x ~ ) ~ this , ~ will , lead to the generation of a probability measure P defined on '13(wd) such that its measure of the Borel cylinder set is given by (ii) in Proposition 65. Such a probability measure is called a Wiener measure with the initial measure (or say, the initial law) p. Conversely, if we have a Wiener measure P with initial measure p on % ( w d ) ,let ( f l , z , P ) = ( w d , % ( W d )~, ) , x ( w t ,) = w(t),W 2 0, w E W d ,then we obtain a BM { ~ t ) defined ~ , ~ on the probability space ( f l , z , P ) .So the BM is in one to one correspondence with the Wiener measure. Now a natural question arises: does the BM, that is, the Wiener measure exist? The existence of the Brownian motion is established by the following theorem.

<

Theorem 66 For any probability measure p on ( R ~%(!X3)) , the Wiener measure P, with the initial law p exists uniquely. Proof. Uniqueness. Since the totality of Borel cylinder sets generates the a-field % ( w d ) ,any two measures that coincide on Borel cylinder sets still coincide on % ( w d ) . Existence. We shall show the result by using the Kolmogorov extension theorem (Theorem 379 in Appendix A). In fact, by this theorem there exists a random process { x ~ ) , ,such ~ that it has the finite probability distribution defined as (2.1). since (xi - 28) N(0, ( t - s ) ) , i = . 4 1,2,. . - ,d. Hence, by calculation, E - xi1 = 3 It - s12. So by the Kolmogorov continuous version the0rem[~4{x~),,, has a continuous version, i = 1,2,. . - ,d; and so does { x ~ -) , ,. ~Let U S Gain denote this continuous

oreo over,

1%:

42

2. Brownian Motion, Stochastic Integral and Ito's Formula

version by { x ~ ) , , ~ .Thus such a d- dimensional random process is a BM with the initial law p. Definition 67 If a d-dimensional BM { ~ t )is, such ~ ~that P ( x o = 0) = 1, that is, p = So- the probability measure concentrated at the single point {0), then it is called the standard BM and denoted by { w ~-) ~ , ~ . From now on we always discuss the standard Brownian motion, and it is simply denoted by BM. Brownian motion has some nice properties. For example, its trajectory is continuous, i.e. {xt),,,- is continuous. Moreover, it can be a square integrable martingale. Corollary 68 If { ~ t )- , ,is~ a d-dimensional Zt-BM, and E 1x012 < oo, then 1) { x ~ ) is, a~ square ~ integrable gt-martingale; 2) xE4 - Sijt is a adapted "). &-martingale. (Here "Tt - " means

'(st-

Proof. By the definition of a BM we have that as 0 5 s

< t , E[xf -

x:13.] = 0 and E [ ( x $- x:) ( x i - 2:) 151 . = S g ( t - s). Since the left hand side of the above equality equals ~ [ x f x15i.1 -x:x;. So the proof is complete. However, a BM also has the following strange properties.

~ ~ BM, then P - a s . , T h e o r e m 69 Suppose that { ~ t ) is, al-dimensional for any given a > {xt)tlo is not Holder-continuous with indexa for each t 2 0.

i,

Definition 70 We say that { ~ t ) , ,is~ Holder-continuous with index a at to > 0, if VE > 0,3S > 0 such thatas It - to1 < S,Ixt - xtoI < E It - t O l f f . Theorem 69 actually tells us that the trajectory of BM is not Lipschitzian continuous at each point t , so it is nowhere differentiable for t 2 0. Hence it is also not finite variational on any finite interval of t, since each finite variational function of t should be almost everywhere differentiable for t . Thus we arrive at the following corollary. Corollary 71 1) The trajectory of a BM is nowhere differentiable fort 2 0, P - a.s. 2) The trajectory of a BM is not finite variational on any finite interval o f t , P - a.s. Now let us show Theorem 69. Proof. Take a positive integer N such that N ( a > 1. For any positive integer T > 0 denote w : there exists a s E [O,T] such that V t E [O,T], A; = It - sl < N +Ixt(w) - X.(W)[ < E It - sl" Obviously, A: f, as n 1 . Let A" = Ur=lAk. If one can show that VE > 0, P(A:) = 0, then one finds that the conclusion of Theorem 69 holds true on the interval [O,T]. Now set

4)

{

,

I

2.2 Spaces Lo and C2.

43

Zk = r n a x l l i < ~1 x ( F ) - ,(-)I , k = 0,1,. . - ,nT; BE = {w : 3k such that Zk(w) 2 ~ ( N / n ) ~ ) . Let us show that A: c BE. In fact, if w E A,: then 3s E [O,T] such that

<

w E P,1,

+*

It - SI < Ixt(w) - xs(w)I < &It- s y . 2 ~ ( N / n ) since ~ , in this case Set ko = max {k : k/n 5 s) . Then Zk,(w) s E [%, -1, and each point in this interval has a distance from s less Hence w E BE. Thus p(AE) = lim,,,P(A~) < lim, ,,P(Bz). than However, P(BE) = P(u:z0 {Zk 2 ~ ( N / n ) ~ I )) P(Zk 5 2&(N/n)") = nTP(Zo 5 2~(N/n)") 5 n~[P(lx(:)l 5 2&(N/n)")lN = n ~ ( e e-nx2/2dx)N,(I = EN^), where we have used the fact that since { ~ t )-, >is~a BM, it has independent increments with the same probability law such that x(%) - x(-) N(0, ;),Vi,Vk. Let y = nax. One finds that as n + co, T ?42/(2n2"-1)dy)~ + 0. P(A:) 5 5 ( 2 ? r ) N / 2 n ~ ( e - ( l / 2 ) ) - 1 (.f-T eThus p(AE)= 0. The proof is complete. The trajectory of a standard BM { W ~ ( W ) ) ,is> ~nowhere differentiable on t 2 0, P - a s . This means that 3A E 5 such that P(A) = 0 and as wo $ A, the function wt(wo) cannot be differentiated at W 2 0. This means that we cannot simply define the stochastic integral :J f (t, w)dwt(w) for each w E A in terms of the usual integral. That is why Ito had to invent a new way to define this completely different integral which now is known as Ito's integral.[501J511

<

2.

<

x;zo

~117:~

2.2 Spaces 130 and L2. To discuss Ito's integral we first need to consider its integrand processes.

Here IIvillw = esssup lp,(t,w)I. NOWlet us discuss the relationship between L2 and Lo.

44

2. Brownian Motion, Stochastic Integral and Ito's Formula

Lemma 73 For f = {f ( t , ~ ) )-, >E~L2 let

llf 112 = C r = o zfr(llf ll2,n A 1) Then 1) 11-112 is a metric, and L2 is complete under this metric, if we make the idenfication f = ft,Vf, f' E L2, as [If - f '112,n = 0,Vn. 2) Lo is dense in L2 with respect to the metric 11.112. Proof. 1) is obvious. We only need to show 2)..Suppose that f = {ft)t>o is a bounded left-continuous St-adapted process. Let k & f n ( 0 , ~ )= f(Olw), fn(t,w) = f($i,w), as t E (F;;, zn I, Ic = 0, l , . . . Then fn E Lo and 11 fn - f [I2 -' 0, as n -+ co, by the left-continuity of f and the Lebesgue's dominated convergence theorem. Now collect all f E L2 to form a family H such that each f E 7-1 can be approximated by some c Lo under the metric 11.112. Then H contains all bounded leftcontinuous St- process, and obviously it is closed for the non-negative increasing limit under the metric 11.112.SO,by Theorem 3 9 2 , a contains all bounded St- predictable process. However, for each bounded f E L2 one t can let fn(t,w) = nSt-l f(s,w)ds to form a bounded sequence such that each fn is a boGded St- continuous process, and by real function theory for each w E fl, f,(t,w) -t f(t,w) for a.e. t. Hence one sees that for each k , as n -t oo, Ilfn - f l12, k = EJ: Ifn(t,w) - f (t,w)12 0, by Lebesgue's dominated convergence theorem. Hence H contains all bounded f E &.Finally, for any f E L2 let fn(t, w) = f (t, w)Il - Then I fnl 5 n, and for each k, as n -+ oo, 11 fn - f -,0. So we can show that 7-1 > L2.

-

{fn)r=l

{fn)r==,

-+

rn

2.3 Ito's Integrals on L2 First, we will define the Ito integral for Lo. Suppose that a St-Brownian motion {wt)t>o - (Wiener process) is given on (fl, 5,P).

Firstly, it is easily seen that the stochastic integral also has an expression, which is actually a finite sum for each 0 < t < co, I(f)(t) = Czo~ , ( w ( t i + Al t) - w(ti A t)), moreover, I(f)(t) is continuous in t. Secondly, it has the following property.

Proposition 75 1) I(f)(O) = 0, a.s. and for any a,P E R; f ,g E Lo I ( a f + Pg) = 4 f + PI(g).

2.3 Ito's Integrals on

L2

45

Proof. 1) is obvious. 2):Since BM w(t)is a square integrable martingale, vs 5 t , A t ) - w(ti A t))I$s]= ~i(w(ti+l A s) - w(ti A s)). E[~i(w(ti+l So E[I(f)(t)13s] = I ( f ) ( s ) This . means that {I(f)(t)),,, is a gt - martingale. Note that

Hence I ( f ) ( t )is uniquely determined by f and is independent of the particular choice of {ti) . Now suppose that f E L2. Since { w ~- ) is~ , ~ nowhere differentiable, we cannot define Jot f (u,w)dwu(w)pathwise for each or a.s. w E a. However, by (2.3) one sees that for f E Lo, and for each given T > 0, {ft)t,-[o,Tl has a norm 11 f = f 2 ( u ,w)du = E [ I ( f ) ( T ) 2 ]So . {f(t))tEIO,T] E L$ is in 1 to 1 correspondence with

EG

{ I ( f) ( t ) 2 } t ~ f o , ~E, M:', and both with the same norm. This motivates to define { ~ ( )(t)2)t,lo,Tl f E for f E L2 through the limit of 2 E if IIfn - f l12,T 0. {I(fn)(t)2}t,[0,TlE M?, where {fn)tEC,TI

~2~

US

4,

-+

That is exactly what Ito's integral defines. Let us make it more precise. For any f E L2 since Lo is dense in L2 with metric 11. 1 12 (Lemma 73),3f , E Lo such that Ilfn - f 112 0, 00- SO III(fn)- I(frn)II2= IIf* - fmI12 0, as n, m + oo.This means that {I(fn))r=lC M27' is a Cauchy sequence. However, M2>' is complete under the metric 11. 1 12 (Lemma 63). Therefore there exists a unique limit, denoted by I( f ) , belonging to M21C.Let us show that I ( f ) is uniquely determined by f and is independent of the particular -+

+

+

2. Brownian Motion, Stochastic Integral and Ito's Formula

46

choice o f {fn);='=, c Lo. In fact, let there be two {fn);=l, such that both

11 fn - f [ I 2 ,

l f;.

-f

1

-+

2

-

m

c Lo 0 , as n + m. Construct a new { f n L

-

sequence {gn);='=,C Lo such that gzn = fn, gzn+l = fn. Then one still has IlI(gn) - I(9,)112 = IIgn - h 1 1 2 + 0, 11% - f 11, 0 as n , m m. So the limits should satisfy the conditions that limn-+mI ( f n ) = limn+m I ( f n ) = limn-+m I(gn) = I ( f ) . -+

-+

Definition 76 I ( f ) E M2>= defined above is called the stochastic integral , ~ ,it is or the Ito integral o f f E L2 with respect to a BM { ~ ( t )-) ~ and denoted by I ( f ) ( t )= J: f (s)dw(s)= J: f ( s ,w)dw(s,w ) . Beware of the fact that the integral is not defined pathwise. So, actually, f (s,w)dw(s,w))(wo),P-a.s.wo. I ( f )(t)(wo)= (J: f (s)dw(s))(wo)=

(J,t

Proposition 77 (i) All conclusions in Proposition 75 still hold for V f E

L2.

(ii) More generally, i f T > a are both stopping times, then W > 0, V f E L2, E [ ( I ( f)(t A 7 )- I ( f ) ( tA a))lSul = 0, a.s. tAr 2 a's' E[('(f A - I ( f ) ( t A a))218u]= E[JtAu f (u,w)du18C7], (iii) Furthermore, V f , g E L2;W > s 2 0; for all stopping times r > a, -W(f )(t)- I ( f ) ( s ) ) ( I ( g ) ( t) I(g)(s))lSsl = E[J,t(f .g)(., w)d.l3sl, E [ ( I ( f)(t A 7 )- I ( f )(tA a ) ) ( I ( g ) ( tA 7 )- I ( g ) ( tA a))I3ul = . g)(u,w)d4Su1, a.s. (iv) For any stopping time a,Vf E L2, I ( f )(t A a ) = I(f')(t), vt > 0, where f ' ( 4w) = Itlu(,)f (t,w).

~[$";(f

Proof. ( i ) is true for V f f Lo, and so is true for V f E L2 through the limits. (ii): Since {I(f)(t))t,o is a St-martingale, and so by Doob's stopping time theorem, the first conclusion of (ii) holds true. On the other hand, by (i) for each t > s > 0,V f E L2,

W f)W2 - ~

()(s)21S*l f

=

W (Kt) f - ~ ()(s))213s1 f (2.4)

Thus { ~ ( f ) ( t -) J: ~ f

Z ( U ,w

) d ~ ) is~also , ~ aSt-martingale, and by Doob's

stopping time theorem, (2.4) stillholds true when t and s are substituted by the stopping times t A 7 and t A a , respectively. So (ii) is proved. (iii): The first conclusion is true for f , g E Lo, and so is for f , g E L2 through the limits. The second conclusion follows from the first one through Doob's stopping time theorem, as in the proof o f (ii). Finally, let us establish (iv).

2.4 Ito's Integrals on c

~

* 47

We still show it first for f E Lo. This can be done by evaluation using the standard discretization of the approximation to the stopping time a, followed by taking the limit. In fact, suppose that f (t,w) = ~o(w>I{t=o> (t) + CEO '~i(w)I(ti,ti+il (t). Introduce {sl)&, which is the refinement of subdivisions {ti): and Now on this partition we re-express f as f (4 W)= cpo(w)I{t=o}(t)+ CEO cpl(w)I(s;,s;+l](t), where cpl(w) = cpj (w), as t j < sl 5 tj+1, and make a standard discretization of the approximation to the stopping time a as follows: Let an(w) = s?+~, if a(w) E (sr, s:+~]. Then as in the proof of Theorem 35, for each n an is a discrete &-stopping time only valued on { s ~ ) z oand , an 5 a, as n t co. SO,if we let fA(s, w) = f (s,w), then fA E Lo. In fact, 3{s7)F0 such that fA(s,w) = (~O(w)I{s=o}(s) + C Z o (Pin1( ~ ) I ( s ~ , s ~ + ~ l ( ~ ) , where cpll(w) = cpl(w), if s E ( s l , s:+~], s an(w) ( s E (32, ST+",,, an(w) cpll(w) = 0, if s E (SF,sY+J,s > an(w) ( s E ( s l , s;+",], an(w) s l ) . Obviously, cpll(w) E since {an(w) sl) E and cp;(w) E Zs;. Now by evaluation we are going to show that 1) I ( f X t ) = I ( f )(t A 4, 2) IlI(f3 - I(f')ll,,t 0, as n -+ 00If these results can be established, then (iv) is proved for f E Lo. However, as n --+ oo, I l I ( f 3 - ~ ()ll& f = IIfA - fllli,t = E f 2(s7~)~(a(w),an(w)](s)ds -,0. Thus, 2) is proved. Notice that if s E (sl, s:+~], then IS<,n(,) = I(fA)(t) = CEO cpll(w)(wt~s;+,- wt~s;) M = Ci=o ~ ~ ( ~ ) I s < a ~ ( w ) (-~wt~s;) tAs~+~ = CEO 'P~(w)Is;
~

{&)Lo.

<

> <

*

zs;,

<

zsr,

-+

Sot

=

tAan CEO( P ~ ( W ) ( W ~ A U-"wtAanAsY) A S ~ + ~ = So f (3, w)dws.

Thus 1) is also proved. Hence (iv) is true for Vf E Lo, and so is also true for Vf E L2 by taking limits. w

2.4 Ito's Integrals on ,C2,'0c First, let us introduce the concept of local martingales. Definition 78 1) An st-adapted real random process { ~ t )-, >is~called a 10caE~~-martingale and denoted by {xt)t>OE Mloc,if 3a, t co,an < co is a zt-stopping time for each n, such that ~xtAun)t10is a &-martingale for each n. 2) In addition, if {xtAan)t>ois a square integrable 5t-martingale for each n, then {xt)t>o is called atocally square integrable St-martingale, 3) Write and it is denoted by Gt)t>o - E M21zoc.

~

~

2.5 Stochastic Integrals with respect to Martingales

49

2.5 Stochastic Integrals with respect to Martingales In this section we are going to discuss the stochastic integral f ( s ,w)dMs, where {Mt),,o- E M2.Recall that the procedure for defining Ito's integral f ( s ,w)dws is as follows: First we define it for f E Lo which is a simple process, and find that f (s,w)dw, E M2>'. Then, after establishing the 1 to 1 correspondence between space C2 and M21Cwith the same metric, for each f E L2, we can take a sequence of C Lo which tends to f

Jl

{fn)zl

in C2. So the corresponding sequence of integrals

{s: fn(s,w)dw,} n=l E 03

M2>"will also tend to a limit in M2", which we denote by

f (s,w)dw,, and define it to be the stochastic integral for f . Note that for a BM { ~ t ) , , ~we have that w: = a martingale t , '82 0, and we establish a one to one corresvondence as follows: for each T > 0.

+

[c

with the same norm 11fl12,T = f 2 ( s , ~ ) d ~ ] 1NOW / 2 . for a {Mt)t>o E M2 we want to do the same thing. So first we need a D-M decompo&ion for its square. For simplicity we discuss the 1-dimensional processes.

Proposition 81 1) If {Mt)t>o - E M2, then {M:),>~ has a unique D-M dewmposition as follows: M; = a martingale + (M), , where (M), is a natural (predictable) integrable incresing process, and it is called the (predictable) characteristic process for Mt. 2) If {Mt)t20, {Nt),,o E M 2 , then {MtNt),,o has a unique decomposition (it may be still called the D-M decomposit~on)as follows: MtNt = a martingale + (M, N), , where (M, N), is a natural (predictable) integrable finite variational process, i.e. it is the diflerence of two natural (predictable) integrable incresing processes, and it is called the cross (predictable) characteristic process (or (predictable) quadratic variational St - adapted process) for Mt .and Nt . q a n < co is a stop3) If {Mt)tlo, {Nt)t20 E M21zoc, then (i) 30, ping time for each n such that {MtAun)t20,{NtAun)t>OE M2 for each n; (ii) there exist a unique predictable process {(M, N),)ylo such that (M, N),,,, = (Mun, Nun),, V n and V t > 0, where we write M,"" = MtAun,and NFn = NtAon. Proof. 1): By Jensen's inequality {M;),,~ is a submartingale. Since it is also non-negative, it is of class (DL). So by the D-M decomposition theorem we arrive at 1).

50

2. Brownian Motion, Stochastic Integral and Ito's Formula

2): Note that in this case {MI (t))t,o , {M2(t))t10 E M2,where MI (t) = Hence by the D-M decomposition theorem ) ~a martingale A2(t), Ml(t)2 = a martingale Al(t), M z ( ~ = where Al(t) and A2(t) both are natural (predictable) integrable incresing processes. So MtNt = MI(^)^ - M2(t)2= a martingale Al(t) - A2(t). Let us show the uniqueness of this decomposition, In fact, if there are two D-M decompositigns for MtNt : MtNt = f i t +At, MtNt = 6: +A;, where fit and 6; are martingales, and At = Alt - A2t, = Ait - A& such that all Alt, A2t,A:,, A& are natural (predictable) integrable incresing processes, then G t - 6; Alt A;, = A:, AZt. However, by the D-M decomposition theorem we must have f i t - 6; = 0,Alt A& = A'lt Azt, since A:, Agt is also a submartingale of class (DL). So fit = 6:,At = A:. 3): (i) is true by the definition of (ii) We only need to show that the equality in (ii) is well definied. In fact, if m > n, then (MU,, Nan)t = (Mum,Num)tAu, . So it is true. H For the continuity of (M, N), we have the following proposition.

MthNt,M2(t) = Mt;Nt.

+

+

+

-

+

+

&

+

+

+

-

+

- -

~

~

1

'

~

~

.

Proposition 82 Any one of the following conditions makes (M, N)t continuous in t : (i) {St)tlois continuious in time, i.e. if CT, f a and they are all stopping = VnzU,,; times, then (ii) M, N E M21C.

zU

Proof, (i): In this case by Levi's theorem Mtr\,, = E[Mt I$tAu,] --E[Mt I ~ ~ A u = ] M t ~ uas, aMn 1a . However, since 0 5 M&,, 5 E[M: lzt~a,1, by Corollar~27 {M&,, is uniformly integrable. Hence limn,, EM&,, = EM;,, i.e. {M:),,~ is regular. Similarly, {N:)t20 is regular. Moreover, by the same token, one also has that MtAu, NtAu, + Mtha NtAo, as a, f a. Hence {(Mt ~ t ) ~ ) , , , , is also regular. However, one easily sees that ( M + N), = (M), + (N), + 2 (M, N), . Thus the continuity of (M, N), follows from the continuity of the other three, because they are all regular. (ii) is similarly proved. For stochastic integrals with respect to the martingale {Mt)tlo E M2, we need to introduce the space of integrand processes as in the case with respect to BM {wt)tlo E M27c.

+

+

+

Definition 83 1) Write Lb = {{f (t, w ) : it~ is zt-predictable ~ ~ such that VT > 0 M

2

(Ilf ll2,T) =

E S ~ Tf2(t,w)d

< 0~i.1

( ~ ) t

2.5 Stochastic Integrals with respect to Martingales

51

For f = { f (t,w ) ,-, ~ E L& set

l l f l?

=

C= :l ~ 1 ( l l llz,, f M A 1).

2) L y = {{f ( t , ~ ) ): ,it~is~st-predictable such that if don is a st-stopping time for each n, and f2(t,w)d ( M ) t < m, V T > 0,Vn). 3) Lo is defined the same as i n Definition 72.

T w,o,,

EST

Note that if E J:'~" f ' ( t ,w)d ( M ) t < w, V T > O,Vn, then P - a.s. J ~ N / \f~' "( t ,w)d ( M ) t < w,V n ,V N = 1,2, . - . . Therefore, J: f ' ( t , w)d ( M ) t < w , V T > 0 , P-a.s. In general, the inverse is not necessary true. However, if ( M ) , is continuous in t , then the inverse is also true. Reasoning almost completely in the same way as in Lemma 73, one arrives at the following lemma.

Lemma 84 Lo is dense i n L L with respect to the metric

II.II~.

f ( s ,w)dM, with respect to Now we can define the stochastic integral { M t ) t 2 0 , first for f E Lo, and then for f E L L , and finally for f E ,Cg•‹C in completely the same way as when defining f (s,w)dw,. However, we would like to define it in another way, even if it is more abstract and different, because it is then faster and easier to show all of its properties. Definition 85 For M E M2>loC and f E L g o C (or M E M 2 and f E L h ) i f X = {xt)t>O E M2>loc(X = {xt)t20 E M 2 ) satisfies that

V N E M 2 ~ l o c( N E M 2 ) , V t 2 0, then set xt = I M ( f ) ( t ) ,and call it the stochastic integral o f f with respect to martingale M . In the rest of this section we always assume that M E M2>""".First let us show the uniqueness of X E M2>locin Definition 85. In fact, if there is another X' E M2>locsuch that (2.5) holds, then ( X - X', N ) = 0, V N E Hence by taking N = X - X' one finds that X = X'. Secondly, we need to show that such a definition is equivalent to the usual one, which was explained before this definition. Proposition 86 I f f E L h is a stochastic step function, i.e. f E L L , and 3an uo = 0, on is a st-stopping time for each n such that f (t,W ) = f 0 ( ~ ) 1 t = + 0 C:=o f n ( ~ ) I ( o ~ , (ot~) ,+ ~ ] where fn E , then

r,

son

52

2. Brownian Motion, Stochastic Integral and Ito's Formula

Proof. In fact, VN E M2?loc ( ~ ~ ( fN) ), = ~ ( x m of n ( w ) [ ~ ~ n + livUn )J), = C,"==, fn(w)((MUn+l ,N)t - (Men, N)t) = C,"==, fn(w)((MUn+l, - (Mun,Nun),) = J,"f (s, w)d (M, N),

.

+

BY this lemma if f E L L , f(t) = fo(w)It=o C ~ = o f n ( ~ ) I ( t n , t n + l l ( t ) , where 0 = to < tl < -.. < t, -+ oo, and fn(w) E Ztn, and all fn are bounded, then IM(f >(t)= CZo fn(w)(Mt,+l - Mtn). That is just the usual way to define the stochastic integral J , f (s, w)dM, E M 2 , for f E L k . Following this usual way for f E L& one can take a sequence {fn) E L& such that 11f n - f llf -+ 0. So there exists a limit J, f (s, w)dMs E M2 such that f (s, w)dM, - J, fn(s, w ) d ~ , ~ l ; + 0. Therefore the stochastic integral J,"f (s,w)dM, E M2 is also defined for f E L&. Let us show that it is equal to the stochastic integral IM(f)(t) defined in Definition 85. To show this we need to use the Kunita-Watanabe's inequality (see the lemma below). In fact, by this inequality VN E M2 I(IM(fn) -IM(f),N),l 6 G I ( f R - f ) ( ~ , ~ ) l d l ( M , N ) I , 5 (J," I(fn - f ) ( s , w ) ~ ~ d ( ~ ) ,()( lN/ t~Y. 2 . l - f 11;)~ -, 0. On l+?nce E 1 ( 1 ~ ( f ~ ) ( t-) ~ ~ ( f ) ( t ) , ~5) ,(lllll,N)2(11fn the other hand, one also has that (J, f(s,w)dMs - J; fn(s,w)dMs,N), = (J,(f(s,w) - fn(~,w))dMs,N), = S,"(f ( 8 , ~ ) f"(s,w))d(M,N),. So in the same way we can show that the left hand side of the above result tends to zero as n -+ oo. Since by Proposition 86 (IM(f n), N)t = (& fn(s,w)dMs,N),. We have (IM(f),N), = (&f(s,w)dM,,N), ,VN E M2. Therefore IM (f)(t) = Jot f (s, w)dMs, Vf E L&. Furthermore, we can also easily obtain the same equality for all f E c Y .

I J,

Lemma 87 (Kunita- Watanabe's inequality). If M, N E M 2 , f E L L , g E LL, then

Proof. First, we see that VA E R, O<(MX-N),MA-N), = ((W,- (M),)A + W t - (N),) - 2((M, N)t - (M,N),)A. So I (M, N), - (M, N),I 6 J(M), - (M), .J(N), - (N),. This intuitively However, to make it rigortells us that d [(M,N),I 6 ous we need to introduce a measure as follows: Let p(t) = J,"d 1 (M, N)l, +

i

ddo,.

+

2.5 Stochastic Integrals with respect to Martingales

53

+

(M), (N), . For any given rational number X let v(t) = X2 (M), 2 1 (M, N), ( N k , then 4 s, 4 t ) - 4 s ) = ((M)t - (M),) +2X((M,N)t - (M,N),) + ( W t - (N),). Notice that I(M, N), - (M, N),I 5 J M ) , - ( M ) , d ( N ) , - (N),, we have v(t) - v(s) 2 0, i.e. v(t) is increasing in t. Hence dv = x 2 W + 2 ~ 2 0, dp - a.e. ~ C L &e above mequality holds for all X E Q , since Q is countable. Hence one has that 5 dp - a.e. ~ o i i c that' e now the abdve inequality is rigorous, so we can use it to obtain the final result. Actually, by Schwarz inequality

1

>

+

1 JqJi,

So' If 5

+

(u)g(u)l Id (M, N)I, = So' If (u)du)l

~ o 'MU) I JF

5

1-

d~

JZKFGKJEiGFGK.

The proof is complete. Now, for the convenience to the reader, we write out the properties of stochastic integral with respect to martingales M E M2 as follows:

Proof. (i) is obvious. Since I M ( f ) E M2, SO (ii) is true by Doob's stopping time theorem. (iii) and (v) can be checked first for f , g, h € Lo, and then for f , g E L&, h E L$ by taking a limit. (vi), (vii) and (viii) are easily proved by Definition 85. Finally, let us show (iv) by checking that definition 85 is satisfied. In fact, writing X u ( t ) = X ( t A a), by Doob's stopping time theorem VN E M2, ( I M ( f ) u ,N), = ( I M ( f)O, NU), = ( I M ( f ) , = Sy'f ( 4 d (M, N), = So' f ( 4 I u l u d (M, N), .

NtAU

54

2. Brownian Motion, Stochastic Integral and Ito's Formula

For the properties of stochastic integral with respect t o martingales M E M 2 we have the following result: Theorem 89 For M , N E M2>'"",f , g E CL, h E C$ the stochastic integral I M ( f ) has the following properties: (1) Properties (i), (iv), (vi), (vii) and (viii) in Theorem 88 still hold true. (2) There exists a sequence of stopping times a , T co such that (ii), (iii) and (v) hold true for t and s substituted by t A a, and s A a,, respectively.

2.6 Ito's Formula for Continuous Semi-Martingales In calculus i f f ( x ) ,x ( t ) E C 1 and both are non-random, then df ( x ( t ) )= f 1 ( x ( t ) ) d 4 t ) . However, this formula for a random process x(t,w) is not necessary true. For example, one can show that

or symbolically, we can denote it by d lw(t)12= 2w(t)dw(t) dt, where w ( t ) is the 1-dimensional standard BM. Actually, this is a special case of the famous Ito formula. So one can understand how important and useful Ito' formula is in stochastic analysis and stochastic calculus. Let us see how (2.7) holds true. In fact, let O = t o < t l <..-
+

+

E 121: w(s)dw(s)- IF(BI = 2E Cy=l

Iw(s)

Iw(~)

-

+

2

2

=E

12 c:=~ J:-l(w(s) - w(ti-l))dw(s)l

ti-1)12 ds

=2 ~ S , t - Cy=l ~ ( t i - l ) ~ s ~ ~ t ~ - ~0,, t ~ ~ l ~ d s as [An[ = maxl
2.6 Ito's Formula for Continuous Semi-Martingales

55

where for the second, third, and fourth equality sign above we have used the result that for the cross term as i < j, E[(& - law(ti)12)(&i- lAw(tj)l2)] = E[(& - l a w ( t i ) 1 2 ) ~ ( (&lnw(tj)1~)15t,)] j = 0, and Aw(ti) N(0, Ati), SO E lAw(ti)12= Ati, and E lAw(ti)14= 3 ( ~ t ~ ) ~ . So (2.7) holds. By the above proof we see that approximately, by Taylor's formula n(lw(t)12)= 2w(t) ~ ( t )( A w ~ ( t ) ) ~ . However, E [ ( A w ( ~-) )n~t ] = o. So letting A t + 0 one can guess that in the L2(R)-limit it is possible to obtain the result that d lw(t)12= 2w(t)dw(t) dt. However, this result would not happen in the determinate case. Because if x ( t ) is non-random, then o ( A x ( t ) )= o(xl(t)A t o ( A t ) )= o ( A t ) is a higher degree of infinitesimal relative to At. So by the definition of the first degree of a differential one has that A(lx(t)12)= 2 4 t ) A x ( t ) o ( A x ( t ) )= 2x(t)dx(t) o ( A t ) = d(lx(t)12)+ o p t ) . Therefore, d(lx(t)12)= 2x(t)dx(t). The idea of showing (2.7) now motivates us to show the general Ito formula for continuous locally integrable martingales. Suppose now that M = {Mt),,o E M2,c.Naturally, comparing to (2.7) we want to show that

+

+

+

+

+

+

[ ~ ( t= ) l2 ~f M ( s ) d ~ ( s ) ( M )(t). 0

Proceeding as above, only substituting w ( s ) and ds by M ( s ) and d ( M )( s ) , respectively, one sees that similarly, lM(t)12= ~ L 1 ( 1 ~ ( t i ) lti-1)12) ~ = EL12M(ti-1) A M ( t i ) ~ y [AM(ti)12 = ~ = IF(t) I z ( t ) , where A M ( t i ) = M ( t i ) - M(ti-1). However, to use the Lebesgue's dominated convergence theorem we need to introduce the stopping time: for each N let

+

+

56

2. Brownian Motion, Stochastic Integral and Ito's Formula

Then as lAnl 4 0,

E 12~:"'" M ( s ) ~ M ( s-) ~ r (Atr N ) l

2

l2

= 2~ Jot"'" IM(s) - EL1 ~ ( t i - 1 ) 1 e ( t , - , , t , ~ d ( M )( s ) 4 0. 2 However, to show E I ( M )(t A T N ) - I;(t A T N ) ~ 4 0, as lAnl 4 0, requires more discussion. One cannot simply copy the proof of (2.8), because now the property of Normal random variables cannot be applied here. Not ice that

E I(M) ( t )- 1;(t)l2 = ~ [ C r = l ( l A ~ ( t-i )Al (~M )(ti))I2 =~ [ ~ b l ( l ~ -A ~( M ( t)(ti))')] i)l~ < 2 ~ [ ~ y = t = , ( l A ~+( (t Ai )(lM~ )(till2] 5 2Em=l
+

<

Lemma 90 Assume that M E M~~~~~~~ such that IM(s)l where C is a constant. Let

< C,Vs E [0,t ] ,

= E [ C ( l ~ l ~A( Tt Ni ) I ' - A ( M )(ti A T N ) ) ] -t ~ 0.

i=l

(2.11)

2.6 Ito's Formula for Continuous Semi-Martingales

57

Thus we have roved that for each N 4' IM(t A T N ) ~= 2 SotATNM(s)dM(s) (M) (t A TN). Letting N f oo, one obtains (2.9). Furthermore, by taking a sequence of stopping times o n f oo, one immediately obtains that for M E M2>10C.C, (2.9) still holds. Keeping the proof and result of (2.9) in mind, one easily deduces the following proposition.

+

Proposition 91 (Ito's fornula). Iff (x) E C2(R), and M E M2*'""+, then

Proof. Note that if we put f (x) = 1x12,then Ito's formula (2.12) becomes (2.9). So we can proceed almost completely as in the proof above. We only show the result for M E M21c. In fact, by Taylor's formula, f (M(t)) - f (Mo) = C X f UWti)) - f (Wti-1))) = EL1fl(M(ti-1)) M(ti) Cy=l ff'((E-l) l A ~ ( t i )= l ~I r ( t ) +12(t), where = M(ti-1) +0i-l A M(ti),O 0i-1 1. Hence as IAnl -+ 0,

++

+

<

<

Furthermore, by (2.1 1) I&(t A TN) 2 SUPlzl<~If (x)I E[CLl(lAM(ti A rN)12 - A (M) (ti A T N ) ) ~ ) ] 0So (2.12) is true for W A 7 N . Letting N f cq the proof is complete. H Now let us consider Ito's formula for the continuous semi-martingale. Suppose that xt = X O +At Mt,

+

2. Brownian Motion, Stochastic Integral and Ito's Formula

58

where xo E $0, {At)tlo is a continuous finite variational (&-adapted) process with A. = 0, {Mt),,, E M2'"c'C.We will call such an xt a continuous semi-martingale. The game proof will show the following result. Theorem 92 If f (x) 6 C2(R), then

where lAlt is the total variation of A,, s E [0, t].Then one easily sees that by Lebesgue's dominated convergence theorem as lAnl + 0,

E

1~:"'"

2

f1(Ma)dA8- I r ( t A rN)I

~~otr\7~

= If'(Ms) - Cy=lf l ( ~ ( t i - l ) ) 1 s c ( t ; - l ~ 1 ~ d(s) l ~ l 0The rest of the proof is completely the same. So the proof is complete. +

2.7 Ito's Formula for Semi-Martingales with Jumps In the practical case we will always encounter some stochastic perturbation with jumps. So we need an Ito formula for a semi-martingale with jumps. Consider xt = $0 At Mt Jz f ( 5 z,w)Np(ds, dz) J:+ JZg(s, z, w)fip(ds, d.4, where xo, {At)tlo and {Mt)tlo are the same as in Theorem 92, and p is a &-point process of the class (QL), f E .Fp, g E such that f (s, z, w)g(s, z, w) = 0. We will call such a xt a semi-martingale with jumps. Here the last two terms are called jump terms. All jumps of xt are caused by them, and the condition f (s, z, w)g(s, z, w) = 0 means that the last two terms do not have the same jump times. We have the following general Ito formula:

+ g+

+ +

+

~

2

7

'

~

~

2.7 Ito's Formula for Semi-Martingales with Jumps

59

Theorem 93 If F ( x ) E C 2 ( R ) ,then

holds. Moreover, (2.15) also can be rewritten as F ( x ~-) F(xo) = F1(xs)dAS Ff(xs)dMs $ F1'(xs)d( M ) , Jz F t ( x S - ) g ( ~2,,w)fip(ds,d z ) + r & F1(xs-)f(s7z , w)Np(ds,dz) + J,[F(xS-- + f ( s ,z , ~ )-) F(x,-) - F1(xS-)f(s,z,w)lNp(ds,dz) + J", JZ[F(x.- + g(s,2 , w ) )- F(x,-) - Ft(xs-)g(s,z, w)lNp(ds,a), or, more szmply,

+r

Jot

+ Jot

+

Jot

Proof. To show the result our scheme is as follows: We first rewrite xt as the sum of two parts: a continuous semi-martingale part and a pure jump part, then apply Ito's formula to the continuous semi-martingale part, and after that add the jump part at all jump times to obtain the general Ito formula. 1) Assume that F ( x ) E C i ( R ) ,i.e. F has up to second continuous derivatives and all derivatives are bounded. To simplify the notation we may first assume that g E $. Let Un T 2,Un E !Bz such that ENp((O,t ] Un) , < oo,Vt,Vn. For each n let xr = xo Mt At Jz f n ( s , z , w)Np(ds,dz) Jz gn(s, z, w)Zp(ds,dz) = $0 Mt At

+ + + Jot+

+ Jotf

+ +

Jot+ 9% ~ , w ) G p ( dd.4~ , +Jot+ J,(f + g>(s, w)Iu,,(z)Np(ds,dz), where f n(s,z , w) f ( s ,z, w)Iun( z ) , and gn is similarly defined. For the -

JZ

2,

=

pure jump term (the last term) the counting measure only counts the numbers of the point p(s) = z falling in U,. Since

2. Brownian Motion, Stochastic Integral and Ito's Formula

60

ENp((O,t],Un) = E(#{O < s I t : ~ ( s E) Un)) < m , where # means the numbers of counted in the set {.) , which we introduced in Chapter 1. So # {0 < s 5 t : p(s) E Un) < co, a.s. That is to say, for a.s. w, there are only finite points of s E (0, t] such that p(s, w) E U,. So we can denote these points by 0 < q ( w ) < . . . < a,(w) < . . . . Let us show that for each m, a, is a stopping time. In fact, {am(w) 5 t) = {Np((O, t], Un) 2 m ) E 8t. For convenience we also write a0 = 0. Then XT = xo Mt At JZ gn(%wJ)Gp(ds, dz) + Cam5t(f + g)(am(w), P(am(w),w),w), F ( G ) - F(xZ;.)= C,IF(xE,r\t) - F(xE,At-)l + Crn[F(x:,At-) - F(xE,-~A~)I= Il(t) + I2(t> t; and F(xEmAt-) = F(xy-) = Note that F(xEmAt-)= F(xEm-), as B , F(xy), as a,-1 < t < a,. Since as s E (am-1 At, a, ~ t )x:, has no jumps. So we can apply Ito's formula for the continuous semi-martingale to obtain u,At a,At F(xEmAt-) - F(xEm-lAt) = F1(x:)dA, + Jum-,~t F1(x:)dMS

+ +

Jgt+

<

JU,-,~t a At

+%.f,"i-TAtF1l(x?)d (M)s - J u _ " _ l A t

F1(x?)dA: where AT = Jz gn(s, z, u)Gp(ds, dz), and we have used the fact that by the continuity of A,, ;S :-"I, Ff(x:)dA, = m-l F1(xy)dA,, etc. Taking the summation we find that Ff(x:)dM, Iz(t) = F1(x:)dAs F" (x:)d (M), - F1(x:)dA:. Now let us evaluate Il(t). By assumption f g = 0, so 1' (t) = Crn[F(xE,At) - F(~E~/\t-)IIu,
SzmA:t

+

Jot

+; Sot

+

+I,'+

E

SI?

+ +

+

2

Jz(gn(~,z,w) - S(S,z,w))fip(d~,dz)I

SOT SZ

=E Ign(s,z, w) - g(s, z, w)12Gp(ds, dz) + 0. Hence, if necessary, by taking a subsequence we m_ay assume that a.s. g n ( ~z,, w)Np(ds, dz) =rt s(s, 2, w>Np(ds,d ~ )on , any 10, TI, where " 3 " means uniformly convergence. Similarly, Sgtf JZf n ( ~z,, w)Np(ds, d 4 =f Sgt+JZ f (s, Z, w)Np(ds, dz), on any [O,TI. Hence xT- and x; converge uniformly to xt- and xt on any [O, TI, respectively. From this, letting n + co, one easily obtains that (2.15) holds true.

Jgt+ SZ

Sot+Sz

2.7 Ito's Formula for Semi-Martingales with Jumps

For example, let us show that as n

-+

61

oo,

+

where hn(s, z, W) = F(x: gn(s, z, w)) - F(x:) h(s, z, w) is similarly defined; and

- F1(x:)gn(s, z, w), and

(2.18) where Xn(s, z, w) = F(x:- +gn(s, z, w)) - F ( x E ) , and x(s, z, w) is also similarly defined. In fact, Vs, z,and a.s. w, hn(s, z, w) -+ h(s, z, w). Moreover, since F E C t , Ihn(s, z, w)l i SUP% IF1'(x)l Ids, z,w)12 i z o I ~ ( s , ~ , w E3 ); I.~ Hence applying Lebesgue's dominated convergence theorem, one obtains (2.17). . , Notice that

I; = E and

JFjz I;ys,

l2

w ) - K ( ~z,,w) ~ ~ (dz), d ~ , 2

i supx 1 ~ ~ 6 l g3 ( 1~z,w)12. ,~ So (2.18) is also similarly obtained. Thus (2.15) is true under the assumption that g E 3'. However, for g E p>"c, 30, f oo, a, is a stopping time for each n such that I p u n l(t)g(t, Z,W)E 3'. Hence one easily sees that (2.15) is true for t substituted by t A a,, W,Vn. Letting n f oo, one derives (2.15), Vt 0. 2) Now assume that F(x) E C2(R). First let us show that each term in the right hand side of (2.16) has meaning. In fact, since { x ~ ) ~is , ~ a right continuous with left limit (RCLL) process, { x ~ - ) , , ~is a locally bounded process, i.e. 30; f oo, a; is a stopping time for eaEh n such that {x(tAaL)-)t,o is bounded, for each n. Indeed, let a; =inf{t 2 0 : lxtl > n ) . Then 4 n,Vt 2 0. So J:""' F1(xs)dMs = J:"" F1(xs-)dM, makes sence, so also does F1(x,)dM,. On the other hand, since g E

,

,

>

Ix(~, - , ~ )-~

Sot

32,loc

Jz

2

NP(ds, dz) < m, a.s. because by 1) E Jz ids, z , 4 1 2 Np(d~,d.4 = E J:"~" JZ lg(s, z, w)12 Sp(ds, dz) < oo,Vn. Notice that for each w E R, { x , ( ~ ) ) , ~is~RCLL, ~ , ~ ~so it is bounded; that is, 3ko(w), such that Ix,(w)l 5 ko(w),Vs 's [0, t]. Hence 3io(w) is such that IF1(x)l IF1'(x)l 5 &(w), as x E [-ko(w), ko(w)]. Thus, Cs,D ..,[F(xs+ g(s, p(s), w)) - F(G-1 - F1(xs-Ids, p(s), w)l c ~ ; D ~ I ,~ ~ ( sP(s), ~, ~ w)12=

J,tAUn

+

P I

z,

2. Brownian Motion, Stochastic Integral and Ito's Formula

62

c.,Dp.,,t

5 ;kcw) Similarly, by

w ) ~< 2 m.

JZ

C,,Dp,.st If (3,p(s), w)l = J,'c If (s, z, w)l NP(d%dz) < one has that + f (Q)P(s),w))- F(xS-)l CSEDp,85t[F(~s-

as-

5 5 ( 4 E,,DP,,,t If (%P(.),.)l < m. Therefore one easily deduces that the right hand side of (2.16) makes sense, and so also does (2.15). Now let WN(X)E Cm(R) such that WN(X)= 1, as 1x1 5 N ; WN(x) =0, as 1x1 > N + 2 ; and [W,!,,(x)l 51,IW{(x)1 1,Vx~ R, where W,!,, and W{ are the first and second derivatives, respectively. ). IFN(x)I IF(x)I , (F,!,(x)~ IF1(x)l set FN(x) = F ( X ) ~ N ( X Then IF(x)I, IF{(x)I IF1'(x)1+2 IF1(x)l+lF(x)l. Moreover, FN(x) = F,!,,(x) = F{(x) = 0, as 1x1 > N 2. Hence for each N, FN(x) E Ct(R). Thus by 1) Ito's formula (2.16) holds true for FN.Letting N f oo, by Lebesgue's dominated convergence theorem one easily obtains that (2.16) still holds for F.

<

<

<

<

+

+

2.8 Ito's Formula for d-dimensional Semi-Martingales. Integration by Parts. The above Ito's formula for one-dimensional semi-martingales with jumps is easily generalized to that for the n-dimensional case. Consider a ddimensional semi-martingales with jumps as follows: xt = (xi, . - - ,z t ) , where for i = 1 , 2 , . . . , d xf = xb Af Mi

+So"+

+ +

Sot+

f "s, z, w)Np(ds, dz) + Jz sib,z, w)fip(ds, d.4, where xo E 30, {At),,, is a finite variational (St- adapted) process, and 2 1 are d- dimensional, and p is a &-point process {Mt),,, E M2*loctc, such that fi(s, z, w)gj(s, r , w) = of the class (QL), f E FP, g E 0, Vi, j = 1,2, - - . ,d. Then we have the following theorem, but as the proof is completely the same we omit it.

~ t * ~ ~ ~

Theorem 94 (Ito's formula). If a real function F(x) E C 2 ( ~ d )then ,

2.8 Ito's Formula for d-dimensional Semi-Martingales. Integration by Parts.

63

Remark 95 1) If we denote [xi,xjlt = (xic,x ~ c+)&~, ( A x ~ A x ~ ) = ( M ~~3, ' ) t + C s 5 t ( n x i n x % which is called the cross quadractic variational process (cross characteris, (2.19) can be rewrittics) of semi-martingales { x i ) t 2 o and { x ~ )-, , ~then ten as FLi(xs-)dx: c & = ~ J: F : ~ , ~ ( X , - ) ~ [ X ~ ,xi], F ( x t ) - F(xo) = J,"+ JZ[F(xs- f ( s ,z , w ) ) - F(xs-) - C;=l FIi (%a-)fi(s,z,w) x t j = , F:ixj (~s-)fi(s,z,~)fj(~>z,~)]Np(d~,dz) J:+ Jz[F(xs- g(s,2 , ~ )-) F(xs-) F',i(~s-)gi(s, z,w) x t j = l F.*,j (xs-)gi(~,z,w)g'(~,~,w)]Np(ds,dz)]. 2) Ito's formula (2.19) can also be written symbolically in differential form as dF(xt) = FJi(xt-)dx; x & = , F;ixj (xt)d ( M i , JZ[F(xt- f (t,2, w)) - F(xt-) - c:=, FLi(xt-)fi(t, z,w)lNp(dt,dz) Jz[F(xt- + g(t,z l w ) )- F(xt-) - c:=, FLi(xt-)gi(t,z,w)1Np(dt, dz), or,

+

+

+ -;

+

-4

+ +

+

,&

+

c:=,

+

By Ito's formula one easily derives the formula of integration by parts for semi-martingales with jumps. Suppose the semimartingales { x ~ ) t t oare given as above, i = 1,2,. . . ,d. Then we have the following thoerm.

Theorem 96 (Integration by parts). d x ~ = x ~xfdxi or equzvalently, - X ~ X ;= Jot x ; ~ x :+ x : ~ x ;+ [xi,xjIt.

+ ddxf + d[xi,xjIt,

Ji

Proof. Let F ( x ) = xixj, for x E Rd. Applying (2.20) one arrives a t the conclusion of the theorem.

64

2. Brownian Motion, Stochastic Integral and Ito's Formula

2.9 Independence of BM and Poisson Point Processes As an application o f Ito's formula we can prove the independence o f BM and Poisson point processes, which is very important in stochastic analysis. For simplicity let us first discuss the independence o f a 1-dimensional Brownian motion and a Poisson point process. Theorem 97 Assume that { x ~ ) is ~ a, 1-dimensional ~ 5t-semimartingale, and p is a gt-point process of cl& (QL). If ( M ) t = t; 1) Mt = xt - xo E 2) the compensator fip(dt,dz) of p is a non-random a- finite measure on [O, co) x 2; then { x ~ ) , ~is, a 1-dimensional St-BM, and p is a 3t-Poisson point process such that they are independent. ~

~

,

~

~

~

l

~

,

>s >0

Proof. I t is enough t o prove that W

m

E[exp(iX . (xt - ~

exp(-

~ i ~ p ( t(] sx ,Ui)ISs]

3 ) )

i=l m

(2.21) = exp[-(t - s ) 1Xl2 /2] e ~ ~ [ C ( e --' l~) f i p ( ( s , t ]Ui)], , i=l

yhere X E R,Xi > 0 , i = I , . . . ,m; and Ui E S z , i = 1,2,... ,m such that N,([O,t],Ui) < co,Vi, and Ui n U y = 4, as i # j . Let F ( x , y l , . . . ,ym) = eiAxe-CZ1 Aiy'. Apply Ito's formula t o the 1 + m-dimensional semi-martingale ( x t ,y:, - . . ,y r ) , where xt = x t , y; = N,((O,t],Ui) = s:+S~ Iui(z)NP(dt,dz),i = I , - - . ,m, and denote Nt = ( N , ((0,t ] ,U l ) ,. . . ,N,((O, t ] ,Urn)),then A F = F(xt,y:,- ,YF! - F ( ~ s , ~ s l , . .,YF) . eihxYe- C Z l A i v : d ~ - 2 eih,e-CZ1 &yidu

~ X L ~

u

st

+ sSt:J Z [ F ( x u Nu, + I u i ( z ) )- F ( x u , ~ , - ) l & ( d u , d z ) + Sz[F(xu,Nu- + I u i ( z ) )- ~ ( x ~ , ~ ~ - ) l f i ~ ( d u , d z ) . 2

3

s::

However, F(xu, Nu- + ILli(z))- F(xU,Nu-) = e i A ~ u e - C Z ~ A i N ~ ( ( O ~ " l ~ u i ) . .(e- C Z i AiIui(z) - 1). Therefore,V A E 5,, multiplying all terms in the expression o f A F by e-iAxu&bl A i N ~ ( ( o ~ u l ~ u i )and ~ A taking the expectation, one finds that ~ ~ i * ( x t - x * ) ~ - Z* Zi N l ~ ( ( % t l l u i )-~P3((At ) = -k$ E I ~ v ' ( ~ ) ~ ~ A"ui(Z) - l)GP(du,d z ) + J: E I A V 3(u)S,(e= -x" 2 J st E I ~ V ~ ( U ) Sat ~U EIAV'(U) x z l ( ( e - A i - 1)fip(dU,Ui), where V " ( t ) = e i A ( x t - x a ) e - C E i N ~ ( ( 3 ~ t l ~ u ~Note ) . that, the ordinary differential equation

sSt

+

2.10 Some Examples

65

So

~ ~ i x ( x t - x . ) CZI ~h i N p ( ( s , t ] ? U i )= P(A)e- $ ( t - s ) e ~ ~ l ( e - " - l ) f i p ( ( s , t ] , ~ j ) The proof is complete. rn The above theorem is easily generalized to the d-dimensional case.

Theorem 98 Assume that { x ~- ) is~ a, d-dimensional ~ ~t-semimartingale, where xt = ( x i , . . . ,x f ) , and pi, i = 1,2, . . - ,n , are &-point processes of class (QL) on state spaces Zi, i = 1,2, . . . ,n, respectively. If E ~ 2 l l " c t c , ( M ~ , M=~d i)j t~; i,j = 1,2,. . . ,d, 1) t - x i t 2) the compensator N,, (dt,dz) of pi is a non-random a- finite measure on [O,oo)x Z , i = 1,2,... ,n; and the domains Dpi(w),i= 1,2,.-. ,n, are mutually disjoint, a.s. Then {xt)t20 is a d-dimensional St-BM, and pi (i = 1,2,. . - ,n) is a St-Poisson point process such that they are mutually independent. A

Proof. The only thing we need to do is to combine all point process pi, i = 1,2, . . . ,n, into one new point process p on a new space Z. Then we can prove (2.21) in exactly the same way as for xt and p, only with X E R -d substituted by ;i= ( X I , . . . ,X ) E R ~and , denote the inner product o f two d -i x; that is, X . x = x i = , X xi. For d-dimensional vectors and x by this we introduce a space Z such that Z = Uy=lHi, where Hi, i = 1, . - - ,n, are disjoint such that for each i, Hi is in 1 to 1 correspondence with Zi. For simplicity let us identify Hi and Zi. So we can set Z = ULIZ.i. Note that B z = Uy=CBzi, and all Bz,,i = 1,2, . . . ,n are mutually disjoint. Now let D, = Uy=l Dpi, and set p(t) = pi(t), as t E Dp,. Then we have a point process p on Z, an! it is easy to see that p_is a point process of the class (QL). Moreover, N,(dt, dz) = EL1Iz, (z)Npi(dt,dz). Obviously, @,(dt, dz) is a non-random a- finite measure on [O, co) x Z. Hence by the proof of Theorem 97 { x ~ ) , ,is~ a d-dimensional St-BM, and p is a &-Poisson point process suchthat they are independent. Note that for each i, pi is also a &-Poisson point process by Theorem 97. However, we still need t o prove that all pi, i = 1,2,. . . ,n are mutualy independent. In fact, i f Ai E Bzi,i = 1,. . . ,n, then they are mutually disjoint. Moreover, {N,, ((0,ti]x Ai))y=l = {Np((O,ti] x Ai))y=l is an independent system o f random variables, since p is a Poisson point process. The proof is complete.

-

rn

2.10 Some Examples In Calculus if x ( t ) = f ( y ( t ) )= e ~ ( y(t) ~ ) ,E C 1 , y(t) is non-random, then

66

2. Brownian Motion, Stochastic Integral and Ito's Formula

dx(t)= e ~ ( ~ ) d ~ ( t ) . However, by Ito's formula we will see that some extra terms will occur in the expression for d x ( t ) , when y(t, w ) is some random process.

{ - Jt20

Example 99 Assume that { ~ t ) is, a~dl-~ dimensional BM, Nt is a stationary 1- dipensional centralized Poisson process with the compensator At; that is, Nt = Nt - At, where X > 0 is a constant, and Nt is a Poisson process such that E N t = At. Suppose that b(t,w) is a St-adapted Rlmdl -valued process and c(t,w ) is a Zt -predictable R' - valued process such that for each T < oo ~ ) l ~ ]
+

+ + +

+ - +4 ++ +

+

+4 P

+

+

+

+

Proof. 1): Let yt = J, b,dw, c,dfi,, f ( y ) = eY. Applying Ito's formula to f ( y t ) ,we obtain the result. t 2): Let yt = b,dw, - J, lb,12 ds. From Ito's formula applied to eYt result 2) then follows.

So

3

2.11 Strong Markov Property of BM and Poisson Point Processes

3): Let xt =

Sot bSdws +

67

csdfis. Applying the Ito formula:

to F(xt) = x: and F(xt) = x:, one obtains the expressions for x: and x?, respectively. Notice that s t is a martingale with xo = 0 and by Ito's formula again one has that =E

EX!

SI',

2

bsdwsl

+ E l~,'c.d?l

2

=~

+

S i [ 1 b ~ hlcs12]ds. 1~

Moreover, s t = Nt - At, and ENt = 0, EN:

= At. From these results one easily deduces the expressions for Ex: and Ex:. Finally, applying the Schwartz inequality one obtains the last inequality. rn

2.11

Strong Markov Property of BM and Poisson Point Processes

The martingale characterization of BM and Poisson point processes (Theorem 98) can be used to show that a BM is still a BM if it starts again from any stopping time, and this property is also true for a stationary Poisson point process.

Theorem 100 If xt = (x: , . . . ,xf) is a d- dimensional St -BM, and a is a ;Ft-stopping time with P ( a < oo) = 1, then {x;),,~ = {xt+u)t20 is a d-dimensional 3; = 3t+,-BM. In particular, { w , * ) , ~=~ { ~ t + xu)t2o ~ is a standard BM independent of 3; = z u . Proof. Let a, = a h . Then a, f a , and for each n, a, is a bounded stopping time. By Doob's stopping time theorem {MT)t2p = {xtfo, E M2*cwith respect to {3t+u,)t20, and MFMp3 - (Mni, Mnj)t is a ~= &(t j )a, -~a,) = Sijt. So ~t+u,-martingale. However, (Mni, ~ for each n, by Theorem 98 {M,")t20 is a {$t+b,,)t20 -BM. Furthermore, since St+,, f 3;. By Levi's theorem one also has that Vt > s, E[w; I3:] = E[xt+u - xu $I]: = xs+u - XU = w:. 2 In fact, sup,E[lMpl ] = t < oo,K = 1 , 2 , . . . , d. Hence { M n=l ~ }is ~ uniformly integrable for each i. Therefore, as n 4 oo, E E [ M~ w,*13~+,,,]E ~p - W; + 0. On the other hand, by Levi's thoerem as n -+ oo, E[w;l3t+Unl E[w,*I3t+ul= E[w;l3;1, a s . Hence as n 4 GO, the equality E[M,nZ (3t+,,]= M y will tend to E[wzi IS;] = w,*i in L1(fl, P), where t > s;Vi. Similarly, one sees that

+

I

-

1

I

1

68

2. Brownian Motion, Stochastic Integral and Ito's Formula

(w*~,~ * j=)bij(t+a-a) ~ = Sijt holds. Hence, by Theorem 98, { ~ f ) = , ~ ~ {xt+, - x , ) ~ , ~ is a standard BM independent of 5: = 5,. Moreover, { x ; ) ~=~{x~+o)t20 ~ = {w: + xu)t20 is a d-dimensional s f = Zt+,--BM with the initial random variable xu E 3;. The proof is complete. Theorem 101 If p is a stationary &- Poisson point process on some , a is a {&}t,o -stopping space Z with the characterictic measure ~ ( d z )and is time with P ( a < oo) = 1, then p* = {p*(t))t,Dp, = {p(t a stationary = { ~ t + o ) t > o-Poisson point process with the same characterictic measure ~ ( d z ) .

+

Proof. The proof is similar to that of Theorem 100. Let a, be the bounded stopping time defined in Theorem 100. By Doob's stopping time theorem -martingale for each {Np((a,, t D,] x U) - ~ T ( U ) is) a~ ~ ~ n, where U E I?,; that is, ENp((O,t] x U) < oo, W. Furthermore, by Levi's theorem and taking the limit as in the proof of Theorem 100 one easily sees that {N,((u, t a] x U)- ~ T ( U ) )-, , ~is a {5t+u),lo -martingale. So the proof is complete. rn By the previous two theorems one immediately sees that a standard BM and a stationary Poisson point processes are both stationary strong Markov processes. That is to say, if { ~ t ) ~is ,a~d-dimensional &-standard BM, then for any zt-stopping time ;with P ( a < m) = 1, it satisfies VA E %(R~), w > 0, p(wt+, E At??,) = P(wt E Al30), a.s. In fact, by Theorem 100 one has that it is equivalent to P(wf E A) = P(wt E A) P(w; E A[%) = P(wt E Al&) where { ~ f ) ~ , is , a 3;-BM. The last equality is obviously true. It is naturd to define the strong Markov property of a stationary Tt-point process p as follows: If p satisfies that Vt > 0, Vk = 1,2,. . , P(Np((O,t + a ] x U) = kl&) = P(Np((O, t] x U) = kl30), a s . , then p is called a strong Markov st-point process. Since the proof is the same we omit it. Thus we arrive at the following corollary.

+

+

zt

zt

Corollary 102 The - standard BM and - stationary Poisson point processes are both stationary strong Markov processes.

2.12

Martingale Representat ion Theorem

a:7P-

In this section we are going to show that any square integrable Martingales can be represented as a sum of an integral with respect t ~the ! BM { w ~ ) and , ~ ~an integral with respect to the martingale measure N,(dt, dz) generated by the Poisson point process p. Here $TrP is the smallest a- field such that makes all w,, s 5 t; and all fi,((0, s],U), s 5 t, U E !BZ mea-

2.12 Martingale Representation Theorem

69

surable. Such a representation theorem is very useful in the mathematical financial market and in the filtering problems. More precisely, we have the following theorem. Theorem 103 . (Martingale representation). Let m(t) be a square integrable Rd-valued -martingale, where $y'k is the a- algebra generated (and completed) by {w,,k,,s 6 t ) , and {wt),,, is a dl- dimensional BM, {kt),,,- is a stationary d2- dimensional Poisson point process %lk

of the class (QL) such that the components {k:)tLo,. .

,{k?)+,"

disjoint domains and disjoint ranges. Then there exists a unique L&, (RdBd1)x F&, (RdBdz) such that

m(t)= m(o)+ J; q,dw,

". -

have E

+ J; Jz P . ( Z ) ~ ( ~d S~ ,) .

Here we write L ; ( R ~ @=~{ f~( t), w ) : f ( t , w ) is zt-adapted, ~ ~ @ ~ l - v a l usuch e d that E r l f ( t , w ) l 2 d t < co, for any T < co ) and F $ ( R ~ @ ~=" { f (t,Z , W ) : f (t,Z , W ) is Rdmd2-valued, Tt-predictable such that E If (t,Z , w)12r(dz)dt < m, QT < co.) T h e following two lemmas are useful and interesting. Lemma 104

.$7$

=

c ' w~2 ,0.

Proof. Let Hn(tl,tz,.. . ,tn; f i , f 2 , ... , f n ) = Hn-l(tl, t27 .. . tn-1; f i , f 2 , . . . fn-2, fn-lHtn-t,-lfn), H i ( t ; f )( x ) = H t ( f ) ( x )= JR, P ( ~ ,-x ~ ) f ( y ) d y , VEf C o ( R d ) , exp[- 1x12 /at], p(t, x ) = and_ H$itl,t2,... , t n ; f i , f i , . . . , f n ) = HA-l(tl,t2,... , t n - l ; f i , f i , . . . ,fn-2,fn-lHtit,-t,-lfn), @(t; f ) = @ ( f ) , Q f E Co(Rd), 00

( @ ( g ) ) (m)= =

m

C

C g ( n +m)e-tn(uj) (tr( U j ) ) n/n!

n=O g(n)e-t"((.'j)(tr(Uj))n-m / ( n- m ) ! ,

n=m where 0 = to < tl < t2 < . . . < tn, f l , f 2 , . . . ,f n E Co(Rd), Ui n U j = q5,i # j,r(Ui) < m,i = I,..-m;Uzn=,Ui= 2, and C o ( R d )= { f : f is a continuous function defined o n Rd such that limlz~-t, I f (x)l = 0 1. Hence, i f t k - 1 t < t k , by the stationary Markov property and the independent property o f BM and Poisson process ) gl(Nk((O,tl]rU j ) ) E [ f i ( w ( t l ) ) f i ( w ( t 2 )' )' ' f n ( w ( t n ) ny=1

<

2. Brownian Motion, Stochastic Integral and Ito's Formula

70

'9dNk((07t'21,Uj)) =

-

gn(Nk((0,tn],Uj))]

nE fi(w(ti))ngl g i ( ~ k ( ( 0ti],,

~ j ) )

nGl gi(Nk((0,ti - t ] ,

- E ( " ~ ' ~[n:=, ' ) fi(w(ti - t ) )

nk:

n&

~ j ) ) ]

fi(w(ti)) gi(Nk((0,ti],U j ) ) ~ ( w t '[ny=k k t L ) fi(w(ti - t ) ) ] .E("~'") gi(Nk((0,ti - t],Uj)) = - fi(w(ti)) . gi(Nk((0,ti],~ j ) ) Hi-k+l(tk-t,tk+l-t,"' , t n - t ; g k , g k + l , ' " ,gn)(Nk((O,t], Uj)) .~i-k+~(tk t , tk+l - t 7 . . ' tn - t ;f k , f k + l , . . ' ,f n ) ( ~ ( t ) ) . Therefore, =

nLk

n!~:

ngl

nGl

Lemma 105 . For any real-valued increasing sequence a, of ~ ~ ' k - s t o p p i n g times,

vnc: where a

=

=

Elk,

lim a,.

n-+w

?im Nk((0,~ ~ l ] U = jNk((0, ) a ] ,U j ) ,P - a.s.

n

-03

(2.22)

Indeed, if not, then as n' T oo Nk((O,anr]uj) 1Y < Nk((0,a ] ,Uj), and P(nJim --roo(Nk((0,a ] ,Uj) - Nk((0,en1], Uj))> 0) 2 P(Nk((0,a ] ,U j )- Y > 0) > 0,

Sek

lim (Nk((O,o], Uj) - Nk((0,on)],Uj))> O } . Thus by Fatou's lemma 0 < E (Nk((0,a ] ,U j )- Y )Ir = lim E[Nk((O,a ] ,Uj)-Nk((0,ant],Uj)]Ir =

{W

:

nl-w

nl--too

2.12 Martingale Representation Theorem

71

= ?r(Uj) lim E[a - anl]Ir= 0.

9%'-+w This is a contradiction. Therefore (2.22) holds. Now it is easily shown that P -as. 9i(Nk((Olti], uj)) !im E[II:=l fi(w(ti)) n -+w

n;"=,

I%'k]

nGl

1%$,

= E[ndlfi(w(ti)) gi(Nk((0, ti], Uj)) . For this, for any given w E fl, let us show a special case: a(w) = tn,ant(w) < ~ ( w )unt(w) , f a(w), as n' T oo. All the other cases are similar or even much easier to handle. For simplicity let us omit the w. For n' large enough, we have tnwl < ant < a = t,, and

]!%:1

E[II:=l fi(w(ti)) IIy=l gi(N~((O,ti],Uj)) =

fi(w(ti))

nzl

gi(Nk((0, ti], Uj))

. (Ha-ant (fn)) ant)) (&-on, (gn)) (Nk((0, ant],Uj)). However, as n' oo -+

In fact, one notes that (2?r(t - Sn,))-d/2 SRde-1~-~12/2(t-ant)f ( Y P Y= I(Rd) =I ( ~x yl < 6) I ( ~ x- yl 2 6) = Il 12. However, by the continuity of f at point x one has that for arbitrary given E > 0 there exists a 6 > 0 such that for ly - XI < 6 If (Y)- f (x)l < ~ 1 3 . Hence as ly - XI < 6, Vn'

+

+

1

111 - ( 2 ~ (t sn1))-di2 J ( ( ~ - ~ ~f<( xJ ) e - ~ ~ - ~ ~ 2 / 2 ( t - s n < ~ )&/3. dY On the other hand, one can also choose a large enough N such that Vn' ( 2 4 t - s n ~ ) ) J;I(-xI>N ~ ~ ' ~ e-l~-~la/2(t-snt)If ( Y Idy < €13, because f is bounded. Now by Lebesgue's dominated convergence theorem as snt -+ t (2?r(t - snf e-J2/2(t-snl) S(N>lx-vl>d) f ( Y ) ~ Y 0Hence there exists an No such that for n' 2 No 1 2 < &/3. Thus (2.23) is proved. Therefore in this case as n' -+ oo +

E[n:=l fi(w(ti)) ny=l gi(Nk((o,til,uj)) [%,$I E[n:=Jfi(w(ti)) gi(Nk((O,ti], Uj)) 137kI. Now denote the totality of square integrable 3$"'k- martingales M with M(0) = 0 by Dlz, and let Dl; c Dl2 be defined by m; = { W t ) = cp(s)dws + $(s1 z)Gk(ds, d.4,

npl

+

S,t

S,t S,

72

2. Brownian Motion, Stochastic Integral and Ito's Formula

(cp, $) E L;~( ~ ~ @ x F& ~ l( ~ ) ~@~a).) We have the following lemmas. Lemma 106 Every M E tM2 can be expressed as M(t) = Ml (t) M2 (t), where MI E ?JJt; and M2 E M 2 satisfies the condition that (M2, N) 0,VN EM;.

+

=

Proof. We only need to prove this result on t E [O,T] for any given T < co. We will use the following scheme: 1) We construct a (closed) subspace 7-1 ={Ml(T) : Mi E M;) c L2(R,5,P), where L2(R,5 , P ) = {[(w) : ((w) E 5 , E l((w)12 < co . Then we obtain

)

its orthogonal (closed) -complement subspace 7-lL c L2( a , 5,P). Any M E Dl2 has a decomposition for its terminal (random) value M(T) = Ml(T) Hz, where Ml(T) E 7-1, Hz E NL.2) Since M and MI are square integrable - martingales, ' ~MI ) (t) M2(t). Obviously, M2 E !TI2. M(t) = MI (t) ~ ( ~ 2 1 % = Finally, the last step is 3): We prove the last equality. Obviously, steps 1) and 2) are easily seen. Let us show 3). By Lemma 107 below we only need JZ $ ( s , z)%k(ds, dz) E tM;, to show that for any N(t) = cp(s)dw, and any ~ ~ l k - s t o p p i ntime g a T, E[M2(a)N(a)]= 0. In fact, E[Mz(a)N(a)l= EIN(a)E(Mz(T)IX1k)l= E[N(a)Mz(T)I = ~ [ ~ 2 (cp(s)l,<,dw, f l + & +(S, z)L<,Nk(d~,dz))] = 0. The proof is complete.

+

zr>k

+

+

Jot

+ Jot

<

f

Lemma 107 Assume that {Xt)t,o, {Yt)tlo with Xo = Yo = 0 are two zt-adapted random process suchthat E [Xtl < co, E < co, Vt Jt 0. Then the following assertions hold true: 1) {Xt)tlo is a zt-martingale For any bounded zt-stopping time a , EX, = 0. 2) If, in addition, E 1xtl2 < co, E 1&l2 < co,Vt 2 0, then E (X, Y), = O,Va, which is a 5t-stopping time {Xt&)t,o is a gt-martingale E[X, Y], = 0, Va, which is a &-stopping time,

1x1

*

Proof. 1): The one way implication " ==+ " is true by Doob's stopping time theorem. Now for any s < t 5 T and A E 5, let a,(w) = s, as w E A; and a,(w) = T, as w $ A. Similarly, define ut(w). Then a, is a gt-stopping time, and so is at. In fact, 'du E [O,T] {w: a,(w) 5 u ) = A E 5, c C U ,a s T > u 2 s; and {a, < u ) = $ E 5,, as u < s; and {a, 5 u) = E &, as u = T. Hence by definition a, is a 5t-stopping time. Therefore, by assumption, EX,. = EX,,, that is, E X I A EXTIA== EXJA EXTIAC. Thus {Xt)tlo is a St-martingale. 2): By the D-M decomposition theorem

+

+

2.12 Martingale Representation Theorem

73

+

XtK = a martingale ( X ,Y ) , . So the first equivalence is easily seen. (Actually, the sufficiency is derived by Doob's stopping time theorem, and the necessity is concluded from 1)). On the other hand by Ito's formula one has that YsdXs [ X ,Y ] t =a martingale + [ X ,Y ]t . XtK = 1: XsdYs So the second equivalence is also easily seen.

+

+

Lemma 108 . Let (Mt,$"'IC)t>~E 9R2. If M satisfies that EMtNt = O , V N E M ; , W > O , then M = 6 We will prove this lemma by showing the following two propositions.

Proposition 109 Under assumptions of Lemma 108 for any 0 5 s VAj E R,VUj E % ( 2 ) , i ~ ( U< j ) CQ, U i n U j =q5,i# j , i , j = 1,2,.-a , ~ ; v X E R E [ n L 1ftgt~T1z.k] = Mae- 2 ( t - ~ ) ~ ( ' - sC;l[(ei*' ) where ft = e i i ( w ( t ) - w ( s ) ) , gt = eiC$l &fik((~>tl,u,i).

-1)-Aj]r(U,)

< t,

7

Proof. Applying Ito's formula to frgr on r E [s,t ] ,where AjNk((a~rI~u,j), one finds that t llla ftZt - 1 = iA.fsfrZrdwr Jst f&dr JZ fr&(eiE?=l 4 r u j ( z ) - l ) r ( d t ) d r Jz f& (eiZE141u' )(. - l)Gk(dr,d i ) . Now for arbitrary A E 3y.k multiplying both sides of the above equality by MJA, and taking the mathematical expectation, one finds that

gr -- ,iC&

-

+ +

yt +

= EMt f t & I ~= E[M.IA] -

J&

Jz

yr(ei Cj"=1' j r u j ("1 -

J: yrdr

I 4 J: yrdr + J: y, ~ ~ = l ( e i l)a(Uj)dr, A~ where we have applied the result that by assumption for any N = E[M,IA] -

-

E M;, EMt(Nt - Ns) = 0 EMt Ns = E[E(MtlC>"N ~=] EMsNs = 0. Solving the above linear ODE, one obtains that yt = E[MsIAle- $( t - 3 ) e(t-S) Cynl( e i A j-l)r(ui). That is, ~[~,,iCj"= X jl N k ( ( " > t l > ' / i )=l ~M, k, ~ ] - ( t - ~ ) ~ ( t -Zj"=,(eiXj s) -l)r(~j) Mmultiplying both sides of the above equality by e-i(t-s)xj"=lX j " ( U i ) the conclusion of this proposition follows.

+

Proposition 110 Under assumptions of Lemma 108 for any 0 5 tl < t2 < . . . < t , 5 T,Vp= 1,2,... ,n; v A , j E R , vUp,j E %(Z),@p,j) < CQ,

-

Up,inUp,j=q5,i#j,i,j=1,2,~~~,mp;VAp~R

2. Brownian Motion, Stochastic Integral and Ito's Formula

74

Proof. The proof is a repeated application of Propositions 109. In fact, EIn;=i fpgpMT1 = E [ E ( n ; = ~~ P ~ P M T I ~ : ' ~ ) ~ = E [ ~ ; z :fpgp~(fn9nMT )Ig'k] - qnn p=l -1 f pgp M n-1 l g , k l e ( t n - t n - l ) [ - ~ + ~ ~ ~ [ e i ' n s -j l - ~ ~ , , l + J ~ , , ) l

Ici"

- . ...

.. . . .

=~

M

[

o

e

~

~

~

l

~

~

2

p

~

~

p

-

l

-l-Ap,i]~(up,j)] [ ~ ~ ~ ~ ~ ] ~=(A l [

)

~

i

'

~

~

because Mo = 0. rn Now one easily completes the proof of Lemma 108 by using Proposition 110. In fact, since BM { w ~ ) and , ~ Poisson ~ point process {Ict)t,o have independent increments, by Proposition 110 one finds that for all Founded r] E c ' k E[r]MT]= 0. Thus MT = 0, a s . for any T 2 0. The proof of of Lemma 108 is complete. Now we are in a position to prove Theorem 103. Proof. Combining Lemma 106 and 108 the result follows. rn

Corollary 111 In Theorem 103 if m(t) is a locally square integrable Rdvalued 3r'k--martingale, then the conclusion is that there ezists a unique (qt,~ tE )3Tlk with S, 1p,(z)1~r(dz)ds < co) = 1, p ( g lqs12d~ < m) = 1, V T E [O, oo), such that Vt 2 0, m(t)= m(o)+ qsd% + Jz pS(x)fik(ds,dz).

P(C

.&:

Jot

Proof. By the definition of a locally square integrable martingale there exists a a , f co, with an < co a stopping time for each n such that {mtA,n),lo is a square integrable martingale for each n. Applying Theorem 103, we have that for each n there exists a unique (qr,pT) E L%w,k(~d@'dl) x F ; , , , ( R ~ @ ' ~such ~ ) that W 2 0,

zt-

+

+ Jot

m(t A a n ) = m(0) $ q:dwS J? p:(z)lj*(ds, dz). By the uniqueness property we can, without confusion, define qt = and ~ t ( z=) P T ( ~ ) as , t E [(),an). For any given 0 < T < co since Vn,E Jz lpr(z)12s(dz)]ds< co. Hence P - a.s. Vn,C [ ~ q , " l+JZ ~ lpr(z)12r(dz)]ds< co. Therefore, P - a.s.

~r

C[lqrl2 +

S,T[lqs12+ J Z lps(412r ( d 4 l d s = C,"l S ~ ~ ~ ~+ J, - ,K [( z )K1 2 rI( d~4 1 d s< because for each w E fl the middle term is only a finite sum, where we define a0 = 0. The proof is complete. rn

~

Stochastic Differential Equations

In the practical case a dynamical system will always be disturbed by some stochastic perturbation, one type of which is continuous, and it can be modeled by some stochastic integral with respect to the BM, and the other is of the jump type, which is usually modeled by some stochastic integral with respect to the martingale measure generated by a point process. In this chapter we will discuss such kinds of stochastic differential equations (SDE) with jumps and some of their applications.

3.1 Strong Solutions to SDE with Jumps 3.1.1 Notation Suppose we are given a probability space (a,S , P ) with a a-field filtration {St)tlo. Consider the following SDE with jumps in d-dimensional space :

xt

= xo

+

1 t

b(s, x,, w)ds

<

+

1 t

o(s, x,, w)dw,

where w r = (w,', - . - ,wtl), 0 t, is a dl-dimensional St-adapted standard Brownian motion (BM), w r is the transpose of w r ; kT = ( k l , - . .,kd,) is a dz-dimensional St-adapted stationary Poisson point process with inde-

76

3. Stochastic Differential Equations

pendent components, and generated by Ici satisfying

Nki(ds, dz)

N k i (ds, dz) = Nki(ds, dz)

is the Poisson martingale measure

- r(dz)ds, i = 1,- . - ,d2.

Here r ( . ) is a u-finite measure on a measurable space (Z,%(Z)), and Nki(ds, dz) is the Poisson counting measure generated by ki. From now on, for simplicity, we will always denote the integral J : = s(o,tl= t+ .

So

Definition 112 . { ~ t ) ~(or, , ~ simply, xt) is said to be a (St-) solution of , ~ ~ (3.1). In the case that xt E Vt 0, where (3.1), iff { ~ t ) satisfies 3y'k is the a-algebra generated (and completed) by w,,k,,s < t, and then it is called a strong solution.

c.', >

From Definition 112 it is seen that for discussing the solution of (3.1) we always need to assume that the coefficients satisfy the following assumption (A), b a n d u : [O,co) x Rd x R + R ~ , C:[O,~)XR~XZXR-+R~ are jointly measurable and St-adapted where, furthermore, c is St- predictable. Moreover, to simplify the discussion of (3.1), we will also suppose that all Nki(ds, Z) , 1 i d2, have no common jump time; i.e. we always make the following assumption (A)2 Nki({t), U)Nk,({t), U) = 0, as i # j, for all U E %(Z) such that r ( U ) < oo. Now for the uniqueness of solutions to (3.1) we have the following definition.

< <

Definition 113 We say that the pathwise uniqueness of solutions to (3.1) holds, if for any two solutions {xi)tto , i = 1,2 satisfying (3.1) on the same probability space with the same BM { ~ t ) -, , ~and Poisson martingale measure Nk (dt, dz), P ( ~ u p-, , 1 ~st - x: = 0) = 1.

-

I

3.1.2 A Priori Estimate and Uniqueness of Solutions Now let us introduce some notation which is useful later.

,$loc(~d) =

f (t, w) : f (t, w ) is St - adapted, Rd - valued such that E sup lf (t,w)12 < oo, VT < oo, tE[O,T]

3.1 Strong Solutions to SDE with Jumps

77

where 0 < c(t) is non-random, such that CT = J: c(t)dt < m, for any 0 < T < m. Then

where kT 2 0 is a constant only depending on CT and E 1x012. Hence one has that under the assumption of this lemma the solution of (3.1) always satisfies { x ~- ) E~S, ~~ ' ~ ~ ( R ~ ) . Proof. Let

TN

= inf { t 2 0 : Ixtl

> N ) . By Ito's formula

Since for any T < m, as t E [0,TI,

Hence

{J;"'"

x,

- O ( S , x,, ~

JotArN

sion holds for gale inequality

JZ

x, . C ( S , 2,-,

where we write IIxII,2 = supslt

2Esup,,,

) d w ~is a) martingale. ~ ~ ~ ~A similar , ~ ~concluZ , W )~

k ( d sdz). , Hence by the martin-

I x , ~ 2 , and we have used the fact that

IgATN x, . o(s,x,, w)dw,l 5 ~ ; E + k & 2Jot~ArN lo(s,x,, w)12ds

I +E IIXII:,, 2 5 $ E llxlltArN + k;2~J,t"7Nc ( s ) ( + ~ IIxIl:)d~,

and a similar inequality also holds for

J J Ix, ~ N . o(s,r,,w)12 ds

78

3. Stochastic Differential Equations

+

where Ic;. = 2E 1 x 0 1 ~2 finds that

flc(s)ds. Letting N T co, by Fatou's lemma one E 11x11;

+

where ICT = (2E 1 x 0 1 ~ 2 c(s)ds only.

S,T

< IcT,

c ( s ) d ~ ) eSoT~ ~c ( os ) d s dependS on E lzo12

and

Lemma 115 Assume that b(t,x,w) and a ( t , x , w ) are uniformly locally bounded in x, that is, for each 0 < r < co, Ib(t,x,w)l+ la(t,X , w)l 5 k-,as 1x1 I r, where k, 2 0 is a constant depending only on r ; and assume that for each N = 1,2, . . . ,T < co there exist non-random functions $ ( t ) and &(u) such that as 1x11, 1x21 5 N ; and t E [0,TI

where $ ( t ) is non-negative such that c ~ $ ( t ) d t< co, and &(u) defined on u 2 0, is non-negative, increasing, continuous and concave such that So+d M ( 4 = 03Then the solution of (3.1) is pathwise unique. Proof. Asume { ~ f ) ~,i, =~ 1,2 are two solutions of (3.1) with the same -

BM { w ~ )and , ~Poisson ~ martingale measure gk(dt, dz). Let A

Xt = xi - x;, b(s,x t , xa, w) = b(s,x t , w) - b(t,x:, w), etc. and 71\1= inf { t 2 0 : lxil+ 1x:I > N ) . Then by Ito's formula as in (3.2) one sees that ZthTN= E [ I X ~ = AE ~ C~AIT~N [ 2 x.S(S, S xi,x:,

W)

+ IS(S,x ~ , x ~ , w ) ~ ~

+ Jz 12(s,x:,x:, z,w)I2n(dz)]ds< E J , " " ~ *$(s)& ([xSl2) ds

5 J," c ~ ( s ) &(ZSATN)d-91 as t E [O, TIHence by the following Lemma 116 for any T < co, P -a.s. ZtATN= 0,Vt E [O,T].Letting N 4 oo one finds that P - a.s. Zt = 0,W E [O,T].

3.1 Strong Solutions to SDE with Jumps

79

By the RCLL (right continuous with left limit) property of { x ~ ) ~i ,=~1,2 , the conclusion now follows. Lemma 116 If Vt 2 0 a real non-random function yt satisfies p(ys)ds < m, 0 I Yt I where p(u) defined on u 2 0, is non-negative, increasing such that p(0) = 0, p(u) > 0, as u > 0; and so+ dulp(u>= 00, then yt =O,tr't 2 0.

J,t

Proof. Let zt = J;p(yS)ds(2yt 1 0 ) . Obviously, one only needs to show that W 2 0 zt = 0. Indeed, zt is absolutely continuous, increasing and a.e.

Set to = sup{t 2 0 : zs = 0,Vs E [O,t]). I f t o < oo, then zt > 0, as t > to.Hence by assumption and from (3.5) for any 5 > 0 = J(o,z(to+q)dulp(u) = J(to,t0+a) dztlp(zt) L. S(to,to+qd t I 5. This is a contradiction. Therefore to = oo.

3.1.3 Existence of Solutions for the Lipschitzian Case In this section we are going to discuss the existence and uniqueness of solution to SDE (3.1). First, we introduce a notation which will be used later. .L;(Rd) =

{

f (t,w) : f (t,w) is St - adapted, Rd - valued such that ~ J ~ ~ f ( t , ~< m ) l ~ d t 0

Theorem 117 Assume that lo b a n d o : [O,oo)x Rd x R-+ R d , C:[O,W)XR~XZXR-+R~ are jointly measurable and St-adapted, where furthermore, c is dictable such that P - a.s. Ib(t,x,w)l < c ( t ) ( l+ IxI), 1 4 4x,w>12+ JZ 2, z,412 r ( d 4 < c(t)(l+ 1x12), where c(t) is non-negative and non-random such that c(t)dt < oo; 2" Ib(t,x1,w) - b(t,2 2 , w)l < c(t)1x1 - x2l ,

St- pre-

3. Stochastic Differential Equations

80

lu(t,X I , & ) - ~ ( x2,w)l2 t, + JZ Ic(t,x l , z , w ) - ~ ( 5t2,, z , w)12~ ( d z )

< c(t) 1x1 - 2 2 1 2 ,

where c(t) satisfies the same conditions as i n lo; P Xo E 8 0 , ~ l ~
st- adapted solution {xt)t20 E s2''loc(Rd). -

e'Nk

In the case that b(t,x , w ) and u(t,x , w) are - adapted, and c(t, x , z , w ) is 3ylEk- predictable, then the solution is also 5?lfik- adapted, i.e. it is a strong solution. Proof. Let us use the contraction mapping principle to prove this r e s ~ l t . Introduce a new norm as follows: For any given T < cx, and for (x.) E B = L"$.( R.~let ) 11(x.)1& = sup e - b O A ( t )llxllt ~ , tE[O,T] where llxllt = supsgt lxsl, and bo 2 0 is a constant, which will be determined later, and

1 t

A(t) = Write

H

= {(x.) E

c(s)ds.

E : ll(x.)llnr < m ) .

Then H is a Banach space. For any 1,2, by the following SDE

( ~ . E) H , i = 1,2, denote (xi.) ,i =

By assumption loone easily sees that (x") E H, i = 1,2. Let

Similarly, define Xt. Then by Ito's formula as in (3.2), and discussing similarly as in (3.3), one has that

3.1 Strong Solutions to SDE with Jumps

81

>

where ko 1 is a fixed constant, and we have applied Lemma 118 below. Note that 0 ( A(t) is increasing, so that

where kg > 0 is a constant depending on 2 u(s) = E IX,I , then

SoT c(s)ds only. Thus, if we write

After appropriately choosing y and bo to make max(k$y-', kg($ - y)-') < 1 by the contraction mapping principle one finds that there exisits a unique E L ~ ( Rsatisfying ~) (3.1). Let us show the following result: solution There exists a version {xt)t>o of {Zt)tlo , that is, for each t E [0, TI P(xt # xt) = 0, such that {xt)Go is RCLL (right continuou and with left limit) and {x~)~,, - is a solution of (3.1). In fact, write b(s,Ts, w)ds xt = xo a(s,zS,w)dws Jz c(s,TS-, z, w)fik(ds, dz), t E [O, TI; ~t = xo b(s,xs,w)ds ~(s,~s,w)dws + JZ c(s,xS-, ~ , w ) f i k : ( ddz), ~ , t E [O,T]. Then E lxt - Tt12dt = 0. SO,there exists a set A1 x A2 E %([O,T]) x 5 T such that EJo IAlxA2(t,w)dt= 0, and xt(w) =Zt(w), as (t,w) @ Al x A2. Hence, for each t E [0,TI,

+

+ Jot

+

+ Jot

+ Jot

fl

3. Stochastic Differential Equations

82

+E J:

S,

Ic(s,T,-,

Z)

- c(s,x,-, z)12A ( ~ z ) ~=s0.]

So, the above fact holds true. Now, by Lemma 114 and 115 the solution is also pathwise unique such that E( sup lxt12) 6 kT < oo, for each T < oo, where k~ is a constant tWV1 T depending on T and So c(s)ds only. So we have show that there is a unique for each given T < oo. By the uniqueness of the solution solution {xt)tEIO,Tl we immediately obtain a solution { ~ t ) ~, ~ which , , is also unique. When all

-

coefficients are g'Nkadapted, etc., then by construction one easily sees that {xt)tlo is also gy'Nk- adapted, i.e. it is a strong solution.

Lemma 118 . (Gronwall's inequality). If 0 6 yt 6 yvt +Sot c(s)y,ds, W 2 0, where y > 0 is a constant, and c(s) 2 0, then W 2 0 yt 6 yvt

+y

exp

([

c(r)dr) c(s)v,ds.

6 y J; exp (J: c(r)di)c(s)v,ds. Hence

+

+

yt 6 yvt ~t 6 yvt y J: exp ( J c(r)dr) ~ ~ c(s)v.ds. The above Theorem 117 is easily generalized to the locally Lipschitzian case.

Theorem 119 If the condition 9 in Theorem 117 is weakened to 2" for each N = 1,2,. . - there exists a non-random function c N ( t )such that as lxll and 1x21 < N ,

where c N ( t )2 0 satisfies the condition that cN(t)dt< co for each T < oo; and all other conditions remain true, then the conclusion of Theorem 117 still holds.

Proof. Let bN(t,x, W ) =

b(t,N f i , w ) , as 1x1 > N.

3.1 Strong Solutions to SDE with Jumps

83

o N ( t , x , w )and cN(t,x,z,w)are similarly defined. Then by Theorem 117 there exists a unique zt-solution - E S : ' O ~ ( R ~ ) solving the following SDE

Set

T ~= inf > {t 2 ~ 0 : IxrI > m) . Then we have that

xr

= xo

+

b(s,x:, w)ds

+f

t

~ ( sx: , ,w)dws

By the uniqueness theorem xp+" = xF, as t E [O, T Hence, .rNtNf , as N f . Let us show that lim T

N+m

~ > v~m)=, 0, 1,2,

~= ca, ' P~- a.s.

.. . .

(3.6)

In fact, i f this is not true, then there exists a To < ca such that P ( A ) > 0, where A = {T < T o ) , and T = l i m ~ - rNrN. + ~ Hence, Elimp? ,m IA 2 l i m ~ N + P~( A ) = ca On the other hand, by Fatou's lemma and by the a priori estimate 2 E N +, IA 5 EEN,,E IA 5 S U P N E s u ~ t g /x?l2 , 5 k,r0 < 00. This is a contradiction. Therefore, (3.6) holds. Now let xt = x y , as t E [ O , T ~ , ~ ) , then by uniqueness it is well defined. Moreover, since T. oo, as N f ca, { x ~ )is,a~unique ~ solution of (3.1). H The above Theorem 117 is also easily generalized to a more general case, which is useful in the non-linear filtering problem.

l ~ cl2 ~ , ~

I X ~ N , N I2

~~y~

Theorem 120 If the condition 3' in -Theorem 117 is weakened to be 3" there is a given process tt E such that - E 1&12 < ca; - adapted, and c(t,x , z, w) and assume that b(t,x , w) and ~ ( xt ,,w) are is - predictable, then there exists a pathwise unique strong solution xt, t 2 0, satisfying the following SDE:

c'Nk c'Nk

c'fik

xt

=

6+

6 t

b(s,xslW M S

+

1 t

4 9 , x S ,w)dwS

84

3. Stochastic Differential Equations

The proof of Theorem 120 can be completed in exactly the same way as that of Theorem 117. Now let us give an example to show that the condition on c(t) in Theorem 117 cannot be weakened.

c(t)dt < oo cannot be weakened). Consider Example 121 . (Condition the following BSDE i n 1-dimensional space

where st,wt, and kt are all 1-dimensional, and U E Bz such that x ( U ) < co. Obviously, if a < 1, then by Theorem 117 it has a unique solution T { x ~ ) , However, ~ ~ . if a 2 1, (in this case So I,zos-uds = co), then it has no solution. Otherwise for the solution { x ~-) ~one > has ~ that

Hence E x t = oo, V t > 0, as a 2 1. This is a contradiction.

3.2 Exponential Solutions to Linear SDE with Jumps In Calculus it is well konwn that a 1-dimensional linear ordinary differential equation (ODE) dxt = atxtdt, xo = co; t 2 0, has a unique solution xt = coelotaads,t 2 0. However, for a 1-dimensional simple linear SDE dZt = atZtdwt, xo = Q; t 2 0, even if { ~ t ) is~a ,1-dimensional ~ BM (Brownian Motion process), write txt = c-,eJo t 2 o, xt does not satisfies the above linear SDE. Actually, it satisfiess the following SDE: dxt = i a i x t d t atxtdwt, xo = q;t 2 0. In fact, let f ( y ) = ey, and yt = s:a,dw,. Then by Ito's formula dxt = df (yt) = f1(yt)dyt !jf"(yt)d ( y ) , = coateYtdwt $coeyta?dt = $a?xtdt atztdwt. So xt satisfies another linear SDE. What is the solution of the above original given SDE? In this section we will discuss this kind of problems.

+

+

+

+

3.2 Exponential Solutions to Linear SDE with Jumps

85

First, we can use Ito's formula to verify that

solves the following SDE

where wt is a d-dimensional BM, and we assume that 8: is a d-dimensional &-adapted process such that

In fact, f (x) = ex E C2(R1). SOif we let .dw,- ~ ~ 1 0 : 1 2 d s , xt then by Ito's formula we have that ex* = 1 ex*O: dw, ex. 18:12 ds J: ex. 101:2 ds =1 exad: . dw,. Therefore (3.9) holds. Furthermore, let us use Ito's formula to verify that

=GO:

+ + Jot

4Jot

+

solves the following SDE dz:

= z~

JZ

+

~:(~)lj*(dt,dz),

= 1,

(3.12)

where for simplicity we assume that &k(dt, dz) is a 1-dimensional Poisson martingale measure, (for the n-dimensional case the discussion is similar), and 8; is a 1-dimensional &-predictable process such that

For this we first show that the right hand side of (3.11) makes sense. In fact, by assumption 30, t oo a is a stoping time for each n such that N (R1). So ~ , ~ ~ . Nk(ds,dz) ~ y ~ 0makes f sense. On the 0t I t l o , E F2,l0c IC other hand, since Jz 8: . Nk(ds, dz) is RCLL in t, it has only a finite number of discontinuous points in t for each w . So as the product is a finite product, it also makes sense. Therefore zKon has meaning for each n, and so also does zg,. Now let

JotAon

3. Stochastic Differential Equations

86

By the formula for integration by parts (Theorem 96) dxt~lt= xt-dyt yt-dxt d[x, ylt, where [x, y]t = (xC,y C ) t + ~ o , , , t Ax,Ay,, x: is the continuous martingale part of xt. However, we easily see that [x, y]t = 0, since y,C and Ayt are zero. Hence t t xtYt - = SO~ 3 - + = Sox S - ~ Y S ~ CO<s
+

+

+

Sz

SZ

Theorem 122 Under assumption that p(Sot 1 0 ~ds 1 ~< m) = 1, and:6 E n F: (R1), let Zt = z p z y , where zp and z y are defined by (3.8) and (3.11), respectively, then zt uniquely solves the following SDE with jumps:

3.3 Girsanov Transformation and Weak Solutions of SDE with Jumps In the previous section we have already known the solution of a linear SDE: dzt = ztbtdwt, 20 = 1. This motivates us to think about the following question: Could we use this simpler known result helping us to solve a more complicated SDE? For example, if we have su_ch a SDE: dzt = ztbtdGt zt btbtdt, z, = 1. then we could write it as dzt = ztbtdwt, =J, where_dwt = dGt btdt. Could we solve the s_econd SDE first, then let dGt+btdt = dwt, or equivalently, let dGt = dwt -btdt, to get back to the first SDE and get the solution of the first SDE? Here, obviously, some problems arise: 1) If wt is a BM, can we see dGt = dwt - btdt or equivalently, see

+

+

3.3 Girsanov Transformation and Weak Solutions of SDE with Jumps t

87

-

Gt = wt-So bsds as a new BM maybe under some new probability measure? 2) If it does, then the same solution zt from the second SDE will also satisfy the first SDE but obviously under a new probability (measure), or say, in a new probability space. What is the meaning of such solution which exists in a new but not the original probability space? So, the solution of this kind of problem actually leads us to study two direction of problems: 1) Find t a new probability and appropriate conditions such that Gt = wt - Sob,ds becomes a new BM under this new probability. This is exactly the content concerned by the Girsanov type theorem. 2) Under a new probability space find the solution of SDE. This is the discussion of the scope of a weak solution to a SDE. In this section we will concern with such kind of problems. First, by using the exponential solutions of the linear SDE we can establish some Girsanov type theorems, which are useful when discussing the (weak) solutions to SDE (3.1), where the coefficients may not be Lipschitz continuous. Moreover, they are also helpful in estasblishing some martingale representation theorems, which are the basic tools when deriving the non-linear filtering SDE in estimation problems involving stochastic processes. Now in Theorem 122 let us make an additional assumption that 0; > -1. Then the exponential solution zt = e x p [ x 8: . dw, - S,t le:12ds]

-

g

fl (1+ Sz ~ ~ ( z ) N ~ ( {dz))es), O<s
$2

B:(")"(d")ds

of the linear SDE (3.14) is non-negative, i.e. 0
Lemma 123 Let VO < r < t z: = zt = zpztN, 2: = exp[s; 8: . dw, - S : 10:12 ds]. (1 0; (3Nk({s) ,dz))e- 1; $2 r<s
n

+ sZ

<

(z)"(~')~'.

3. Stochastic DifferentialEquations

88

T

where c(t) 2 0 is non-random such that So c(s)ds < m, then E lzt12 < e2-fotc(s)dS,~t E [O,T], PT is a probability measure, VO < T < m; and zt is a P-martingale.

-

Proof. 1): By Ito's formula

+zt

e p ( z ) f i k(dt,dz) = 6:

~ h e n V 0< < ;

ztdwt

+ zt-

S,

8 r ( z ) f i k ( d t ,dz).

(3.17)

t

E(z&?;; 1%) = 1. Hence by Fatou's lemma letting N -+ m one obtains (i). Notice that zrz; = zt. Hence by ( i ) E(zt I3r) < zr. (ii) is obtained. Now notice that for 0 < r < t E ( z t ) = E(zrE(z,T I3'r)). Hence E(z; IST)< 1 implies that E ( z t ) < 1. Therefore (iii) is derived. 2): By Ito's formula 2 I ~ t / \ ~ , l= 1 2 2s-dzs [z], = 1+2 J, e : ( z ) ( ~ , - ) ~ f i ~ ( ddz) s , 2 J":' e: . ( z . ) ~ ~ w ,

~,t"'~

+

JotArn

+

+

+ gArn~ e : d~ ~ +~ jiATn l ~ jZ le:(z)b-

1

2

N ~ ( ~ dz), S ,

where r, = inf { t E [0,T ] : lztl > n ) ;T , Thus E lztArnl2 = 1 E le:zsl2 ds

+

+ ~,t

JiArn

= T , for

i nf

(4).

+ E JiATn Jz 18:(1)4-

1 Nk(ds,d z ) 2

< 1 2 c ( s ) ~ l2 ds. By Gronwall's inequal!ty and Fatou's lemma W 2 0 E [zt12< e2l o c ( s ) d s . Recall that t , zo = 1. dzt = 6: . ztdwt + zt- Sz @ ( z ) ~ ( d dz), Hence zt is a P-martingale. I t yields that Ezt = E z ~ z ?= 1. Therefore FT is a probability measure. rn Now we can give the following interesting and useful Girsanov transformation theorem.

3.3 Girsanov Transformation and Weak Solutions of SDE with Jumps

89

Theorem 124 (A Girsanov type Theorem). Assume that ( Q Z ) is a standard measurable space. (For standard measurable space see the definition below). 1) If for each 0 < T < co,

where c ( t ) 2 0 is non-random such that c(t)dt < co, for each T < co, then PT, defined by (3.15), for each 0 T < co is a probability measure, and there exists a probability measure F defined on ( R ,3 ) such that P IgT = PT, for each 0 < T < co. Furthermore,

<

-

rt

is a BM under probability P; F ( d t , d z ) = fi(dt, d z ) - Br(z)n(dz)dt = N ( d t , d z ) - ( l + B ~ ( z ) ) n ( d z ) d t ,t, ~

-

<

is a k-martingale measure with the new compensator ( 1 + ~ r ( z ) ) a ( d z ) d t . In particular, if @ ( z ) 0 , then under the above condition we have that %(dt, d z ) = fi(dt, d z ) = N ( d t , d z ) - n(dz)dt, that is, fi(dt, d z ) is still a Poisson martingale measure with the same compensator x(dz)dt under the new probability measure F. 2) Assume that Ft defined by (3.15) with 0 y ( z ) r 0, i.e. z y = 1 and for each 0 < t < co, dFt = zz,WdP is a probability measure, i.e. and there exists a probability measure F defined on ( R , 5 ) such that F PT, for each 0 < T < co. If for any given 0 < T < co,

-

IaT

=

then w:, t 2 0 , defined i n I), is still a F-BM; and the original s ( d t , d z ) , t 2 0, z E 2, is still a F - ~ o i s s o n martingale with the same compensator n(dz)dt. Definition 125 A measurable space is called a standard measurable space, i f it can be Bore1 isomorphic to a measurable subset of a complete separable metric space with the induced topological a-field, that is, there is a bijection map between these two sets such that both the map itself and its inverse map are measumble.

3. Stochastic Differential Equations

90

Many measurable spaces are standard measurable spaces. For example, Rd with its Borel sets is a standard measurable space. C([O, co), R1), the totality of all continuous R1-valued functions defined on [0, co) with the metric llXll = Czl IIX$A1, where llXllT = sup,,^ IX(t)l, V T < w,VX E C([O, co), R1) such that the Borel field is by all open sets of the topology from this metric, is also a standard measurable space. Furthermore, the space D([O, co), R1), the totality of all RCLL (right continuous with left limit) R1-valued functions defined on [0, co) with the Skorohod metric, (see Lemma 388 in the Appendix), such that the Borel field is generated by all open sets of the topology from this metric, is also a standard measurable space. The standard measurable space has many nice properties. For example, we have the following lemma about the existence of an extension probability measure for a family of consitent probability measures. (See 4.1 in Chapter 4 of [491).

Lemma 126 Suppose that (Q, 5 ) is a standard measurable space, {St)t,o c 5 is an increasing a- field fitration such that 5 = Vt>05t. If {pt,5trtlo is a family of consistent probability measures, that is, as s < t,,utI5. = ,us, and they are absolutely continuous with respect to a probability measure P, and P is defined on 5; then there exists a unique probability measure p such that pis, = pt, Vt 2 0. Now let us prove Theorem 124. Proof. 1): Under assumption 2) of Lemma 123 one has that PT, defined by (3.15), for each 0 < T < co, is a probability measure. Since {.&, 0 < T < co) is a system of consistent probability measures, which is \ I absolutely continu_ous with_repect to the original probability measure P, that is, as t < T, PT = Pt , PT << P, and (Q,5) is a standard measurable space, by Lemma 126 a unique extension probability measure defined on (Q,5) exists. Now by Ito's formula for any A E Ss, rl E R~ d(eiT(~l -w' a ) ztIA) = IAei9.(w;-w:) [.ye: . ,qdwt +zy Jz e ? ( ~ ). ~,N_&(dt,d ~ ) i~. (dwt - eydt)$~y lr112 z,Nzrdt i q . eyz,Nzrdt] = dMt - lVl2 zt~Aeidwi-w:)dt, where Mt is a local P-martingale, and we ha;ve used the fact that Cr ~ d[te i ~ . ( w : - - ~ : ) z(~ZTN- + n Z N ) - e i ~ . ( ~ v - - ~ : ) Z y - Z , N _ ~ -eiV.(~:--dlz;- A z,N_] = 0. 1): If (3.16) holds, then by Lemma 123 Mt is a P-martingale. EMt = 0. Hence after solving an o~dinarydifferential equation one finds that E [ ~ ~ ' ~ ' ( w ~ - w : ) ~= I ~~] ( ~ ) ~ - 4 I d ~ ( t - s ) . Or,

+

-3

+

~-[~iV(w:-w:) P

I Ss]= e-31q12(t-s).

3.3 Girsanov Transformation and Weak Solutions of SDE with Jumps

91

Therefore wi, 0 G t , is a BM under probability P. Now for any given U E B(Z), with r ( U ) < oo,and A E 5, write = N ( ( O , ~ I , U ) ,= %Z ( ( O , ~ ] ,=UN((O,t],U). ),N~ andlet xt = zt (N,' - Ni)IA. Then by Ito's formula

- -

+~t-

8:(t)fi(dt, dz)

+ IA%- J, e : ( z ) ~ ( d tdz) ,

+ ~ ~ e ~ ( z ) z t - ~ ~ ( dz). z))~(dt, Again by Ito's formula

Set r, =inf { t 2 s : lxtl > n ) . By 2) of Lemma 123 as t E (0,T ]

+

where cl(r) = c(r) 1. Hence by Lemma 123, Gronwall's inequality and Fatou's lemma VT 2 t 2 s E lXtl2 kge% S: c l ( r ) d r , where 0 kg is a constant depending on T , r ( U ) and ko only. Therefore xt, s t < T , is a P-martingale. This shows that VA E Z,, E[(% = 0.

<

< <

R)z~IA]

3. Stochastic Differential Equations

92

-

Therefore N,' is a P-martingale. Since - a.s. @(dt, dz) = @(dt,dz) - 8r(z).lr(dz)dt = N(dt, dz) - (1 8r(z)).lr(dz)dt,0 t. Hence @(dt,dz) is a F-martingale measure with the compensator (1 ~ r ) n ( d z ) d t In . the case that @(z) = 0 we immediately find that F - a.s. %(dt, dz) = fi(dt, dz) = N(dt, dz) - n(dz)dt, 0 t. Hence in this case @(dt,dz) is a F-~oisson martingale measure with the same compensator .lr(dz)dt. 2): Now suppose that 18;12 dt < W,VO t T, P - a.s. 8 r ( z ) = 0,VO t, P - a s . Let 8z,t = G"Ip;l
+

<

+

<

<

< <

4

cis;

k,

~ [ ~ i l ) . ( w-wl l , n~. a ) Z n , t ~ A ] -+

E[~+(wI-w')

~ A I ,

for Vs < t, A E 5,. First, since w;,, converges to wi in probability, ] , 00. ~ [ ~ ~ q . ( ~ b , t - ~ -+L ,~ e[ )~ di V~ (]w i - w : ) ~ ~ ~n~ + Secondly, since by assumption Ezt = Ezntt = 1, Vn,VO t T, E Izt - zn,tl = E(Izt - zn,tl zt - zn,t). However, 0 Izt - zn,tl zt - zn,t 2zt, and it is not difficult to see that as n -+ oo for a fixed t E [s,T ] (take a subsequence, if necessary) Izt - ~ ~ zt, - ~zn,tl --, 0, in probability. Applying Lebesgue's dominated convergence theorem one finds that for a fixed t E [s,T] as n + co E Izt - zn,tl = E(Izt - zn,tl zt - zn,t ) -+ 0. Therefore, VA E g8,for a fixed t E [s,TI, by Lebesgue's dominated convergence theorem

+

+

<

<

+

+

< <

3.3 Girsanov Transformation and Weak Solutions of SDE with Jumps

93

Recall that ~ ~ i V . ( " k , t - " h , s ) ~=~e-$1'?12(t-s)~zn,tIAa ,~I~ By letting n -+ oo one finds that E ~ ~ v . ( " : - " : )= ~ ~e -I $~1 d 2 ( t - s ) ~ Z t ~ =AF T ( ~ ) e - t I V 1 2 ( t - s ) , where we have used the result that EztIA = E ( z t I A E ( 4 Igt)) = EzTIA = F T ( ~ ) . Therefore, wi, t E [O,T],is a &-BM. Similarly, by ,qeiCZl %N((sd,Uj)zn,t~A] = ~ [ ~ ( t -C$l(e"j-l)+j) s ) %$IA] and letting n -+ oo one has that q e i C& x ~ N ( ( ~ ~ ~ I = ~ u,qe(t-s) ~ ) , I ~~ jIm = l ( e-1)du.) j 3 ~ A I , where Qs < t , A E gs, U j E B z , i = 1,2,. . . ,m such that Ui n U j = 4, as i # j, and r ( U j ) < oo,Qj, i.e. U j E r , Q j . Therefore, N ( ( 0 ,t ] ,U ) ,t E [O,T],U E r is a PT-Poisson counting measure with the compensator n(U)t. That is to say fi((0, t ] U , ) = N ( ( 0 ,t ] ,U ) - r ( U ) t is a FT-~oisson martingale measure with the compensator r ( U ) t . Since the results obtained here are tr5e for any-0 < T < oo. By assumption is a probability measure such that P = PT,V T > 0. Hence the final conclusion is derived. Girsanov type theorems are powerful tools for obtaining the existence of weak solutions to SDE (3.1) under very weak conditions. Let us first introduce the concep of a weak solution for a SDE. For this we rewrite the d-dimensional SDE as follows:

-

and suppose that xo has a probability distribution law p, i.e. p ( A ) P(xo E A ) , Q A E B ( R ~ ) .

=

Definition 127 If there exist a probability space (with a a-field filtration)

(6,

(&)-t>O ,F ) and a gt- adapted BM Z t ,t > 0 , a Poisson martingale gt

measure x ( d t , d z ) (also - adapted) with the given beforehand compensator r ( d z ) d t , and a gt-adapted processes { x ~ ) define ~ , ~ o n this probability space such that (3.18) holds, F - a s . , andP(x0 2 A ) = p(A),QA E B ( R ~ ) , then we say that

,

-

( t )

t)O

,F; {wt}t20 ,%t,

is a weak solution of (3.18).

d z ) , {"t)t20) (0'. simply, {xt)tlo)

3. Stochastic Differential Equations

94

From - the definition the "weak" sense is that it may happen that xt $

zy'

Remark 128 Given a Poisson martingale measure W ( d t ,d z ) with the compensator n(dz)dt on a probability space (6,% ,F ) means that there t>O exists a Poisson random counting measure r ( d t , dz) with the compensator n(dz)dt such that N ( d t ,d z ) = N ( d t , d z ) - n(dz)dt, where Tlf((0,t ] ,U ) and X ( ( 0 , t ] ,U ) ,t > 0 , U E B Z , are - adapted RCLL (right continuous with left limit). However, from this one easily constructs a Poisson point process ij that generates this Poisson counting measure. In fact, let DP(,) = { t > 0 : S,W({t), dz,w) = 1 ) = { t > 0 :there exists a unique point, denote it by z ( t , w ) E 2, such that r ( { t ) , { z ( t ,w ) ) ,w ) = 1). Define ij(t,w) = z(t,w),Vt E Dii(,), . Then we get a &- Poisson point process - p such that N ( d t , d z ) = Nii(dt, d z ) .

(gt)

st

gt-

For the weak solution of (3.18) we sometimes also need to discuss its weak uniqueness.

Definition 129 If for any two weak solutions

(

)

-

t>O

~";{W;)~~~,N~(dt,dz),{x1)~)~),i=l,2,

of (3.18) having the same initial law p, i.e. F 1 ( x i E A ) = F 2 ( x i E A ) = p(A), V A E % ( R ~ ) @implies ~ , that they have the same probability distribution, that is, W1 < t2 < . . . < tnl-VA E B ( R ~ ) @ ~ , F1((z:,,xL,..- ,xL)E A ) = p2((x:,xt2,,... ,x:J E A ) , then we say that the weak uniqueness of weak solutions to (3.18) holds, where ) %(R~)= @% ~ ( R d ) x . . . x % ( R ~ (n-times). Now we easily answer the question put out at the beginning of this section.

Proposition 130 Assume that for each 0 < T

< oo,

where By(w) is a d- dimensional gt -adapted process, 8 r ( z ,w ) is a 1 - dimensional 5t-predictable process, and c(t) 2 0 is non-random such that c(t)dt < oo, for each T < oo. If bt(w) is a d-dirnensional5t-adapted

3.3 Girsanov Transformation and Weak Solutions of SDE with Jumps

95

process, such that Ibt(w)l 5 ko, where ko 2 0 is a constant, then the following SDE has a weak unique weak solution: V t > 0,dzt = 6: . ztbt(w)dt 6: - ztdwt zt- Sz ~ y ( z ) ~ k ( d zt),,

+

+

= co, where q, 0 is a constant, wt is a d-dimensional BM, and i k ( d t ,d z ) is 20

>

a 1-dimensional Poisson martingale measure. Proof. Let

Then by Lemma 123 P - a s .

Moreover, zt is the unique solution of the above SDE. Now since lbt(w)l 5 ko, by Theorem 124 Gk(dt,d z ) and t Gt = w t -So b8ds,0< t , are a Poisson martingale measure (the same original one) and a BM under the new probability (measure) with T T 2 %lzT = ex~[Sobs . dw8 - S, Ib81 dsl, respectively. Hence, P - a.s. dzt = 6: - ztbt(w)dt 0: . ztdGt zt- Sz 6;(z)Gk (dt,dz). That is, zt is a weak solution o f the above SDE. T h e weak unqueness is derived from the fact that any weak solution o f the above SDE can always be returned t o the expression (3.19) with some BM and the same Poisson counting measure under some probability measure. So the probability distribution o f the weak solution should be the same. rn Furthermore, from Theorem 124 it is immediately seen that the following theorem holds.

+

+

l2 <

Theorem 131 Assume that bo(t,x ) is Rd@'-valued, and IbO(t,x ) kT, VO t < T , for any given T , where is a constant depending on T only, and a-'(t, x ) exists such that, la-l(t, x)l 5 c l ( t ) , where c l ( t ) > 0 is non-random and J: cl(t)dt < CO, for each T < CQ. Then the following statements are equivalent: 1 ) (3.18) has a weak solution { x ~- ) with ~ , ~the initial value xo = x , where x E R~ is a constant vector. 2) SDE

<

3. Stochastic Differential Equations

96

has a weak solution { x ~- ) with ~ , ~the initial value xo is a constant vector.

= xo,

where xo E Rd

In fact, (3.21) can be rewritten as dxt = b(t,xt)dt a ( t ,xt)dGt J, c(t,xt-, z ) ~ k ( d dt ,z ) , where dGt = a-'(t,xt)bO(t,xt)dt dwt, and the rest involves applying the Girsanov type theorem. Beware of the fact that the coefficient b(s,x) bO(t,x)in (3.21) now can be very discontinuous in x. Moreover, if we let b = 0 , then under the condition of Theorem 131, (3.21) always has a weak solution. To relax the bounded condition on bO(t,x ) we only need to weaken the sufficient conditions for Girsanov theorems to be satisfied. The following Theorem is useful.

+

+ +

+

Theorem 132 (Girsanov type Theorem). Assume that xt solves the following d- dimensional SDE dxt = b(t, xt)dt a(t,xt)dwt J, c(t,xt-, z)fik(dt,d z ) , with the initial value xo = xo, where xo E Rd is a constant vector and, moreover, assume that IbO(t,x)I2 + lb(t,x)12+ 1 4 4x)12 + JZ I+, 2, z)12 4 d z ) 5 k o ( l + 1x12), and a-' exists such that la-'(t, x)1 5 c l ( t ) , where c l ( t ) > 0 is non-random and cl(t)dt < CCI, for each T < m. Write zt(a-'bO) = exp[J;(a-'bO)(s,x,) . dw, - J; (a-'bo)(s, ds], dFt = ?dp, Gt = Jo (a-'bO)(s,xs)ds wt, Then Ft is a probability measure, and for each 0 < T < co, Gt, t E [0,TI, is a BM under the new probability measure FT, and ~ ~ ( t( ]0d, z,) , t E [0,TI, is a Poisson martingale measure under the new probability measure FT with the same compensator ~ ( d z ) tNotice . that here the space R and the a-filtration {&}tao do not change. Furthermore, for each T < m,

+

+

1

%, ) I 2

+

-

(a,57 { z t } t € P , ~';P

~{ x; t } t p [ ~ ,{Gt?t.[~,~] ~l { R ~ ( (t~>dz))tt[O,Tl,drEBZ o, 1 9

is a weak solution of the following SDE: FT - a.s. W E [O,Tl, dxt = (bo(t,xt) b(t,xt))dt a(t,xt)dGt J, c(t,xt-, z)Nk(dt,d z ) , with the initial value xo = xo, where xo E is a constant vector. Furthermore, if (f2,g) is a standard measurable space, then a probability P exists such that PisT = &,VT < co. Thus the result is generalized to all t 2 0. Or, simply speaking, { ~ t- ) is~ a, weak ~ solution of the above SDE, W 0, F - a s .

+

>

Proof. Let

+

+

3.3 Girsanov Transformation and Weak Solutions of SDE with Jumps

b N ( t , x )=

-

97

( bO(t,x), as 1x1 5 N , 0, otherwise.

a Then by ~ e m k 123 d P p = %(a-'bN)dp is a probability measure. Applying Theorem 124 for any given 0 T < oo asO N ) , and T N = T , for inf(q5) Then xt solves the following SDE: as 0 5 t 5 T ~ATN (bO(s, x S ) b(s,xS))ds a ( s ,xs)dwy X ~ A T = ~xo SO J r T NJZ C ( S , xs-, z)fik(ds,dz), i)TN - a.s. By Lemma 114 2 E';(suP~~ Ixt~rN ~ I ) k ~ . Hence 2 N2@(7N < T ) 5 E"(IXTATN I ITN
<

+

+

+

+ ~otr\7~

<

<

Theorem 133 (Girsanov type Theorem). Assume that xt solves the following d- dimensional SDE dxt = b(t,xt)dt a ( t ,xt)dwt Jz ~ ( xt-, t , z)fik(dt,dz), with the initial value xo = xo, where xo E R~ is a constant vector and, moreover, assume that J, I+, x , 4124 d z ) 5 ~ l ( t ) , [ ( x. bO(t,X I ) V ( x . b(t,x ) ) V I+, x)l21 5 cl(t>(l+ 1 x 1 ~ gk(x)), a-l exists such that la-'(t, x)l cl(t), where gk(2)= 1 l n ( l + l n ( l + . . .l n ( l + ~ x l ~ ~ ) ) ) ,

+

+

<

+

k-times

3. Stochastic DifferentialEquations

98

(n is some natural number), and c l ( t ) 2 0 is non-random such that for each T < oo, /?cl(t)dt < oo; furthermore, bo(t,x) is locally bounded for x , that is, for each r > 0, as 1x1 5 r, IbO(t,x)l< kr, where Ic, > 0 is a constant only depending on r . Let zt(u-'bO) = exp[$(a-lbO)(s,x,) . dw, - $ I(a-1b0)(s,x,)12ds], d& = ydp, @t = So (0-lbO)(s,x,)ds wt, Then all conclusions in Theorem 132 still hold true.

Sot

+

Proof. Let bN (t,X ) = bO(t,x),as 1x1 N , 0, otherwise. By using the notation in Theorem 132 we still have that xt solves the following SDE: as 0 t T N x,) f b ( ~x , ) ) ~ + s U ( S ,x , ) ~ w ; X ~ A T=~X O SO~ A T (bO(s, J:TN JZ C ( S , 2,-, z)fik(ds,dz), i)TN - a s . Notice that by evaluation

{

<

< <

+

+

/YN

n;, g i l ( x )+, axisxjgm+l(x) llT=i gkl ( X I [ 4n2xix.lx14n-4 (l+I:I.ny I nE1g i 1 ( 4

,&gm+~( x )= a2

anxi x

2n6ij 1~1~"-~+4n(n-l)xixj Ix~'"-~

=

-

'"-'

l+lx12n

ZL0El~ ' ( 4 ,

4n2xix (l+l:12n,2 .lx14n-4

-

where we write go(x) = 1. Denote gk+l(x) =gradgm+l(x), and g;+,(x) =

a2

d

[mF@+l(x)] i,j=l . Then by assumption we find that &&+I(.) . bO(t,x ) gk+,(x) . b(t,x) 5 koc1(t), 19;+1(~)11 4 4x)12 b ~ l ( t ) , I&+l(~)l 5 ko, where ko > 0 is a constant. Now applying Ito's formula to g(xt) = gm+l(xt) we have that as 0 5 t T ~ A T NI 9 (xs)(bO(s,xs) + b(s,xs))ds 0 I s(xt*r,) = d.0) + So g l ( x s ) * ( ~x , ) ~ w ; SOtArNJZ g l ( x s - ) c ( ~xs-, , ~ ) f i k ( dd ~~, ) + &[$(xS- +c(%X S - , z ) ) - ~ ( x s - )- ~ ' ( X S - ) C ( S , X S - , t)]Nk(ds,dz) = c:=~ 12,P P - a s . Taking the mathematical expectation with respect to the probability measure in both sides we see that 0 < ~ " ( g ( x t ~ r NI ) )k ~ , where we have ap lied the facts that R T N J, Ig"(x,- + W s ,xs-, z))l xs-, z)12.~r(dz)dsI 5 @J0 iko cl(s)ds. Hence S(N)@(TN < T ) I J @ ' ( ~ ( x T A T ~ ) L ~ < I Tk ) ~ ,

+

<

<

+ JFTN

JIATN

+

PP

1~1~1

J?

I+,

3.4 Examples of Weak Solutions

Therefore, as N

P ~ ( T
--t

99

co,

0. Thus all of the conclusions in Theorem 132 now follow. rn --t

3.4 Examples of Weak Solutions For the existence and weak uniqueness of a weak solution of SDE with jumps we have the following examples.

Example 134 Assume that Ib(t,x)I 1 ko, la(t,x)12 JZ Ic(t, 2 , z)124 d z ) ko(l + 1 x 1 ~ 1 , and for each N = 1,2, . . . there exist a non-random function c N ( t ) such that as 1x11 and 1x21 5 N , ) ( ~ ( t , 5 2 , ~ ) 1JZ~ Ic(t,x1,2,w) - c(t,x2,z,w)12 r ( d z ) Ia(4 2 1 , ~ 2 c N ( t )1x1 - x21 , cN(t)dt< co for each T < oo; where c N ( t ) 2 0 satisfies that and a-'(t, z) exists and is bounded la-' (t,x ) ko, where x , b, c E R ~a ,E

+

+

<

1<

~ d @ d .

Then the following SDE with jumps i n d-dimensional space on t E [0,T ] :

.

,

where xo E Rd is a constant vector, has a weak unique weak solution. Proof. Since the coefficients a(t,x ) and c(t,x , z ) are less than linear in growth and are locally Lipschitzian continuous, by Theorem 119 there [ ~ , ~ ~the following exists a pathwise unique strong solution { ~ t } ~ ,satisfying SDE: P - a.s.

on the original given probability space (O,Z, {&}tEIO,Tl,P ) . However, by the Girsanov type theorem (Theorem 132) if we let

then P is a probability measure, and 'f& = wt -

SOt

0

-1

( t ,x s ) b ( ~X ,S ) ~ S t, E [O, TI,

is a BM under the new probability measure F, and fik(ds,dz) is still a Poisson martingale measure under F with the same compensator x(dz)ds

3. Stochastic Differential Equations

100

such that fik(ds,dz) = Nk(ds,dz) - x(dz)ds. Hence we can rewrite (3.23) as follows: ?; - a.s. t E [O,T],

xt

= xo

+

I

t

+ & a(s,x.)~u?I+ t

b(s,X+S

lJZ

C(s,2.-, z)fik(ds,dz).

(3.25) Thus we have obtainzd a weak solution (O>3,{3t)tE[0,T P; Gt, f i k ( d t , d z ) , { X ~ } ~ ~ [ O , T ] ) ' Now let us prove t e weak uniqueness. Suppose we have another weak solution (01, 3l,{3't)tE[0,Tj ,PI;Gi, fiw (dt,dz),{xt )tEIO,Tl),where kt(.) is a 3:Poisson Point Process, which generates the Poisson martingale measure but with the same compensator n(dz)dt such that fip (dt,dz) = Nkl (dt,dz) n(dz)dt. Then we have i3' - a.s. t f [O,T],

1:

So we see again that by the Girsanov type theorem (Theorem 132) if we let

.

.

then P' is a probability measure, and wi = 63; a-l(t,xb)b(s,x',)ds,t E [O,T], is a BM under the new probability measure P', and fik,(dt,dz) is still a Poisson martingale measure under P' with the same compensator x(dz)dt such that fikJ(dt,dz) = Nkl(dt,dz) - n(dz)dt.Hence we can rewrite (3.26) as follows: P' - a.s.

+

xi

= x,

+

I

t

a(s,x:)dw:

+

l

C ( S , 2:-,

Z)fikl(d~, dz),t E [O, TI. (3.28)

Notice that the solution for SDE (3.23)is a pathwise unique strong solution. Hence there exists a measurable function F such that xt = F(w.,fik:((~,s],U);Vs 5 t,VU E % z ) , P - a . s . This means that for (3.28) we must also have xi = ~ ( w ; , f i k r ( ( ~ , ~ ] , U5) ;t,vU t / S E % z ) , P 1- a.s. SOV A E ( R ~ ) @ and~Vtl,... , ,t , E [O,T], P ( ( x ~,.,. . ,~ t , )E A) = P'((x:, , . . . ,xi,) E A). By (3.24) and (3.27) one immediately sees that P ( ( x ~ , ,... ,xt,) E A) = F'((x:,,. . . ,xi,) E A). Hence the weak uniqueness for the weak solution holds. In the case that the SDE (3.22) has no jump term, that is, c = 0, if a = o ( x ) does not depend on t , then we can weaken the condition on a to get a weak solution. However, in this case the weak uniqueness is not necessarily true.

-

3.4 Examples of Weak Solutions

101

Example 135 Assume that a = a ( x ) does not depend on t , and INt, x)l 5 ko, la(x)12 ko(l+ 1x12), ko, where x, b, E Rd,a E and a-'(x) exists and is bounded la-'(2) RdBd. Then for an T < m xt = xo So b ( ~xs)ds , a(x,)dw,, t E [O, TI, has a weak solution, where xo E Rd is a constant vector.

<

1<

+ Y

+$

Proof. Let 5t)tlobe a BM given on a probability P). Write .iZt = Sort ~ - ~ ( w ~ ) dVt w , , 0, where -rt is the inverse function of the strictly increasing non-random con2 tinuous function A(t) = E J la-'(w,)l ~ ds, that is, A(-rt) = t. Set 3t =&,, w L 0. Notice that Vt 2 s, = E J: lo-'(w,)12 du = t - s. E[(Gt-

>

4)21gs]

So that by Theorem 97 {iZt,&}

{

is a BM on the probability space t>o ,P ) . Let xt = xo + ww. Then xt satisfies the SDE: W 2 0,

( 0 , t2o dxt = a(xt)diZt,xo = xo, P - a s . Now for any given T < m write 2 zt(a-'b) = exp[J;(a-'(x,)b(s, 2,)) . dGs la-'(x,)b(s, ds], dFt = gdp, 6; = J; a-yxs)b(s, X J ~ S+ .iiit, Then by Theorem 132 Ft is a probability measure, and for eazh 0 < T < m, G;, t E [0,TI, is a BM under the new probability measure PT. hrthermore, for each T < oo,

4Ji

(a>3, { g t }t€IO,TI

1

FT; {~t}te[O,T] 1 {Gi}tc[O,Tl)

is a weak solution.of the following SDE: FT - a.s. Vt E [0,TI, dxt = b(t,xt)dt a(xt)dG;,xo = xo. w

+

%,)I

Some Useful Tools in Stochastic Differential Equations

In this chapter we will discuss some powerful tools in the theory and applications of stochatic differential equations (SDE). We will establish a Yamada-Watanabe type theorem, which is very helpful in getting a strong solution for the SDE, when we have already got a weak solution and we know that the pathwise uniqueness holds for this SDE. We will derive a Tanaka type formula, which is very useful when discussing the uniqueness, and the convergence of solutions for SDEs. In particular it can help us to get comparison results for solutions of SDE, which are important for the discussion of the properties of solutions and the pathwise stochastic optimal control in a later chapter. We will introduce the local time technique which can help us to simplify and weaken the conditions for a 1-dimensional SDE when disccusing the uniqueness and existence of solutions. Finally, we will establish the Krylov type estimate which is very powerful when discussing the weak solution of an SDE when the coefficients are only measurable.

4.1 Yamada-Watanabe Type Theorem Consider a d-dimensional stochastic differential equation with Poisson jumps as follows: dxt

= b(t, xt, w)dt

+ a ( t , xt,w)dwt +

Jz

c(t,xt-, z,w)q(dt, dz),

4. Some Useful Tools in Stochastic Differential Equations

104

where wt is a d-dimensional standard Brownian motion process (BM), q(dt, d z ) is a Poisson martingale measure with a compensator ~ ( d z ) d T(.) t, is a a-finite measure on a measurable space (2,B(Z)) such that q(dt,d z ) = p(dt, d z ) - ~ ( d z ) d t , where p(dt,dz) is a Poisson random measure with the intensity measure Ep(dt, d z ) = r(dz)dt.

Remark 136 By Remark 128 we know that given a Poisson martingale measure q(dt,d z ) there always exists a St-Poisson point process k(.) such that q(dt,d z ) = %k(dt, d z ) , where fik(dt,dz) is the Poisson martingale measure generated by k(.). Theorem 137 (Yamada- Watanabe type theorem). Assume that the wefficients b(t, x),a(t,x ) , and c(t,x , z ) i n (4.1) do not depend on w , and make the assumption (H):

Then we have the following two assertions. 1) If (4.1) has a weak solution and the pathwise uniqueness holds for (4.1), then (4.1) has a pathwise unique strong solution, t 2 0. 2) If (4.1) has two weak solutions ( x i ,wf, qi(dt,d z ) ) ,i = 1,2, defined on two different probability spaces (Qi,p, P i ) , i = 1,2, respectively, where wf is a Pi-BM, q"dt, d z ) is a Pi-Poisson martingale measure with the same compensator x(dz)dt,i = 1,2, then there exist a probability space ( Q , 5 , (&), P ) and two St-adapted cadlag (right continuous with left limit) processes (?i),i = 1,2, with a BM Gt and a Poisson martingale measure q(dt,dz) having the same compensator r(dz)dt as the given one defined on it such that (?i),i = 1,2, are adapted to St, and the probability law of (Z;,.iiit,q(dt,dz))coincides with that of (xf,wf, qi(dt,dz)),i = 1,2. Moreover, (Zf,Gt, q(dt,d z ) ) satisfy (4.1) on the same probability space (a,5,(&I, PI, i = 1,2.

(z),

Before we establish this theorem let us give an example for the measure such that (4.2) holds.

T(.)

Example 138 Let n ( d z ) = *. Proof. In fact, in this case = sRd-{o) SRd-10) %A(dz) l+lzl

J;zl
IzI

2

dz

i;im f

J;zl?l

Then (4.2) holds.

&$b <

6

To show Theorem 137 let us first establish a Lemma.

4.1 Yamada-Watanabe Type Theorem

105

Lemma 139 Suppose that xt is a solution of (4.1) with the BM wt, and the Poisson martingale measure q(dt, d z ) , which has the compensator n(dz)dt such that q(dt,d z ) = p(ds, dz) - n(dz)dt, where n ( d z ) satisfies (4.2). Denote

qZIZl

-

Ct = Sat w ( d s 7 d z )+ S: ~P(~s,W. Suppose that three 5t-adapted cadlag processes (Zt,Gt, C,) have the same finite-dimensional probability distributions as that of ( x t ,wt, Ct). Write for t 2 0 , U E 'WZ), F((0,tl, U)= C,<s5, 102a~*Ev, q(dt,d z ) = F(dt, d z ) - n(dz)dt. Then Zt is also a solution of (4.1) with the BM Gt, and the Poisson martingale measure q(dt, d z ) , which has the compensator n(dz)dt. Proof. First let us show that such a

Ct

is finite P - a.s. In fact,

$ J: Sz lz12~ ~ ~ l < l ~ ( d5zt )Jdz s& 1 ~ ~ ~ < 1 4 d z )< m. Hence J,' &I l ~ ( dd~z ,) = S: JZ Ijzltln(d4ds = t &I,l $+(dz)

1

n ( d z ) < 00. t4z121 ql+I.%I Hence p((0, t ] ,{lzl 2 1 ) ) < m, P - a.s. Denote the number of jumps, the jump times, and the values of jumps of p((0, t ] ,{lzl 2 1)) by v ( w ) , r i ( w ) < 72(w) < . . . < 7,(,)(w), and { z l (w),. . . ,z,(,) (w)) , respectively. Then P - a s .

);!LE

Jot

&I,l I4 ~ ( d d%z ) = In;(w)l < m, because it is a finite sum for P - a.s. w. Therefore, Ct is finite P - a.s. Now notice that p is the counting measure induced by counting the jumps of It -with compensator x(dz)dt. Hence it is a Poisson counting measure. Since Ct, t 2 0 , and It, t 2 0 , have the same finite-dimensional probability distributions, p is also a Poisson counting measure with the same compensator x(dz)dt. Hence q(dt,dz) = F(ds, d z ) - n(dz)dt is also a Poisson martingale measure. ByJhe coincidence of finite-dimensional probability distributions of (Zt,Gt, C t ) , t 2 0 , and ( x t ,w t , Ct), t 2 0, applying the monotone class theorem (Theorem 390), it is easily seen that the probability laws of (Zt,G t l t t ) ,t 2 0 , and ( x t ,wt, C,), t 2 0 , also coincide.' Notice that there exists a Baire function F such that t t F ( x s ,w s , C,; s t ) = xt - x - Sob(s,xS)ds - So 4 ~ xS)dws , - Sot SZ 4%xs-, z)a(ds,dz). Hence one has that P(F(Z,,~,,~~;s
<

106

4. Some Useful Tools in Stochastic Differential Equations

the weak solutions (xi, wf ,qi(dt, dz)), i = 1,2, are defined on probability Pi), i = 1,2, respectively. By the assumption of 2) in spaces ( R i , p , Theorem 4.1 we can let zpi(ds, dz), i = 1.2. = J;,Ig zqi(dsl dz) + f; Construct space R = Ddx Dd x W$X Dd, where Dd = D([O, m); Rd) (the totality of valued right continuous with left limit functions defined on [0, m)), and W$ is the set of continuous valued functions w(t) defined on t 2 0 with w(0) = 0. Map (xl(.,w),wl(.,w), cl(., w)) and (x2(.,w), w2(., w),c2(.,w)) into the R1 = 1st x 3rd x 4th component space of R and R2 = 2nd x 3rd x 4th component space of 0, respectively. From these two maps we get the probability laws P: and P: on f l l and R2, respectively. Then both marginal distributions ?(Pi) and ?(Pz) coincide with PWx PC,where PWis the Wiener meaor (both of them sure on w$, PC is the measure on Dd induced by have the same probability distribution), and ? : Dd x W$X Dd + W$X Dd is the projection. Let ~ p ' ~ ~ ) ( d and w l ~) p ' ~ ~ ) ( d be w zthe ) regular conditional distributions of (wl) given (w3, w4) and of (w2) given (w3,w4), respectively. Define a probability measure Q on the space R by (dw3)~c(dw4). Q(dwldw2) = Q1(w3rw4)(d~1)~prw4)(d~2)~w Let 5 be the completion of the a-field B(R) by Q, and 5t = f1~>~(!23t+, V N ) , where !Bt = !Bt(Dd) x !Bt(W$) x !Bt (Dd) and N is the set of all Q-null sets. Then clearly (wl, w3, w4) and (xl, wl, cl) have the same distribution, and so do (w2, w3 ,w4) and (x2,w2,c2). NOWlet us show the following facts. Fact A. For A E !Bt(Dd), (w3,w4) E W$ x Dd Q 1 ( ~( 4~ ~ ~ ~ ) and Q2 ( ~ (~ 4' ~ ~ ) (203, w4) E wf x Dd are 23, (wf x D ~ )- measurable, ~ ~ ' ~ where PW>C = PW x PC, and !Bt(Wf x D ~ ) is ~ the ~completion ' ~ of !Bt(W$ x Dd) by pW1C. Indeed, for fixed t > 0 and A E !Bt(Dd x D ~ ) there , exists a conditional probability Q ~ ~ ' " ~ ) (such A ) that (w3,w4) E Wf x Dd + Q ! Z U ~ ' ~ ~ )is( A ) &(wf x D ~ ) ~ measurable, ~ ' ~ - and W for any C E !Bt(w$x D ~ ) . P,'(AxC) = fc Q $ ? ' ~ ~ ) ( A ) P(dw3)P~(dw4), Now let = ( ( ~ 3~, 4 E) W$ X Dd : pt(w3, ~ 4 E) AI, Ot(w3, w4) E A,), Al,A2 E %(W$ x Dd), where Ot is defined by Ot (w3,w4)(s) = (203(t S) - w3(t), w4(t S) - w4(t)), s 2 0,and p, is defined by p, (w3,w4)(s) = (w3(t A s), w4(t A s)). Since 0t(w3, 204) is independent of !Bt(w$ x Dd) with respect to PWICwe have fc (A)PW(dw3)~C(dw4)

(z),

d

6

c1 c2

--

+

+

QP'"~)

Q(w3>~4) - & t ( ~ 3 , ~ 4 ) E It~ ~ ( A ) P ~ ( ~ W ~ ) P ' ( ~ W ~ ) Pw4) ~>C E (O~(W~, = Pi(A x {pt(w3,~4)E A1))Pw'C(Ot(~3, ~ 4 E)A2)

4.1 YamadaWatanabe Type Theorem

107

= P1(xl(.)E = P1(xl(.) E = P1(xl(.) E

A,pt(w,C) E Al)p1(Bt(w,C)E A2) A , pt(w,C) E Ai, &(w,C) E A2) A , (w,C ) E C ) = P ~ ( xA C ) , where we have used the result that { x l ( . ) E A,pt(w, C ) E A1) E St, moreover, &(w,C) and St are independent. Hence it is easily shown that (""'"")(A)= Q ~ " ~ ' " ~ ) a.a.(w3, ( A ) , w4) (pW>C). Qlt (Here "a.a." means "almost all"). Fact A is proved. Fact B. w3(t) is an d-dimensional St-BM on (Q,5,Q ) , and p(dt, dz) is a Poisson counting measure with the same compensator n(dz)dt on ( R , 5 ,Q ) , where ~ ( ( 0t ], U , ) = ~ o < , s tIO#AC.EU,for t > 0 , U E B(2). Indeed, by using Fact A we have that for X E R ~ , AE ~B S ( D d ) , i = l,2,4;A3 E BS(w,d),05 s 5 t EQ[ e i ( X , ~ 3 ( t ) - ~ 3 ( s ) ) ~xAa A 1 xA3 x A 4 ] ( A 2 ) p w(dW3)pC(dW4)

ei(~,~3(t)-~~(~))~p*w4)(~1)~pr

- .!A3 x A4 - e-(lN2/2)(t-s)JArXA4Q ( ; I . ~ ~ ~ ) ( A ~ ) Q ~ ' ~ ~ ) ( A ~ ) P Y ( ~ w ~ ) P c ( ~ w ~ = e - ( l x l a / 2 ) ( t - s ) ~x( ~ A2 1 x A3 x A4). Therefore the first conclusion of Fact B is true. Now for t > s 2 0, disjoint Ul,. - -,Urn E B ( 2 ) such that p((O,t],Ui) < o q V i = 1,. . .,m and X i > 0,i = 1,. . .,m; Ai E B,(Dd),i = 1,2,4;Ag E Bs(W,d),O < s < t EQ[exp[-

czlXiP((s,t],Ui)]IA*

XA,

- JA5XA6~ x P [ -

xA3xA4] ( " 3 ? ~ 4 )( A 1 )

EL1 X i ~ ( ( s , t lUi)lQl ,

. Q ~ ~ ~ " ) ( A ~ ) P " ( ~=Wexp[(t ~ ) P-Cs() CL1(e-" ~ W ~ ) - l)x(Ui)]

J

&("3'"4) 1 (~1)~~'"~'(~2)p"(dw3)p~(d~4)

= exp[(t - s ) C z l ( e - x i - l ) ~ ( U i ) l Q ( Axl A2 x A3 x A4) Therefore the second conclusion of Fact B is also true. Now let us return to the proof of Theorem 4.1. From Fact B it is not difficult to show that ( w l ,w3, q(., .)) and (w2,w3, q(., .)) are solutions of (4.1) on the Same space ( Q ,5, (&I, Q ), where q(dt,dz) = p(dt, dz) - x(dz)dt is a Poisson martingale measure with the compensator .rr(dz)dt.Hence 2 ) of Theorem 4.1 is proved. Now assume that the condition in 1) holds. Then the pathwise uniqueness implies that wl = wa, Q - a.s. This implies that x Q $ ~ ~= W~2 ) = ~ 1, ~pw>C ) -(a.s.~ ~ Now it is easy to see that there exists a function (w3,w4) E W$X Dd -+ FZ(w3,w4) E Dd x D~ such that = Q $ ~= S~{ ~ z~ ( w 3~ 1 w 4~ pw,C ) ) ~) a s . By Fact A this function is Bt(w,dx D ~ ) ~ ~ ' x~Dd)-measurable. / B ~ ( D ~ Clearly, FZ(w3,w4) is uniquely determined up to pW2C-measure0. Now by the assumption of 1) (4.1) has a weak solution. Denote it by (at,wt, q(dt,d z ) ) , where q(dt,dz) is a Poisson martingale measure with the compensator t ?r(dz)dt. By above F,(ws, C,, s t ) , where Ct = zq(ds,dz)

QY*"~)

QP'"~) <

J,

So A,I21

+

zp(ds, dz), is also a weak solution on the same probability space,

108

4. Some Useful Tools in Stochastic Differential Equations

since it has the same probability distributions as that o f F, (w3( s ) ,w4( s ) ,s 5 t ) . Then by the pathwise uniqueness xt = F,(ws, q,, s I t ) .Hence ( x t ) is a strong solution o f (4.1) . Corollary 140 The pathwise uniqueness of solutions implies the weak uniqueness of weak solutions. In general, The pathwise uniqueness of solutions implies the uniqueness of probability laws of ( s fw, f ,qi(dt,d z ) ) from the weak solutions to the SDE (4.1). Proof. Suppose that two weak solutions ( x f ,wf, qi(dt,d z ) ) , i = 1,2, are defined on probability spaces (ai,p,($), P i ) , i = 1,2, respectively. As the . proof in Theorem 137 construct a space R = D d x Dd x W ~ DX ~ Map (xl(.,w),wl(.,w),rl(.,w)) and ( x 2 ( . , w ) , w 2 ( . , w ) , q 2 ( . , winto ) ) the R1 = 1st x 3rd x 4th component space o f R and R2 = 2nd x 3rd x 4th component space o f R , respectively. From these two maps we get the probability laws P', and Pz on R1 and R2, respectively. By the proof o f Theorem 137 one sees that the pathwise uniqueness deduces that ( ~ 1w3, , q(,, .)) = ( W Z ,W S ,q(., .)). This implies that the probability law P', coincides with the probability law P,". In particular, the probaility law o f xl(., w ) coincides with the probability law o f x2(.,w). That is, the weak uniqueness holds. In general, the probability laws o f (xl(.,w),wl(.,w), rl(.,w)) and (x2(.,w),w2(.,w),r2(.,w)) coincide, because P,' = Pz. That is, VA1 E B(Dd),VA2 E B ( w t ) , V A 3 E

?wd),

P,'((xl(.), w l ( . ) ,cl(.)) E A1 x A2 x A3) = P,"((x2(.),w2(.),r2(.))E A1 x A2 x A3). By using the Yamada-Watanabe Theorem and the pathwise uniqueness theorem (Lemma 115) one easily obtains the existence o f a pathwise unique strong solution for a SDE with discontinuous coefficients. Example 141 Assume that (4.2) (that is, Assumption (H)) holds and assume that 1 4 4x)12 + JZ Ic(t,x, 4124 d . 4 I ko(l 1x12), 2 l a ( t , x ) - 4 t , y ) 1 2 I cTN(t)12 - YI , as 1x1, lvl I N , t E [O,TI, 2 Jz Ic(t,x, z ) - 44 Y , 4124 d z ) I ~ ; ( t 1%) - Y I , as I4 ,lyl I N , t E [O, TI, where c;(t) 2 0 is non-random only depending on N and T such that c;(t)dt < 00, and a-'(t, x ) exists and is bounded la-'(t, x)l 5 ko, where x , b,c E R d , a E R ~ @If ~ . b(t,x) = - i y X # o , then for each T < oo the following SDE with jumps i n d-dimensional space on t E [ O , T ] :

+

4.2 Tanaka Type Formula and Some Applications

109

where xo E R~ is a constant vector, has a pathwise unique strong solution. Therefore, (4.3) on t 2 0 also has a pathwise unique solution. Proof. For each T < oo by Example 134 (4.3) has a weak solution. If we can show that the pathwise uniqueness holds for (4.3), then by Theorem 137 we will obtain a pathwise unique strong solution for (4.3) on t E [0,TI. In fact,

+ +

I - 1x1 - 191 1x1 I91 = 0. So Lemma 115 applies, the pathwise uniqueness holds for (4.3). rn

4.2 Tanaka Type Formula and Some Applications 4 . 2 1 Localixation Technique Consider a real semi-martingale xt as follows:

where At is a real continuous, finite variational process with A. = 0, Mt is- a real continuous, local square integrable martingale with Mo = 0, and Nk(dt, - dz) is a Poisson random martingale measure such that Nk(dt,dz) = Nk(dt, dz) - x(dz)dt, where Nk(dt, dz) is a Poisson counting process generated by a -Poisson point process Ic(.) with the compensator n(dz)dt, x(.) is a non-random u-finite measure in some measurable space (2,%(Z)). Obviously, we may assume that s t is a right continuous with a left limit process (denote it by a cadlag process). To make sense of (4.18) let us assume that

at

E yVoc = k

f (t, z, w) : f is

at - predictable such that

Jot Jz I f(s,z,w)12~(dz)ds < oo, P - a.s. for all t 2 0.

(4.5) For simplicity, from now on, we will always use the phrase "by localization" to simplify our proofs or statements. It means that we let

where lAlt = the total variation of A on [0,t], , Nt = Sot Jz C(S,2, ~ ) f i k ( d sdz), 2 (N), = J, I ~ s , z,w)I r(dz)ds, and discuss our problem for processes x t ~ . ,~AtA7,, , MtAT,, and NtA7,. After obtaining the result we then let n -+ oo. The advantage of such a

J,t

110

4. Some Useful Tools in Stochastic Differential Equations

2 localization technique is that IAItAT, ,IMI&,,,, (M)tATn1 and ( N ) ~ Aare ~, bounded by n for each n, and 2 E IN~AT,,~I n, E l x t ~ ~ , 4~ , ~ 1~x 0 1+~ 1%

<

4.2.2

Tanaka Tgpe Fomnula in d- Dimensional Space

Now let us consider some more concrete SDEs with jumps in d-dimensional space

i = 1,2, where At = (Alt,..., A,t), Mt = (Mlt,-..,MVt);Ait is a continuous increasing process, 1 < i < m; Mit is a continuous local square gtegrable martingale, 1 I i 5 r ; such that (Mi,M j ) t = 0, as i # j; Nk(dt,dz) is a Poisson martingale measure, which has the same meaning as that in (4.4) c is a R'-valued vector, and b and a are d x m and d x r matrices, respectively. Recall that (4.11) can also be written as dxt = x i bi(t,xi, w)dAt a ( t ,xj, w)dMt t 0, i = 1 , ~ . +JZci(t,x f - , ~ , u ) N ~ ( d t , d z ) = , x xj b , ~ 2 We have

+

+

Theorem 142 (Tanaka type formula) Assume that l o Ibi(t,x,w)12+ la(t,x,w)l2+ Jz Ici(t,x , z , w ) l % w I g(Ixl), i,j = 1,2, where g(u) is a continuou~non-random function on u 2 0; 9 Iff(t,x,w> - .(t1y,w)12 I G(~)PTN(III: - Y I ) 12 - Y I , as 1x1, lvl < N , and t E [O, TI; where 0 < k E ( t ) is such that EL=, k z ( t ) d ( ~ k <)m, ~ for each T < m; 0 I &(u) is non-random, strictly increasing, and continuous in u 2 0 with p G ( ~= ) 0, and J'?+du/&(u) = m. If xi satisfies (4.7) with coeficients (bi,a , ci) and the initial value x i , i = 1,2, respectively, &b defined on the same probability space and with the same Mt ,At and Nk (dt,dz), then W > 0 = - xgl sgn(xt- - x$-) . d(xi - x$)

EC

1.: I;.-

+ Jot

; .1

lx'-xa +$xdldlr %,j,k=lSgt I(X:#X:) a

a1

2 & . .-(=! -=? *3 rs I X : - X ~ I ~

..)(~4.-~j2,)

. ( ~ i k ( ~ , ~-~~,i wk () ~ , x $ , w ) ) (x~~j,kw()~ g~j k ( ~ , ~ $ , w (Mk), ))d 1 +StO Jz 1[ [ x i -- xs-2 cl(s,xs-, 2 , W ) - c2(s,xi-, Z , w ) ] - 12;- - 2:-sgn(xs- - xS-) . (cl(s,xi-, z,w) - c2(s,x$-,Z , w))]Nk(ds,dz), where

+

1

4.2 Tanaka Type Formula and Some Applications

sgnx

=

111

{ (8,.0,- -,as8). as x # 0, x 0. =

Proof. To show the formula for t 2 0 we only need to prove that it holds on t E [0,T ] for each given T < oo. For simplicity, in the following for an arbitrary given T < oo, let us write k ~ ( tand ) pN (u)for k$(t) and &(u), respectively. Write the equality which we want to prove as Ixt' - x q = 1x6 I~. Let T N =inf {t 2 0 : Ix:l+ Ix:I > N } . Now for any fixed N take 1 2 a, 5 0 such that Jaann-' (u)du = n , and for each n take a continuous function f,(u) such that 0, as 0 u a,, or u a,-l, fn(u) = any number less than 2(npN(u))-l,otherwise; f,(u)du = 1. Set for x E Rd with

I ; . + c;=,

< <

{

>

12-'

dV fn(u)du, n = 1,2,. . .. cpn(x) = Then cp, E C 2 ( R d )and ,

3

( a / a x i ) ~ n ( x=) I(an
fn(u)du,

~d"'

(a2/a~ia~j)~n(~)

+

3).

= I ( ~ , , < ~ ~ fn(u)du ~ ) ( fn~ (1x1) Obviously, I ( a / a x i ) ~ ~I (5x 1,) and as n 4 oo ~ n ( x4 > 1x1 , ( a l a x i ) ~ n ( x4) I(x#o) By Ito's formula

3.

2

(P~(x:ATN

1

- x ~ A T N )- (PT%(XL.O

-

c~,'E~ gATN ( a / a x i ) ( ~ , ( xf x i ) . (b:k

( s , W ) - b:k(s, 4,w ) ) d A h c$=~ (a/axi)cp,(x: - x:) . ( a i k ( ~x,,1, W ) - a i k ( ~x:, , w))dMks +$ SOtATN(a2/axiaxj)cp,(x: - 2:) ' ( u i k ( ~x:,w) , - o i k ( s , x2 S , w ) ) ( u j k (x:, ~ , W ) - ojk(slx:,w))d (Mk), ~ t = J~z ( d / a ~ i ) ~ n (-~ x: f-- ) . (ci( s ,x:- ,z l W ) -c:(s, 5:-, Z,w))Gk(ds,dz) +So"^'" Jz[cpn(x:- -2:- +cl(s,x:-,z,w) - c 2 ( s , x f - , z , w ) ) f-xf-) -cp,(x:- - 2:-) - ~ f = ~ ( a / a x i ) c p , ( x .(c; ( s ,xi.-, Z,W ) - c:(s, 2:-, z , w))]Nk (ds,dz) = c;=~ IF. By Lebesgue's dominated convergence theorem it is easily seen that as n 4 m IF J;"'" q n ( x f - x:) - (bl( s ,x i , w) - b2(s,x:, w))dAs, E I c$=, (a/axi)cp,(x: - x:) . ( u i k ( 3 , x f ,W ) - a i k ( ~x,i , w))dM/cs - J":' sgn(x: - x:) . ( ~ ( xs i, , w) - a ( s ,x f , w))dMs l2 4 0. Note that as n+oo,&E [O,T],VN =

+

pTN

c;& + JotATh'

-

S,tATN

112

4. Some UsefulTools in Stochastic DifferentialEquations

-'Jjk(s,x!,~))I I(aik(s,x;,w) - flik(~,xf,w))(ojk(~,xi,w) I PN(~N)~N(~)Hence by Lebesgue's dominated convergence theorem it is seen that as n+m I; -+ I4 (with t A T N substituting t). Similarly, it is seen that as n m I; -+ I4 (with t A T N substituting t ) . Moreover, by Lebesgue's dominated convergence theorem as n -+ m EI J z ( a / a x i ) v n ( ~ ; -- x f - ) . ( c ! ( s , ~ ; -z,w) , t -c:(s, xz-, r , w))Ek(ds,dr) - Jo Jzsgn(x:- - x f - ) .(c1(s,xi-, Z , W ) - c2(s,xf-, r , ~ ) ) N ~ ( d s , dl2 r ) 0. Finally, notice that limN+ooT N = m. The desired conclusion now follows. w -+

c:=,lFTN

-+

4.2.3 Applications to Pathwise Uniqueness and Convergence of Solutions The Tanaka type formula is a powerful tool when discussing the convergence of solutions for a sequence of SDEs and discussing the pathwise uniqueness of solutions t o a SDE. Actually, we can give the following theorems. Theorem 143 Suppose that in (4.7) b=b1=b2, C = C ~ = C ~ , X ~ = X ~ = X ~ , that is, we have only one SDE. Assume that conditions 1" and .!? in Theorem 142 hold, moreover, 9 pN (u) in 2" is concave, 'dN = 1,2, - . a;

4"

~ ; , ( s P ( x -9))i ' (b(t,x,'J)- b ( t , ~ , ~ ) ) I i kC

YI),

T N ( ~ ) ~ ~-( I X

as1~1,lylIN,t~[O,T],Vk=l,...,m;

where c;(t) and ~ z ( uhave ) the same property as that of k$(t) and pN ( u ) , 50 Jz Ic(~,x,z,w) - c(t,x,z,w)I 4 d ~I) % ( t ) G ( l x - Y I ) , as 1x1 ,191 N , t E [O, TI; where &(t) and E:(U) have the same property as that of k$ (t)and pN (u). Then the pathwise uniqueness of solutions for (4.7) holds.

<

4.2 Tanaka Type Formula and Some Applications

113

-

Proof. Assume that xj,i = 1,2, are two solutions of (4.7) on the same probability space with the same At, Mt ,Nk(dt, dz). Then by the above Tanaka type formula as t E [O, T ]

I 5 C;L"=l ESo CTN ( S ) G N (-I ~x?l)dAks +G E E ' G(S)PS(IX; ~ -xzl)d(Wks +2 C;L"=l E E $ ( S ) ~ ~ ( I X ; - x:I)ds

E

I&,

-T

~ATN

- $ATN

SrTN

JiATN E N ( l x ; - xsl)d&, -T

5E where

TN

= inf

& -T = =;L"=l

{ t 2 0 : lx:I cz(s)Aks +

+Ix:~

>N },

J; k G N(u)= Eg (u)+ pz (u)+ G z ("1

~ ( ~( M ) )dk s+ Cr=l . f i z z ( ~ )+t, d~

-T

and G N ( u )still has the property as that of pN(u) by Lemma 144 below. Let

Ft = .y.

Notice that

EI % L

& > t. Hence Ft = 2;'

TthrN - X ? TthrN

< t. Thus

I ~ E J , L " ' ~ G ~ ( IT ~a $ ~ - X ? I ) ~ ~

lxkaArN- x?T.ArN 1)ds. From this it easily found that P - a.s. Vt 2 0

5 S;

Lemma 144 If pi(u) is strictly increasing, concave, and continuous i n u 2 0 with pi(0) = 0 such that So+du/pi(u) = oo, i = 1,2, then pl ( u ) + p2(u) is still strictly increasing, concave, and continuous i n u 2 0 with p1 (0) p2(0) = 0 and satisfies

+

Proof. One only needs to prove (4.8). The rest of the conclusions in Lemma 144 are obviously true. Let u = inf { u > 0 : pl ( u ) = p2(u)). 1) Suppose that 'Zi > 0. Since pl(u) - p2(u) is a continuous function. It must be true that pl(u) - p2(u) > 0 , for all u E (0,T i ) , or p1 ( u ) - p2(u) < 0, for all u E ( 0 , ~ )Thus, . in the first case, dul (pl(u) p2(u)) 2 So+dul (2pl(u))= oo. For the second case the proof is just the same. 2) Assume that E = 0. Again by the continuity of pl(u) - p2(u) it should be p1 (u)- p2(u) > 0, for a11 u > 0 or pl (u)- p2(u) < 0 , for all u > 0. The proof can still be completed as in 1). We can also use Theorem 142 to discuss the convergence of solutions to (4.7). Suppose that x;, n = 0,1,2, - . . satisfy the following SDE's, respec-

So+

.

+

114

4. Some Useful Tools in Stochastic Differential Equations

tively,

We also denote xf = xt,b0 = b.

c/ m

lim E[exp(-

n4m

k=l

t

F , ( s , w)dAk,

0

Proof. For arbitrary given 0 5 T < r x let us show that (4.10) holds as t E [0,TI. Applying Theorem 142 we find (x?ATN

I

- x t ~ 7=~Ix$ - ~ 0 1

+ .ftATN sgn(x!

+J,""'" Jz[Ix:,:.1

2,-

+ c n ( s , x r l1 , ~-)c(s,xa-, z,w)I

- x,- I ] ~ k ( d sdz) ,

+ J,""'~JJIx; - Ix;

- x,) . (bn(s,x,! w) - b(s,xS,w))dAs

- x, - x,] - sgn(x;

+ cn(s,x;,

z, W)- C(S,x,, z, w)[ - x,)(cn(s, x;, z, W)- C(S,x,, Z, w))]~(dz)ds,

4.2 Tanaka Type Formula and Some Applications

where

115

+

T N = inf t E [O,T]: J i g ( s ,w ) d ( ( M ) , s ) > N } . Write X r = x? - xt, o;(t,w) = o(t,x?,w) - o(t,xt,w), c;(t,z,w) = cn(t,x?-,z,w) - c(t,xt-,z,w). Then by the above I IX$l+ F ( s ,W ) 1x21dA, Fn(s,w)dA, S Gn(s,z , w)r(dz)ds 2 SZ G(s,Z , W ) IX:l ~ ( d z ) d s +2

{

Ixz",

JotATN

I

JotATN

JotATN

+

' t ATN

+ JotATN

+

So

K ( s ,w) 1x21d (Mk), N ~ A T , where Nt = s g n ( ~ ~ ) - u ; " , ( s , w ) d ~ , + J ~cJ~~( [s l , z~, ~ w-) I - I ~ : - I ] ~ ~ ( d s , d z ) . Applying Lemma 146 below we find exp(F(s,w)dAS- 2 JZ G(s,z, w)r(dz)ds)). lxpATr I lX;l+ J ' ~ exp(~ ' J; F(u,w)dAu - 2 J; Jz G(u,Z , w)r(dz)du)) .(Fn(s,w)dAs 2 Jz Gn(s,z, w)r(dz)ds dNs). Hence E H ~ A TI ,X ~ A ~ 5 , . E IX;l +E Hs(Fn(s,w)dAs 2 Jz Gn(s,Z , w)r(dz))ds I E I X ; ~ +E H,(F"(S, w ) ~ A , 2 jZ G ~ SZ , w, ) T ( ~ z ) ) ~ s , where Hs = exp(- S; F(u, w)dA, - 2 S; Jz G(u,Z , w)r(dz)du))I 1. Note that by the assumption for each w, when r is large enough r,(w) = 03. Therefore by the above result letting r + 03, by Fatou's lemma we have E H ~I X P I~ E I X ; ~ E j,t F ~ Sw ,) ~ A , 2 J, G ~ Sz , w , )T(~z)~s. Now letting n + 03, we obtain limn,, EHt IXrI = 0. Notice that the condition 2' here is weaker than the usual Lipschitzian condition. Because now b(t,x , w) can be discontinuous . For example, b(t,x ) = A. - Alsgn(x), where Ao, A1 are constants. f k0 xC;=l

+

JiAT'

JpTr

+

Jprr

I

+

I

+

+

+

+

Lemma 146 (Stochastic Gronwall's inequality). Assume that K , Nt are RCLL (right continuous with left limit) processes, where Nt is a semimartingale, and assume that Bt is a continuous increasing process. If

vt < Nt + Jot V,dB,,Vt 1 0; & = 0 , then e - B a d ~ sVt, 0. e-BtK 5 No Proof. Notice that by Ito's formula d(NtBt) = BtdNt NtdBt, ds(Ns(Bt - Bs)2/2)= [(Bt- Bs)2/2]dNs- (Bt - Bs)NsdBs, etc. Hence by assumption it can be seen that K I Nt VsdBs Nt NsdBs d ~J,", VudBu = No NoBt $ ( ~ t- Bs)dNs 'J dNs V,(Bt - Bs)dBs

>

+

+

+

+

+

< + 1:

+

+ Ji + Ji

4. Some Useful Tools in Stochastic Differential Equations

116

N O ( C ~ = ~ ( & ) " I+~J! ~ ) ( c ~ = ~(& BS)"ln!)dNs

5 ..

+ J ' v , ( B ~- ~ , ) " k ! d ~ , .

Since & is RCLL, it must be bounded locally. (Its bound can depend on w). Hence J: V,(Bt - ~ . ) ~ / k ! d5~JCO . (w)~ f + ' / ( k I ) ! 4 0, as n 4 m. Therefore Vt 5 NoeBt J: e B t - B s d ~ sVt, 2 0. w Conditions 3", 4" in Theorem 143 can be some non-Lipschitzian condition. Let us give an example as follows.

I

1

+

+

Example 147 Let bi(t,x) = -ko3IxZo klxi k2 lxil Then b(t,x ) satisfies the condition 30.

+

In fact, as x , y 1

5 -(-

+

+ k3 1x1

# 0,

1 '1 (-i;a +fi)=Cid=lm(-@1x1 - IYI + + yf9= 0)

yp

$+y+w) lul

XI

lul

where we have applied the Schwarz's inequality to get that ~ : = X1i Y i 5 1x1 IYI Note also that as x #O,y = 0, Hence b(t,x ) satisfies the condition 3".

4.2.4

Tanaka Q p e Forrnual in I-Dimensional Space

Now consider SDEs (4.7) in 1 - dimensional space: i = 1,2,

where d = 1, but At = ( A l t , . . .,Amt) and Mt = (Mlt,. . ., MTt) are still n-dimensional and r-dimensional such that (Mi,M j ) , = 0, as i # j; respectively; so b E RIBm, and a f RIB'. In this case we can prove that the Tanaka type formula will be simpler and the conditions to be satisfied are also weaker. Actually, we have

Theorem 148 (Tanaka type formula) Assume that 1" Ibi(t,x,w)I2 la(t,x,w)12+ Jz 1ci(t,x,z,w)li ~ ( d z5) g(Ixl), i,j = 1,2, where g(u) is a continuous non-random function on u 2 0; FJ 1 4 4 X , W ) - 4 t ,Y , w)12 k;(t)P;(lx - YI), as 1x1 ,191 5 N , t E [O, TI; T where 0 5 k$(t), ELzl E SoT kN(t)d (Mk)t < m, for each T < m;0 5 p%(u) is non-random, strictly increasing in u > 0 with pK(0) = 0 , and

+

<

4.2 Tanaka Type Formula and Some Applications

117

Jq+

WPN (u) = 00. If xi satisfies (4.11) with coeficients (bi, a, ci) and the initial value x i , i = 1,2, respectively, and they_are defined on the same probability space and with the same Mt, At and Nk(dt,dz), then Vt 2 0

: . 1 I: .- =; .1 I;.+ Jot sgn(xt- - x:-)d(x:

- a:)

-1;.

+ J t J- : . I[

I-:.-

-x:-+cl(s,xf-,z,w)-c2(s,x$-,z,w)l O =1 -sgn(x,- x:-) . (cl(s,&, z,w) - c2(s,xz-, Z, w))]Nk(ds,dz), where asx#O, 0, as x = 0. Proof. We only need to point out the difference of the proofs between here and Theorem 142. Checking the proof of Theorem 142 we find that here we have cp, E C2(R1). So

Jix'

d ( x ) = I(an = I(a,<~xl)fn(1x1) 7 and as n -r oo ~ ( ~ (2) n 1x1 P', (x) S S X. Moreover here, after applying Ito's formula to cpn(xt - x:), we see that as n + oo,Vt E [O,T],VN 0 5 ELzlJOtATN cpE(d - 2:) . ( 4 %4,w) - a k (s, x:, wN2d (Mk), = EL=1 J,"^'" f n (12: - x:1) ( n k ( s , x : , ~ )- c k ( ~ , x : , w ) ) ~(Mk), d 5 2k:, kN(~)d ( ~ k )/n , 0, and the rest of the discussion remain true. So we arrive at the conclusion. -+

-+

;

g

-+

N

Conditions for a in Theorem 148 are weaker than that in Theorem 142. Moreover, here c even can be non-Lipschitz continuous. In fact, we have the following example.

Example 149 Set a(t, x) = k I ( t ) , m kz(t)x, where 0 I :kj(t) is non-random such that ki(t)2dt < m, i = 1,2, for each T < m. Then condition .P in Theorem 148 is satisfied.

+

To see this we use the elementary inequalities

pi7-m11m>

and

: .I - YI' I:JiGJdFGi to get la(t,x) - e ( t , y)12 5 2kl(t)' la: - yl+ 4Nk2W2 lx - yl , as 1x1, lyl L N . So, for this example, condition 2" in Theorem 148 is satisfied. Furthermore, let us show that, actually, condition Jz x, z,w>I4 d 4 1

118

4. Some Useful Tools in Stochastic Differential Equations

in l o of Theorem 148 also can be dropped. For this, let us first prove the following theorem.

Proof. 2) is deduced from 1). In fact, 1x1 = x+ + x-. So 11%q s , ~ , w ) l -lZs-I - sgn(%-)?(s, Z , w)l I q s , z,w))+ - I ( B ~ - > O )Z~, WS ), I + q s , z , w ) ) - - (2s-1- + ~ ( z s - 5 0 ) q s , z , w ) ~ . Thus 1) implies 2). Let us show 1). Notice that x- = (-x)+. So we only need to show the first conclusion in 1). Take 0 < a , 1 0 and continuous functions 0 g,(x) such that

+

I(%- + +I(%-

<

w+

4.2 Tanaka Type Formula and Some Applications

119

JP-l

gn(x)dx = 1, gn(x) = 0, as x $ (O,an-I); n = 1,2,. . Let x+ : gn(u)du,n = 1,2,. . . hn(x) = So dy J Then h;(x) = I(r>o)+ J: g n ( ~ ) dh;(x) ~ > = gn(xC). Hence as n 1; hn(x) x+, h ; ( ~ ) -+ h n ( ~E ) C2(R), Ih;(z)l Applying Ito's formula, we find that

<

-+

-+ m.

where

Since hE(x) 2 0, so by Taylor's formula hn(zs) - hn(zs-) - h;(zs-) n = ;h;(zse n z S ) ( n q 22 o. Hence AT is an increasing process for each n. Now let

+

zs

By the localization technique we may assume that Mt and Nt are martingales, moreover, Mt, (M)t , and E (N), are bounded. Then it is easily seen that as n -+ m by (4.12) A; -+ in probability for each t 2 0. In fact, as n -+ oo hn(Zt) -+ 2t, P - a s . for each t 2 0, and by Lebesgue's dominated convergence theorem as n -+ m

xt,

2

J s - d s - ( > O ) d s -+o. From this it follows easily that is also an increasing process. Notice that 0 5 nxt = - 2-; - I(%& >o) A Ztl vt 2 0, where we write 2f_= 2; Hence W 2 0

xt

q

-

0 5 @.At = I(%,>o)% + ro@, where we havcused the fact that I(&>o) n At = I(%,>O) - (2,A%)] -t = I(gt->O)[xt - %] = I ( B ~ - > o ),x ~ and -II(%,-lo) n At = I(~,- At t CO<sP lnxsi,l= C O < ~ < ~ - Co<s&%s- >0)% I(%,-lo)%)

+

[q

A-

-

+

nL

120

4. Some Useful Tools in Stochastic Differential Equations

= 2s: Sz[I(%.->o)(Zs-+z(~, z , w ) ) - + I ( E . - ~ o ) ( ~ s - + E (~s , w ) ) + ] N k ( d s , d ~ ) = Jz[(2sZ , w))+ - (2s-)+ - I(z,- > o ) ~ s Z ,,w)]Nk(ds,d ~ ) .

+z ( ~ ,

.& one sees that L 5 12; -GI + ET=lJilip(~l~)l dAP(s) + EL IS: I(Z.->~)&(S,w ) d ~ k ( sI )

On the other hand by the definition (4.14) of

Remark 151 1) By the proof of Theorem 150 one sees that Co,s,t[I(~,->o)(2)- + I ( Z ~ - ~ O ) ( ~ ~ ) + I +7i, I ( ~ . - > ~ ) ~ M< , I m, P - a.s. 5 ljl: holds for a general real semi-martingale 2t =Z0+71,+Mt, where & 2 0 with 20= 0 is a real continuous increasing process which is locallv inte.qrable, i.e. for each T < m E& <&;and Mt with Mo = 0 is a real RCLL local square integrable martingale. 2) By the proof of Theorem 150 one also sees that Co<sst[I(e->a)(ss- a)- + I ( ~ ~ - ~ a )-( a)+] 2s 5 I(%-a)+ - (20 -a)+[ +& l J , , ~ ( . - ~ - > ~ ) d<~ m, . l P-a.s. holds for a general~ealsemi-martingale 2t = 20 + X t + Mt as that i n 1 ) and for any constant a E R1.

+ I$

+

To see that 2) in the above corollary is true one only needs to write (2-a)+

hn(x)=Jo

dyJ:g,(u)d~,n=1,2,...,

and then to proceed with the proof in almost the same way as in the proof of Theorem 150, as a result of which we obtain 2). Now from Theorem 148, Theorem 150 and their proofs we immediately get the following theorem.

Theorem 152 Assume that the condition 2' i n Theorem 148 holds, and assume that 2 lo' l b i k x1 w)12 + b ( t ,x,w)12 + .fz 2. z,w)I ~ ( d z5) g ( 1 ~ 1 ) ~ i = 1,2, where g(u) is a continuous non-random function on u 2 0. If x i , i = 1,2 are two solutions of (4.11) with the initial value x& i = 1,2, respectively, then Vt 2 0 I(x:-x:)>Od(~:- x:) ( x i - 2:)' = ( x i - xi)+ + ~ , x : ~ , ~ , +~ ) ) + J ~ J z [ ( ~ : - - x : - + ~ 1 ( ~ , x : ~ , ~ , ~ ) - ~ 2 (-(x:--x:-) 1 1 -I(x:-x:)>o(c (51x8-7 2 , ~-) c2(s12 : - 1 x , ~ ) ) ] N k ( d ~~ ,) , (x: - 2:)- = (2; - x i ) I(x:-x:)Iod(xt - $2) +Jo"Jz[(xf- - x i - + c ' ( s , ~ ~ ~ , z , w ) - c ~ ( s , ~ ~ ~ , z , -wI :)- )) -- - ( ~ ~ ~

+ Ji

+

4.2 Tanaka Type Formula and Some Applications

121

+ ~ ( x : - x : ) 5 0 ( ~ 1 xi-, ( ~ , 2 , W ) - c2(s,x!-, 2,~ ) ) l N k ( d dz), ~, and 1x2 - xfl = 1x6 sgn(x; - x?)d(xi - x?) 2 [[Ixf-- xs+ ~ ' ( s , x : - ~ z , w-) C ~ ( S , X : - , Z , " J ) ~ - I x f -x:-1 O Z1 -sgn(xs- - 2:-) . (cl(s,xi-, z,w) - c2(s,x?-, z,w))]Nk(ds,dz). Moreover, the last terms of the above three formulas are absolutely convergent, P - a.s.

+sot +sts

I;.-

4.2.5 Tanaka Type Formula in The Component Form The 1- dimensional result and Remark 151 motivates us to consider a Tanaka formula for the component ( x h - x$) of ( x i - x f ) , when xi and xf are solutions of n-dimensional SDE with jumps (4.7) under some other conditions. For this we write (4.7) in its component forms: for i = 1,2, . - . ,d

Thus we can apply Theorem 152 to obtain the following theorem.

4.2

Tanaka Type Formula and Some Applications

+

123

+

(ii) x 2 y ==+ x c(t, 2, z, w) 2 y c(t, y, z, w). Then the pathwise uniqueness holds for SDE (4.16). Proof. Assume that x f , i = 1,2 are two solutions of (4.16). Write jZ, = xt - x f , T N = inf {t > O : 1x$1+ Ixf > N } : Then by the Tanaka type formula (Theorem 145)

1

I % A~, I

~ATN

5Ed

E where

Jt = J;[,[Ix:-

T

+

k N ( s ) p ~ ( l z s l ) d s EJtnrN,

1 - x : + c(s,x.-,.,LO) - c(s,x~-,z,w)I- I x : - x:-1 - x:-) . (c(s, 2:-, 2,W)- C(S, 2:-, z, w))]Nk(ds, dz).

-sgn(x,In the case that the condition (ii) of 4" holds then Jt = 0. So we have

lzt~~~I 5 Ji~;(S)PE(E lanrN1)dS, t E [O,T]. Thus E IztATN I 0, t [O,T].From this one easily derives that zt 0 , t [ d , ~ ] . Since T < ca is arbitrary given, so zt = 0,Vt 2 0. In the case that the E

88

=

=

E

E

condition (i) of 4" holds then by the same Tanaka type formula

E p t A T N I 5 E J ~kTN(S)P5(Izsl)dS " ~ ~ +2E Jz Ic(t, x: ,z, W)- ~ ( tx:,, z, w)l r(dz)ds 5 3EJiATNk;(s)&(lz,l)ds, as t E [O,T]. The conclusion now follows. Comparing the uniqueness theorem here and Theorem 115, one finds that the condition on a here is weaker than there. Here a can be only H6lder continuous with index However, the condition on c is different. Anyway, if the SDE is without jumps, that is c = 0, and then we have got a really weaker conditions on the pathwise uniqueness of solutions. For an SDE with jumps in d-dimensional space we can also have a uniquenes theorem discribed by the conditions on the components of the coefficients which are different from Theorem 143 and Theorem 115. For simplicity consider the following SDE in d-dimensional space

gATN

3.

Theorem 155 Assume that 1" E lxo12 < ca, and 2 (x , b(t,x,w)) < c ( t ) ( l + 1x12), la(t, x, 412+ JZ1 4 42, z , 4 1 2 4 d z ) 4 t H l + 1x12), where 0 c(t) is non-random, such that for any 0 < T < oo CT = c(t)dt < ca,

<

<

SOT

4. Some Useful Tools in Stochastic Differential Equations

124

. .

besides, p$(u) is concave; 40 X i 2 Yi Xi ci(t,X , 2 , w ) 2 yi ci(t,y, z , w), wherex= ( X I , . . . , x i , . - . , x d ) , y = ( y l , . . . ,yi,..- ,yd). Then the pathwise uniqueness holds for SDE (4.16).

* +

Proof. Notice that for any real number x 1x1 = x+ + x - . Now using the notation in the proof o f Theorem 154 by the Tanaka formula on components (Theorem 153) we have that as t E [0,T ] E xi,t~TN2E S k $ (~s ) ~ n ~(Xl , ds,~ \Ji =~ 1,2,. . . ,d. Hence as t E [O, T ]

I I 1) E laATN 1 4 EL, E la,tATNl 5 d - E S - ~~; (~ ~ ) ~ ~ ( l 2 ~ l ) d ~ . T h e conclusion now follows easily in the proof o f the previous theorem. A

A

as

rn

4.3 Local Time and Occupation Density Formula Consider a real semi-martingale xt which is given by (4.4):

W e can introduce the following definition. Definition 156 The function L:(x) of two variables (t,a ) is called the local time of semi-martingale x t , i f it satisfies: for V t 2 0 and a E R (xt -a)+ = ( X O - a)+

+

1 I(xa->a)dxs -L:(x) 2

+

4.3 Local Time and Occupation Density Formula

125

It is easily seen that (4.19) can be rewritten as (xt - a)+ = (xo - a)+ +So' I(x.->a)dxs ~ L ? ( x ) + Co<s a ) (-~ s(0- + C O < ~I(x.-~a)(xs <~ - a)+. Moreover, by t h ~ form s one easily sees that the following lemma is true.

+

Lemma 157 L?(x) is continuous in t 2 0. Proof. In fact, for each t > 0 1 s(L:(x) - L?-(x)) = (xt - a)+ - (xt- - a)+ - I(,,->,) A xt -I(xt->a)(xt - a)- - I(xt-a)= I(xt->a)(xt- a)+ -I(xt->a)(xt- - a Axt) - I(xt->a)(xt- a)- = 0, and 1 0. However, as we said before we only consider right continuous martingales (or their right continuous versions), so it is easily seen that L:(x) is also right continuous in t 2 0. Therefore, L:(x) is continuous in t 2 0. By Remark 151 one immediately obtains the following lemma.

+

4. Some Useful Tools in Stochastic Differential Equations

126

Proof. The first conclusion is derived by the proof of Theorem 150 and Remark 151. Notice that t xt - a = xo - a Jo dx,, x- = x+ - x , 1x1 = x+ x-, So the second and the third formulas are established. The following occupation density formula on local time is very useful.

+

+

Lemma 159 For any bounded Bore1 measurable function f

Proof. By approximation we may assume that f E C r ( R ) ,the totality of functions with compact support. Let F ( x ) = JR f (a)(%- a)+da, V x E R. Then it is easily seen that F1(x)= J_"m f(a)da, F"(x) = f ( x ) . By Ito's formula

-

l

F'(X.-)~(S,r , w)&(ds, dr) -

l

L I F ( X ~ -

:l

-F(x,-) - F1(x,-)c(s, z , w)]Nk(ds,dz) = -

+ 4%t ,w))

f (xs)d( M ) ,.

(4.21)

By localization we may suppose that IAlt ,Mt, ( M ) ,and ( N ) ,are bounded, JZ C ( S , z, w)Zk(ds,dz). Denote the left side of (4.21) by where Nt = Ii.Obviously, I1 = f (a)(xt- a)+da, -13 = JR f (a) I(%.>a)dAs)da,

c:=~ J,

(.Jot

4.3 Local Time and Occupation Density Formula

127

Theorem 160 (Stochastic Fzlbini Theorem). 1) Assume that { f (t,a, w)),,, is a real random process such that (i) I f (t,a, w)l 5 44, where 0 g(a) is a Borel measurable function on a E R1; (ii) { f (t,a, w))(t,w),jo,clo)xR,a,Rlis a px%(R1)-measumblefunction, where P is the predictable a-field (for the definition of predictable a-field and predictable function see Definition 12). Now suppose that {Mt),>, - is a continuous square integrable martingale,

<

is a % ( R 1 )x&-measurable and for each t > 0 {Jot f ( s ,a, w ) d ~ , } (a,w)€R1 X R function. Furthermore, suppose that p(da) . . is a non-negative Borel measure on R 1 and JRl g(a)p(da) < ca. Then P - a s . V t 0 J,"[JRI f ( 3 ,a, wMda)ldMS = JRl [Jot f ( 3 , a, w ) d M s l ~ ( d a ) . 2) Suppose that E k ( d t , d z ) is a Poisson martingale measure generated by - a St-Poisson point process k ( . ) with the compensator n(dz)dt such that N k (dt,d z ) = Nk (dt,d z ) - n(dz)dt, where n ( d z ) is a a-finite measure on the measurable space ( Z , B Z ) , and suppose that p(da) is a non-negative Borel measure on R1. Now assume that { f (t,a, z , w)),?, is a real random process such that (i) If(t,a,z,w)l 5 g(a,z>, where 0 5 g(a, z ) is a % ( R 1 )x %z-measurable function on ( a ,z ) E R1 x Z such that JRl JZ g(a, z)2n(dz)p(da)< co; is a P x -masurable fine( 4 { f ( 4 a, z , w))(,,,,w)Ep,oo) tion, where P is the predictable a-field (for the definition of predictable a-field and predictable function see Definition 1261). Moreover, for each is a %([0,co))x5-measurable t > 0 {J: f ( 3 ,a , I , w)Ek(ds,d z ) }

>

h

( a , w ) € R 1x S I

function. Then P - a.s. W 2 0 J," Jz [JR1 f ( s ,a, 2 , w ) ~ ( d a ) ] f(ds, i k dz) = SR1 [J," Jz f ( s ,a, z , w ) S k ( d s ,dz)]@(da).

128

4. Some Useful Tools in Stochastic Differential Equations

We still need to show that there exists a set A E 5 with P(A) = 0 such that as v 4 A,Vt 0, f ( ~ a, 7 w)dMS]~(da)(a)' (S,"[SR1 f (s, a, w)~(da)ldMS)(a) = (JRl (We call A a common P-null set for all t 2 0). That is, P - a s . W 2 0 J,"[JR1 f (7' a>w)~(da)ldMS= JR' f (s, a, w)dMSl~(da)' For this let us show that I(t) = JR1 f (s, a, w)dM,]p(da) is a "right continuous with a left limit" (RCLL) process. In fact,

>

[Ji

[Jot

:1

JR1 S U P ~ ~ T f (s, 9' w)dMsI ~ ( ~ ' 51 JR lg(a)l ~ ( ( ~ ) ; ' ~ ) a ) p<( m' ~~) So P - a s . JRl s u p t < ~ f (s, a, w)dMs P ( ~ <~m' ) Hence if t, 1 t, t < T, then one can apply Lebesgue's dominated convergence theorem to find that as n -t w , I(t), qtn) since f (s, a, w)dMs is right continuous in t. The proof that I(t) has a left limit for any t > 0 is the same. Now we have that the both sides of (4.22) are RCLL, so the common P-null set for all t 2 0 is easik constructed. 2): First, by assumption (i) and (ii) JZ f (s, a, a, w) Nk (ds, dz) exists, since E Jz If (s, a , z,w)12r(dz)ds I t Jz d a , z)2r(dz) < w. Now the proof is similar to that in 1). In fact, El Jz[JR,f (s, a, z,w)~(da)l@dds,dz) - JR1 Jz f (s, a, z, w)&(ds, dz)ll-l(da)12 = E [ JSZ(JRZ ~ f (9' a ~', ,~ ) f~ 2 7' 7 w)~(dal)~(da2))r(dz)ds JR2 JZ f (7 ' a~> >' ~ )(s,fa2,z, w)r(dz)ds~(da1)~(da2) = O, -2 JR2 JZ f (s, '1, '7 ~ ) f~ 2', , w)r(dz)ds~(dal)~(da2)1 again by Fubini's theorem for Lebesgue-Stieltjes integrals. Hence, it is similar to I), and we obtains 2).

1s:

I

+

Sot

Ji

J;

[J,"

+

Ji Ji

4.4 Krylov Estimation

129

Let us explain the pysical meaning of the local time. By (4.20) one has that for a.e. a E R

where xC = M. Hence L:(x) can be viewed as the "measure" of occupation at the point a by the process x(.) during the time interval [O,t]. In particular, as Mt = wt is the BM, one sees that L:(z) = JOE-' Jot I[a,a+e)(xs)ds* This is exactly the Lebesgue measure of the time set { s E [0,t]: x , = a ) . Furthermore, by (4.23) we immediately obtain the following corollary.

Corollary 161 L:(x) is an increasing function on t 2 0.

4.4 Krylov Estimation 4.4.1

The case for 1- dimensional space

By means of the occupation density formula one can derive the so-called Krylov estimation with respect to the 1-dimensional SDE with jumps as follows:

Theorem 162 Assume that xt satisfies (4.18) with c satisfying that for each T < m JZ Ic(t, Z , u)12~ ( d z<) m , P - a.s. Let T N = inf{t I 0 : lxt12 I A I ~ .lA4t12 + lIVt12 (N)t> n). I f f E L 1 ( R ) ,then for any 0 5 T < m , one has that E I f (xs)l d ( M ) , 5 ~ N , TJR I f (Y)I ~ Y ^ ~ N ,l lTf l l L l ( R ) , where k N , 2~ 0 is a constant independent o f f such that

+

+

+

+

JpTN

1/2

ICN,T= k ~ E ( I A I T ~f~ (NM ) ~ ? T+~( N ) T / \ r N ) . Hence, if Mt is a square integrable martingale, c satisfies E s,T & I C ( S , k , w)124 d 4 d s < m , and IAIT

< Oo,

then

EST I f ( x s ) ~( M ) s5 k~ [ I f where 0 5 k~ = k o o , ~ .

IlLl(R)

,

Proof. Let f ( x ) 2 0 and f&) = f ( x ) A n. The by the formula (4.20) J," l f n ( x * ) l d ( W S= JR Ifn(a)lL?(~)da,W2 0. By Lemma 158 and Corollary 161

130

4. Some Useful Tools in Stochastic Differential Equations

Corollary 163 If Mt = a(s, w)dws, At = b(s, w)ds, and la(s,w)l 60 > 0, Z , W )4Id~. 4 I ko, lb(t,w)12 + l 4 t , w)12 + SZ IC@, where 0 5 ko, and 60 are constants, and wt, t 2 0, is a standard Brownian Motion process, then for any f , which is a Bore1 measurable function valued in R, EJ: If (xs)lds 5 lvr llf llLl(Rl , where 0 kT is a constant depending on ko, 60 and T only. Proof. If

>

<

Ilf llLl(R) = 00,

then the result is trivial. If

Ilf IIL~(R) < 00,

then the result follows by Theorem 162. The Krylov estimate is a useful tool in Stochastic Analysis, because it transforms the estimate on functionals of solutions for 1-dimensional SDEs into the calculation on the usual deterministic Lp-integrals on R1. Here we use the occupation formula of local time for the solution xt to establish the Krylov estimate, which is different from the usual or original proof for such kinds of formula that make use of the PDE results. We can see that the technique used here has some advantages. First, in Corollary 163 the coefficients b(t, x, w), a(t, x, w), and c(t, x, z, w) of SDE with jumps can depend on w, that is, the coefficients can be random, where when the PDE results are used this cannot be so. Second, in Theorem 162 the Krylov estimate is obtained for a very general SDE, which is hard to use with the PDE technique in general.

4.4.2

The Case for d- dimensional space

However, to derive the Krylov estimate on functionals of solutions for d-dimensional SDEs we have to get some help from the PDE results, so the coefficients a and c cannot be random. Assume that (x(t)) satisfies the following d-dimensional stochastic system:

4.4 Krylov Estimation

4

131

where is a continuous locally - integrable finite variational zt-adapted process, Vi = 1,2, - - . ,d, and At = .. . , that is, each component (of it) can be expressed as the differenceof two $t-adapted increasing 21 < m, for each T < m, where is the total processes, and E 1

(xt, xf);

4

($1

T

-

-

.T

variation o f on t E [O,T];moreover, wt is a d-dimensional BM, and Nk (dt,dz) is a 1-dimensional Poisson martingale measure with the compensator n(dz)dt such that Nk (dt,dz) = Nk(dt,dz) - n(dz)dt. W e quote a result from Krylov ( 1 9 7 4 ) [ ~(or, ~ 1 see [31) as the following lemma, which indicates the existence of a solution for a partial differential inequality.

-

Lemma 164 Suppose we are given a bounded Bore1 measurable function [O,oo))and X > 0, then for any s > 0 0 5 f = f (t,x ) : E: x Rd -+ there exists a smooth function u E ( tx, ) : R x Rd -+ [0,m) such that 1) z t j = , hihj(a2/axiaxj)uE I X U € , v h E R ~lhl, I 1,

x:(=

where k(p,A, d , E ) = e"X(d+1)/ppd/p(~dd!)1/p~d/2p(~(d + l))('/p)-', Vd is the volume of d-dimensional unit sphere, and I l g ( s , ~ ) l l ~ , ( ~ ~= ) ,( ~h i~x ,n ~d dl g ( s , ~ ) I ~ d s d x ) ' / ~ (= IMs, Y ) I I , , ~ ~ 3) lgrad,u"I 5 (X)'12uE, 4) for all non-negative definite symmetric matrices A = (ai, j ) d X d z z j = l a i , j ( a 2 ~ E / a ~i a~~( jt )r . A ) d5 0, c ; ~ = a~ i , j ( a 2 ~ E / a ~ i a ~ j ) - ~ ( t ~ . ~ + i ) d + 5 (-(det a / a t~)) ul / E (d+l)f,, where f E is the smoothness knction o f f , i.e. f E = f * wE = JroO ds JRd f (t - E S , x - E Y ) W ( S , y)dy, = f ( t , x ) , as t 2 0, 0, otherwise, w(t,x) is a smooth function such that w(t,a ) : R1 x Rd -+ [O,m), w(t,x) = 0, as ( t ,x) 4 [-I, 11 x [-I, lId, JroO dt SRd~ ( x)dxdt t, = 1.

y(t,x)

{

By using the above lemma we can prove the following Theorem 165 (Krylov type estimate). Assume that u = o(t,x ) : [O,T]x Rd -+ Rdmd, c=c(~,x,z):[O,T]XR~XZ+R~

132

4. Some Useful Tools in Stochastic Differential Equations

are jointly measurable and satisfy the conditions

where k~ = So cl(t)dt < co, for each T < co. Moreover, assume that E1 21, 5 k T , for each T < m. Write T

vt =- SOt tr.A(s,xs)ds, A ( s , x ) = $aa*(s,x ) . '

Then for all 0 < f , which is a bounded Bore1 measurable function defined o ~ [ o , Tx]Rd, a n d p > d + l

where we w d e 11 f il~,[o,Tl Rd = J, & 1 f (t,x)lp dxdt, and k t 0 b a constant depending only on p, k T , d and T . Furthermore, i n the case that a is uniformly non-degenerate or locally uniformly non-degenerate we have the following corollaries: 1) If, i n addition, for each N = 1,2,. . . there exists a 6~ > 0 such that for all p E Rd, ( A ( t ,X ) P , P ) 2 1/112 6 ~ as 1 1x1 5 N , where (., .) is the inner product i n Rd, and A ( t , x ) = aa*(t,x ) , (that is, a is locally uniformly non-degenerate), and letting T N = inf { t E [O,T] : lxtl > . N ) . then S,TN f ( s , x S ) d s 5 d > G T~ ,, I l f llp,[O,Tlx[-N,N]@d where k 2 0 is a constant only depending on p, ko,d, dN and T . 2) If, i n addition, there exists a 60 > 0 such that for all p E Rd, M t , X ) P , P ) 2 lp12 60, (that is, a is uniformly non-degenerate), then f ( ~ , x s ) 5d ~k ( ~ , k ~ l d , 6 ,( TI f )1 l p , [ ~ , ~7 ] ~ ~ d where k 2 0 is a constant only depending on p, h , d , 6 and T . 3

Proof. Let f (t,x ) = 0 , as (t,x )

E

(T,co) x Rd.

According to Lemma 164, for f 2 0 there exists a u" satisfying 1)-4) in on+ [0, ~ )T ] we find Lemma 164. Applying Ito's formula to u " ( t , ~ ~ ) e - ~ ( % that E u" (T,X T ) ~ - ~ ( ~ TEu" + ~(0, ) -x O )

+

e-X(qa+s)gradzu"(s,x,)d& E J ,e-A(qa+s)((a/&)@ f , a~i , (=d 2~u " / ~ x i ~ xi )X(tr.A, + l ) u E ) d s + E CO<sIT e - A ( q a + S ) ( ~ 2,) E ( ~-, u g ( s ,x,-) 3 -gradzug(s, x,-) . A x s ) = Ci=' I;.. Now we set -E

+~

4.4 Krylov Estimation

133

8 = k ( ~A,, d>€1Ile-'s(dfl)lPf y)llpJOnx Rd . By 2) in Lemma 164 E uE(O,xo)I 8. Notice that by 3) and 2) in Lemma 164 I$ 5 ~ ~ B XJ:? Ee-*qa-'(l-(d+')ir)rd A < kobAf E jkTkOA;8. T On the other hand, notice that le-'r(d+')lp f (r S, y) drdy)llp (JRd - e*s(d+l)/~lle-"r(d+l)/~f (r,y) < xRd = e's(d+l)l~Ile-*r(d+l)lp f (r, Y)

1x1

I -Is

+

lp lp,(r, ,z:

11p,(r,v)dO,TI x Rd

'

Hence by 1) in Lemma 164 and the condition (4.25) we have I$ $ ;E Co,ss, e-'(q~+~) -(dZu"(s,2,~AX~)/~X~~X~)AX~(S)AX~(S) % x ~ ~ ' ~ ( ~ +C' )OI <~ sEg ( A ~ ( ~ $ ) )I2C ; , ~ , ~ c1 ~ (s)~s. Furthermor, by 4) in Lemma 164 IZ T<- E e-'(qa+")(det ~ ( s~ , ) ) ' l ( ~ + f,(s, ' ) x,)ds Thus (4.26) is valid for f,(s, x,). Let E -+ 0. By Fatou's lemma (4.26) is derived for f . Now let us show the corollaries. 1): By (4.26) E J:N (det ~ ( s~ ,,) ) ' l ( ~ + 'f)(s, xs)ds

<

+

c&=~

SOT

I E c ( d e t ~ ( s~,, ) ) ' l ( ~ +(s,' )xf ~ ) I ( ~ ~ ~ ~ ~ N ) ~ s

I k(p, k ~ , d , TIlf) Ilp,[o,Tlx[-N,Nl Therefore, O I J:N f (s, x S ) ~ S5 & k ( ~ 7'T, d, T, Ilf Ilp,[l),Tlx[-N,N] 1) is proved. Corollary 2) obviously follows by (4.26).

.

4.4.3 Applications to Convergence of Solutions to SDE with

Jumps Now we are going to use the Krylov estimate to discuss the convergence of solutions to stochastic differential equations with jumps, which have some weaker conditions on the coefficients in some respect. Suppose that x r , n = 0,1,2,. . . satisfy the following d-dimensional SDE's, respectively,

where wt is a r-dimensional BM, and @k(ds,dz) is a 1-dimensional Poisson martingale with the compensator ~ ( d z ) das t before, so bn, cn E Rd, and on E ~ ~Here8we write ~ x.! = s t , b0 = b. Theorem 166 Assume that

134

4. Some Useful Tools in Stochastic Differential Equations

+ JZ

+

1' 1 b ~ ( s , x ) 1lan(s,x)12 ~ where ko 2 0 is a constant,

5

I C ~ ( S , X , Z ) ~ ~ T ( ~ ko, Z )

5 F(s)p(lx - yl2L where 0 5 F ( s ) satisfies that W 2 0 F(s)ds < co, and p(u) > 0, on u > 0, is strictly increasing, continuous, and concave such that .fn, du/ p(u) = co;

2

lim E[Ix; - xtl ] = 0.

(4.28)

71-00

Proof. By Ito's formula - x,) . (bn(s,x:) - b(s,x,))ds E[lxr - xt12]- E[lxg - sol2]= 2E +E Ilan(s,2:) - 4% xs)112ds +E Jz Icn(s,x:, Z ) - C ( S , x,, z)12~ ( d z ) d= s In(i,t ) . Obviously, by the Krylov type estimate (Theorem 165) I n ( l ,t ) 5 2E - x 8 ) . (bn(s,x:) - b(s,x:))ds +2E J:(x: - x,) . (b(s,x:) - b(s,x,))ds

Jot(%:

Jot Jot

Hence E 1x2" - xt12 5 E

c:~~

15:

- x,12 ds

+6

F(s)p(E1%:

2

- x,[ )ds

4.4 Krylov Estimation

135

Thus -applying Fatou's lemma, one finds that 2 limn,,E Ix; - xt12 I J : Z,,,E Ixy - x,l ds +6 ~(s)p(lim,,,~ Ixy - x,12)ds. Therefore, Gn,,E IX; - x ~ 2I = 0. w The condition on 3O, and the condition that an can change with n, are weaker than those in Theorem 145. However, the condition on lo for bn and cn, and the condition that all coefficients do not depend on w , are stronger than those in Theorem 145. By using the Tanaka type formula and the Krylov type estimate we can also obtain another theorem on the convergence of solutions.

sot

lim E[IxT - xtl] = 0.

n-w

Proof. By the Tanaka type formula (Theorem 142) E Ix; - xtl - E xI: - xol =

136

4. Some Useful Tools in Stochastic Differential Equations

+E

Jisgn(x: - x,) . (bn(s,x;) - b(s,x,))ds

+a cL :

-(a:#-X~~)(X~~-X~~)

E Sgt I(x:+xa) '(ffik(~,x:,W)- ffik(~,xs,w))(ffjk(~,~:,~) -g j k ( ~ , X s , ~ ) ) d ~ + E J ' jZIIx:- - X S c~(s,x:-,z,w) - c(s,x,-,Z,W)[ -sgn(x:- - xS-) . (c"(s,xF-, Z , W ) - C ( S , x,-, z, w))]Nk(ds, dz) = In(i,t ) . Obviously, I n ( l ,t ) I F ( ~ ) P (1Es: - xsl)ds + k~ 1lbn(t,x)- b(t,~ ) l l L . ( ~ o , q ~ R,d ) In(%t ) KT F(s)P(E12: - xsl)ds, and In(%t ) I 2 1: .fz G(s,z)p(Elx: - x,l)a(dz)ds + 2 k ~IIJ. Icn(t,x , z ) - ~ ( Xt , ,z)1"(dz) Lq(p,TlxR d ) . Now the proof can be completed just as in the last part of the proof of the previous theorem. For the convergence theorem on solutions to 1-dimensional SDE with jumps by the Krylov estimate with other conditions we can easily obtain some other theorems. Suppose that the x;, n = 0,1,2,- . . satisfy the following 1-dimensional SDE's, respectively,

1%:-

+

x,-1

c:=~ <

Ji

11

where wt is a r-dimensional B M , and G,+(ds,dz) is a 1-dimensional Poisson martingale with the compensator .rr(dz)dtas before,so that bn,cn E R', and an E [email protected] we write x! = xt, bo = b. Theorem 168 Assume that l o 1bn(x)l2 lun(x)l2 jZ I C ~ ( X , Z ) I ~ T ( ~I Z k0(1 ) 1x12), where ko is a constant, P conditions 2'-50 in Theorem 166 are automatically reduced to the case d = 1, and the Lq([O,T ] x is replaced by the L1(R)-norm. Then we still have that Vt 2 0

+

+

+

norm

2

lim E [ ~ X-;xtl ] = 0.

n-im

Theorem 169 Assume that 1" lbn(4l2 lff(x)l" Jz lcn(x,4124 d . 4 L k0(1+ 1x12), where ko is a constant, P conditions P-50 in Theorem 167 are automatically reduced to the case d = 1, and the Lq([O,T ] x is replaced by the L1(R)-norm. Then we still have that Vt 2 0

+

nor norm

lim E[IxF - xtl] = 0.

n--too

4.4 Krylov Estimation

137

The above two theorems can almost be proved in exactly the same way as in Theorem 166 and Theorem 167. The only difference in the proofs is that the Krylov estimate used in the proof here should refer to Corollary 163. However, we also notice that the condition in lohere is weaker than the boundedness condition.

Stochastic Differential Equations with Non-Lipschitzian ~oefficiknts

In many cases we need to minimize some target functional subject to a controlled dynamical system; for example, to minimize the energy expended by the controlled system during a period of time, like, minimizing E lx~l~ dt, where u(.) is a control, xy is the solution of the system corresponding to the applied control u(.). We will find that the minimal value of the target functional will be obtained when we can apply some extreme solution of the dynamic system. For this example the idea is that at each time when the trajectory of the state process leaves the point 0, we should immediately use a feedback control to fully pull back the trajectory directed towards 0, because if the state xy is closer to 0, then the energy 1xyl2 expended is also closer to zero and so it is smaller, even though it cannot be 0. Such an extreme feedback control is called a Bang-Bang control. Obviously, such a feedback control is not Lipschitz continuous, and so it also makes the caefficients of the system non-Lipschitzian, for example, when the system is linear with respect to the control u(.) : the system coefficient is A(t)xt B(t)ut. However, we need the state of the system, that is, the solution, to exist for such a control, so the system can be controlled . Therefore, discussing solutions for stochastic differential equations (SDEs) with jumps and with non-Lipschitzian coefficients, is necessary and useful from the practical point of view. The interesting thing is also that in the ordinary differential equation (ODE) case, if its coefficients are only continuous then a solution, even when it exists, is not necessary unique. However, in the SDE case we can have a unique solution even when the coefficients are not continuous. This means that a stochastic perturbation can some-

+

140

5. Stochastic Differential Equations with Non-Lipschitzian Coefficients

times improve the nice properties of the solution. Furthermore, in a later chapter, we will see that the stochastic integral term is very important in the financial market. Actually, its coefficient corresponds to a part of a portfolio of investment of the stocks by an investor in the financial market. In the optimal consumption problem the SDEs with non-Lipschitzian coefficients also need to be considered. In this chapter we will use the smoothness method and the Skorohod weak convergence technique to discuss the existence and uniqueness of strong solutions and weak solutions for SDE with jumps and with nonLipschitzian coefficients.

5.1 Strong Solutions. Continuous Coefficients with p- Conditions In this section we will use the smothness method to obtain the existence and uniqueness of a strong solution for a SDE with continuous coefficients, which satisfy some so-called p-condition. Consider the following SDE with jumps:

a d-dimensional BM, fik(ds, dz) is the Poisson martingale where { w ~ -) , is, ~ measure generated by a Poisson point process k(.) such that Gk(ds, dz) = Nk(ds,dz) - ~ ( d z ) d t where , Nk(ds,dz) is the counting measure with the compensator ~ ( d z ) dgenerated t by k(.), T(.) is a a-finite measure on some measurable space (2,B Z ) , and b E Rd, a E RdQDd, c E Rd. In (5.1) if c = 0, we get a continuous SDE. Furthermore, if a = 0, then (5.15) will be reduced to a continuous ODE for each fixed w. So we will call the case a = 0 a degenerate case, no matter if c = 0 or not. We will use the smoothness technique to show the results.

Theorem 170 Assume that 10 ~ = ~ ( ~ , x , w ) : [ o , c o ) x R ~ x o - + R ~ , a = a ( t , x , w ) : [ O , c o ) x RdxO+RdQDd, c = ~-( tx,, z, W): [O, co) x Rd x Z x fl -+ Rd, are - adapted and measurable processes such that P - as. IW, x,w)l ~ l ( t ) ( l 14, b(t,x,w>l + .fz b(t, X, z7w)12~ ( d z ) ~ i ( t ) ( l +1x1~1,

zpNk

p

+

<

5.1 Strong Solutions. Continuous Coefficients with p- Conditions

141

where

is the a-field generated by w and @k up to time t, that is, 3 ~ ~= " "a(w,, Kk((0,s ] ,U),VU E Bz,s 5 t ) , and c l ( t ) is non-negative and non-random such that for each T < co cl(t)dt < oo; 2' b(t,x,w ) and a ( t ,x , w ) are continuous i n x ; and limh+0 JZ Ic(t, x + h , z , W ) - ~ ( xt ,,Z,w)12x ( d z ) = 0 ; 3' for each N = 1,2, . . . , and each T < oo, 2 ( ( 2 1 - ~ 2 1 (b(t, , x1,w) - b(t, x2, w ) ) 2 + I & x1,w) - 4 t ,x2,w)12 JZ Ic(t, x1,z,w) - ~ ( x2,z, t , w)I x ( d 4 < .TN(t>pTN(Ix1- x212), as !xi[< N , i = 1,2, t E [O,T];where cg(t)dt < oo; and &(u) 2 0,as u 2 0,is non-random, strictly increasing, continuous and concave such that So+dulpTN(4 = 00. Then for any given constant $0 E Rd (5.1) has a pathwise unique strong solution.

+

First, let us give an example of the existence of a solution to an SDE in the case that V T < co, cl(t)dt < co, c l ( t ) is unbounded and, moreover, bl is also unbounded and non-Lipschitzian continuous in x.

fl

Example 171 Let b(t, x ) = -It#OIz#Ot-Q1x XI-@ , where a1 < 1,O < p < 1, and suppose that a and c satisfy 1" and 2" i n

Theorem 170, and satisfy the condition 3" i n Theorem 170 with b = 0. Then (5.1) has a pathwise unique strong solution. Obviously, c ( t ) = I,#os-"*, is unbounded in t , and b is also unbounded i n t and x , and is non-Lipschitz continuous in x . Proof. Notice that V x ,2' E Rd ( x - X I , -x 1x1-@ + x' l x l p = - 1x12-@- 1X'l2-@

)

+ l x l p ( x , x l )+ ~xl-@ ( x , x l ) < - 1x12-P -P+1

IX'~~--@

- I X ~ 5 0. ) + 1x'1-@+~1x1 + 1x1 lx I = (1x1 - lxll)(lx I Hence Theorem 170 applies. Before we prove Theorem 170, let us first establish a lemma. I

Lemma 172 Under assumptions 1" and 2" i n Theorem 170 there exist bn, an and cn, n = 1,2, . - . , satisfying the following conditions: [ X I ) , as n 2 NO; 1) Ibn(t,x,w)l 5 lan@,x , w)12 JZ lcn(t,2, z,w)12 ~ ( d . 4< f h ( t ) ( l +1x12), where No > 0 is a constant; 2) a s x , x l E Rd Ibn(t, x , w ) - bn(t,X I , w)l < kncl(t) 1% - 2'1 , [an(t,x , W ) - un(t,x', w)12 JZ Icn(t, x , z , W ) - cn(t,x', z , w)12x ( d z ) < knci(t) 15 - x112, where kn 2 0 is a constant only depending on n,

+

+

+

142

5. Stochastic Differential Equations with Non-Lipschitzian Coefficients

Proof. Let us smooth out b only with respect to x to get bn, i.e. define bn(t, x, w) = JRd b(t, x - n - ' ~ ,w) J(T)&, where for all u E R~ cd exp(-(1 - lu12)-I), for lul < 1, Jd(4 = 0, otherwise, and the constant cd satisfies JRd Jd(u)du = 1. Then Ibn(t,x,w)l I JRd Ib(t,x-n-l- x, w) J(T)& I cl(t) JRd(l 1x - n-lZI) J(B)& I cl(t)(l 1x1 n-I JRd IzI J(Z)&) = c l ( t ) ( l + 1x1 n-'ko) (1 n-lko)cl (t)(l 1x1) I 2cl (t)(l lxl), as n > ko. So bn satisfies 1). On the other hand, (bn(t,x, w) - bn(t, xl,w)l = Ind JRd b(t,T, w) J(n(x - T)& -nd JRd b(t,Z, w) J(n(xl - Z)&l I nd JRd Ib(t,T, w)[1 J(n(x - Z) -J(n(xf -Z)l& 5 ndcl(t) lx -xll - JRd ~ t ( l +IZl)grad[J(n(x - Z 9(x1- x))]dO& I Icnc1(t) lx - xlI. So 2) is established for bn. Now by Heine-Borel's finite covering theorem for any N > 0 and any given E > 0 one can find a 6 > 0, 6 may depend on t and w such that as < 6, Ib(t, x - n-l, w) - b(t, x, w)l < 7,V 1x1 N; because b is continuous in x. Hence, as n suPlX(ghr IW,x1 w) - b(t,x, w)l I I JRd SUPI~I
{

1

+

r +

+ +

+

+

+

+

<

> 3,

ISRd

< Jz JRd Ic(t, x - n-lZ, z, w)l 2 J(z)&T(~z) I cl(t) JRd(l + Ix - n - ' T l 2 ) ~ ( ~ ) &I 2cl(t)(l+ 2 1x12 +2nP1 JRd 1312J(T)&) I 2cl(t)(l+ 2 [xi2+ 2n-l&) < 4 q ( t ) ( l + lx12), as n > 2&. So 1) is proved for en. On the other hand, JZ Icn(t,x, z, W)- cn(t,x', z,w)12~ ( d z ) nd JRd ~c(t,Z,z, w)12x(dz)l ~ ( n ( x T) - J(n(xl - T)I2& < ndcl(t) 1% - x112JRd c ( 1 + lZ12) Igrad[J(n(x - Z 8(x1- x))]12do& kncl(t) lx - X I [ . So for cn 2) is also established. Finally, by a similar proof as in bn one easily derives that 3) is also true for cn. w

<

2

+

5.1 Strong Solutions. Continuous Coefficients with p- Conditions

143

Now let us prove Theorem 170. Proof. For bn, an and cn obtained from the above lemma by Theorem 117, there exists a pathwise unique strong solution ( x y ) satisfying the following SDE

+ItS,

cn(s,XY-, z, w ) ~ k ( d dz). s,

By Ito's formula E [ [ X-~xy12 = 2E[J: (x? - x:) . (bm(s,x?, W ) - bn(s,x:, w))ds Icrm(s,x?, W ) - o ~ ( s , x : , w ds )~~ JZ Icm(s,x y , z, W ) - cn(s,x:, z, w)12r(dz)ds] = E [ J ~ [2(x? - x:) . (b(s,x y - m - l z , w) 2 -b(s, x; - n - l ~w, ) ) /cr(s,x y - rn-lT, w) - a ( s ,x: - n-lT, w) 2 JZ x? - rn-lZ, z, w ) - c(s,x: - nP1T,z, w)l ~ ( d zJ)@ ] )G

+ Ji + Jl

I,,

+ Ic(s, < E J: J

+

I

~{cl, (s)p( I X ?

1

- x:

- (rn-l - n-I)x12)

+2Ocl(s) (m-I - n - l ) TI)~ ( ~ ) & d s . Hence as t E [0,TI,

Since by Lemma 114 for all n and VT < m

Hence, by Fatou's lemma, it is easily seen that lim E [ ~ X T x:12]

m,n+w

where pl(u) = p(u)

< $ ~0~cl(s)pl ( m,n-+w FE

lxT - x:12) ds,

+ u. Therefore, m,n-+m lim E l x r - x?12 = 0. By (5.3) one

also finds that for each T < oo

T

lim E J o IxT - x?12 dt = 0. SO there

m.n-+w

exists an ( x t ) E L ~ ( Rsuch ~ ) that for each T < m T 2 limn,, E So lxy - xtl dt = 0. On the other hand, by the above result one also has that for each t 2 0 limn-+mE Ix? - xt12 = 0. So x? --+ xt, in probability for each t , and one can choose a subsequence { n k }of { n } , denoted by { n } again, such that P - a.s. as n --+ oo, xy --+ x:, vt = Tk,k = 1,2,. . ;

144

5. Stochastic Differential Equations with Non-Lipschitzian Coefficients

{rk)zl

where c [O,T] is the totality of rational numbers in [O,T]. Hence by Fatou's lemma

Now applying Remark 397 in Appendix B to show that as t E [0,TI, when n-+co

lim

11'"

n-+w

Moreover, one also finds that as n

= 0.

-+ co,

(5.7)

5.2 The Skorohod Weak Convergence Technique

145

Hence for this 6 > 0 , 5 ~= '0. Furthermore, ~ for arbitrary given Z > 0 there exists a f i such that as n fi, J T < ~ Z. ~ Thus, for each s, as n -+ m, 2 JZ I C ( ~ ~ X ! , ~ , W ) -c(s,xs,z,w)I I ~ ~ ; ~ ~ k ~ I l ~ ~0 , lin~P.k ~ ~ ( d z ) Hence, Lebesgue's dominated convergence theorem applies, and (5.8) holds. Thus (5.6) follows. By the same token one easily shows that as n -+ m, J; bn(s, x:, w)ds -+ b(s,x,, w)ds, in P ; and t an(s,xy,w)dws -+ So a ( s , x,, w)dw,, in P. Therefore, ( x t ) is a solution of (5.1). The pathwise uniqueness follows from Lemma 115.

>

Jot

5.2 The Skorohod Weak Convergence Technique To discuss the existence of a weak solution for a SDE under some weak conditions the following Skorohod weak convergence technique is very useful and we will use it frequently in this chapter. Let us establish a lemma, which is very useful in the discussion of the existence of a weak solution to an SDE. In the rest of this Chapter let us assume that Z = R~ - {O) , and Sz &n(dz) < m.

Lemma 173 Suppose that Ibn(t,x,w)l s ( t ) ( l +lxl), as n 2 No; lan(t,x , w)12 + JZ lcn(t,x,z , w)12n ( d 4 < Cl(t)(l + lx12), where No > 0 is a constant. Assume that for each n = 1,2,. . . , xy is the solution of the following SDE: an(s,xy)dw, xy = x0 J; bn(s,xz)ds + So' JZ cn(%xy-, M d s , d z ) , where we denote q(dt,d z ) = fik(dt, d z ) the Poisson martingale measure wzth the compensator r(dz)dt such that q(dt,d z ) = p(dt, d z ) - n(dz)dt and p(dt, d z ) = Nk(dt, dz). Then the following fact holds, this fact we may call "the result of SDE from the Skorohod weak wnvergen.ete@nique": There exists a probability space (0,5, P ) (actually, 6 = 10, I ] , % = %([0,1])) and a sequence of RCLL processes ( Z ~ , i Z ~ , ~=)0,1,2, , n .. . , defined on it such that (Ey,i Z y , c ) , n = 1,2,. . have the same finite probability distributions as those of ( x y ,wt, Ct), n = 1,2, . . . , where ct = So' J;,ls,ziSx(ds,W + J;ii>l . a ( d s , d z ) , and a s n - + m , W 2 0 , -7l ijy -+ $, i n probability, as i j y = Zy, iZ;, ,n = 0,1,2, .... Write p(dtl d z ) = CS&tI(OZ,<:Ed~)(s)>? ( d t l d z ) = p(dtld z ) - n(dz)dt,

<

+

+ Jot

Jo'

ct

146

5. Stochastic Differential Equations with Non-Lipschitzian Coefficients

V n = 0,1,2, ... Then T ( d t ,dz) is a Poisson random counting measure with the compensator x(dz)dt for each n = 0,1,2, ..., and it satisfies the condition --n i t = Sof J ; z I l l zin(d'(ds, dz) + Sof J ; z I >zFn(d~, l dz),n = 01 172, ... --Moreover, iZr and 65; are BMs on the probability space (Q,5,P ) and, T ( d t ,dz) and jj"(dt,dz) are Poisson martingale measures with the same compensator x(dz)dt. Furthermore, @?) satisfies the following SDE with and p ( d t , dz) on (Q,5, P ) . ZT = xo + of bn(s,Z:)ds + of an(s,Z:)diZ: ojf jz c ~ ( s , z ) P ( ~ sdz). ,

---

65r

+

z:-,

Proof. By the properties o f bn, an, and cn, applying Lemma 114 one immediately finds that ~ u p , E ( s u p-, <1~~ 2 1 1 ~5) kT. Moreover, as r < t T

E 1s: - x:12 5 3E

1;

<

2

bn(s,x:)dsI

Now write

Ji51,151

+ 3E f' lan(s,x:)12 ds

Ji

it = zfik(d9,dz) + hzl>lz N k ( d ~ , = d + Let us show that italso satisfies (5.9). In fact, by the martingale inequality

E ~ U P1dl2 ~ ~ 5TJ: 4 z 1 5 1 k12*(dz)ds 2 2 T & I I l &"(dz) < m. Hence lirn~,, suptlT ~ ( l d>l N ) - -0. Write = &I,l 1z1 Nk(ds,dz). Since f: is RCLL, { 0 < s < T : Af: > 11) is a finite set, so Iz*(w)l Ili.(u)l>i < mColalT A'fIcntt>1) = Hence P ( s u p t l ~I:] < m) = 1. In particular, limN,, suptlT N ) = 0. Now for arbitrary E > 0

1

p(lCt

- is1 > &) 5

~(1. f :

&151z N k ( d s , d z ) l

>

+ ~(1s: &I>l zfik(ds,dz)l > ~ / 2 = ) Ji + J2.

I t is evident that as (t- sl 5 h + 0

P( I c : I

>

5.3 Weak Solutions. Continuous Coefficients

s,ZIL1

-

147

5 2 ( 2 / ~ ) ~ &7w It - 81 0 . Notice that Nk(dt,dz) is a Poisson random measure with the compensator 0 ?r(dz)dt, as It - sl <_ h J2 5 P(Nk((S,tl, 1.4 > 1) > 0) = 1 - exp(r(dz)dr) < 1 - exp(-+I > 1)h) 0 , where a(lz1 > 1) = n(dz) = 24ZI,, &x(dz) < m . Hence satisfies (5.9), that is, l i m ~ s+u ~p t~5 P((Ctl ~ > N) = 0, and , Ctz > E ) = O-. limh10 s u ~ t l , t z < T , ( t l - t z l < h~ ( I c t Since E Iwt - W, 1' = It - SI . One also easily shows that (5.9) holds for

-

&,,,

sst,s,

(,

I

Wt

.

Hence Skorohod's theorem (Theorem 398) applies to {xT,Ct,wt) and the conclusion follows by Lemma 399.

Remark 174 By this lemma one sees that Intheresult of SDE from the Skorohod weak convergence technique" holds, and we can prove that

16

t

(bn(s,

c)- b(s, 2))d.l

(ft)

-

0, in probability B,

then ( 8 , f , ,B; {$)t,o, - b(dt,dz), {5$)tL0), tgo solution of (5.11) in the next section.

0'.

3,is a

5.3 Weak Solutions. Continuous Coefficients The technique used in proving Theorem 170 motivates us to obtain an existence theorem for weak solutions of SDE with jumps and with u, which can be degenerate. Consider the following SDE with non-random coefficients: w 2 0,

-

Theorem 175 Assume that 1" b = b(t, x) : [0, m ) x Rd Rd, u = u(t, x) : [O, m ) x Rd --+ RdBd,

5. Stochastic Differential Equations with Non-Lipschitzian Coefficients

148

c = c ( t , x , z ) : [O,m)x Rd x Z + R d , are jointly Bore1 measurable such that P - a.s. j, 1 4 4x , 4124 d z ) < ~ l ( t ) + ( l1xI2), where c l ( t ) is non-negative such that for each T < m cl (t)dt < m; lb(t,x)12+ la(t,x)l < c1(t)(l+ 1x12), where c l ( t ) has the same property as in 1"; S b(t,x ) is continuous in x and a ( t ,x ) is jointly continuous in (t,x ) ; and limh,h~,o Jz Ic(t h', z h, z ) - c(t,x , z)12~ ( d z=) 0; 4" Z = Rd - { o ) , & &&~(dz) < m. Then for any given constant xo E Rd (5.11) has a weak solution.

fl

+

+

Proof. By Lemma 172 we can smooth out b,a and c only with respect to x to get bn,an, and cn, respectively. Then we have a pathwise unique strong solution xy satisfying a SDE similar to (5.2),but here all coefficients bn, an, and cn do not directly depend on w. Now applying Lemma 173 "the resul of SDE from the Skorohod weak convergence techniquett holds. So we only need to show (5.10) in Remark 174 holds. However, since W 2 0 , 8 + 8,in probability P, as n -+ m, as in the proof of Theorem 170 one finds that (5.4) and (5.5) hold. So by Remark 397 in the Appendix we may assume that all {ZT,t E [0,T]):'o are uniformly bounded, that is, IZTl ko,Vt E [0,T ] , V n= 0,1,2,. . in all following discussion on the convergence in probability. Now for an arbitrary given a > 0

<

+P ",1 -

jZ c(s,2-,z ) ~ ( d sdz) , - J; J,

c;=,

Z-', z)ud(ds,dz)l > a/3)

c(s,

1;. Obviously, 9 F I;" ;2E S U P I , I < L ~ JZ Icn(s,x , 4 - C ( S , x,z)12 ~ ( d z ) d= s ITi, and 2 I; 5 $EP Ic(s, 5F1z,w) - c(s,$ 8 , z,w)l I1z~l
<

fi fi < <

IF 5

( 5 ) 2 E&<1zlt6 ~ : lc(s,$,t)12*(dz)ds

+B(I L.', Ilqlrkoc(s,e-, z)Gn(ds,dz) - J; J;,12aI l ~ f l < k o c ( ~ z)B(ds,dz)l ,2-, > f ) = I,b + I:'~Notice that as 6 J, 0,

5.3 Weak Solutions. Continuous Coefficients

J,T

149

G,

E ~ O < l i l Clc(s, 6J z)12n(dz)ds < w and {O < lzl < 61 14. So one can take a small enough 6 > 0 such that 1: < 213. Observe that

(e)E 2

5 2--1

- G = o c(%,

J ~

1sup8sTl~:l
giT *im ,z)I( iT

2m' ,+I T

2"'-1

IC(SY z:-, z)

wl(s)12~(dz)ds iT

+ P ~ j = o ~ s u p ~ , , l ~ ~ l
OT

I)

+

- J &'

J I , l- , a c ( ~ , Z "*im i T,z)$)(ds,dr)l> f ) = I g where 0 < < . . - < 2m < ... < T is adivision on [O,T].Since by I g = 0, one can choose a large enough m such conditions l o - 3O lim,, that 1% < 216. On the other hand, by the conclusion 1) of Lemma 400 in Appendix C for these given m,6 there exists a fi such that as n E, < 216. So, we have proved that limn,, IF = 0, and eventually we obtain that limn+m x:=l I: = 0. That is, the third limit in (5.10) holds. The proofs of the remaining results are similar and even simpler. Thus Z: is a weak solution. For that the coefficient b can be greater than linear growth we can establish the following thoerem.

<.&

>

Theorem 176 Assume that conditions l o ,3O and 4' in Theorem 183 hold and assume that 50 IW,% ) I 5 cl ~ ( 1x1 1 gkw, lu(t, x)12 I + 1 x 1~ ~ Ig L kw, where c l ( t ) 2 0 has the same property as that in the condition loof Theorem 183, and gk(x) is such that gk(x)= 1 ln(1 ln(1 . . . ln(1 l ~ l ~ ~ ' ) ) ) ,

+

nr=,

+P + + + k-times

(no is some natural number). Then for any given constant xo

E Rd

(5.11) has a weak solution on t

> 0.

Proof. For each n = 1,2,. . . introduce a real smooth function W n ( x ) x, E R d , such that 0 5 W n ( x )5 1 and W n ( x )= 1, as 1x1 5 n; W n ( x )= 0, as 1x1 n 1. Write bn(t,x) = b(t,x)Wn(x),an(t,x) = u ( t , x ) W n ( x ) . Then by Theorem 175 for eact n there exists a weak solution xr with a BM wF and a Poisson martingale measure Zkn(dt,dz), which has the same compensator n(dz)dt,defined on some probability space (On,r, ,Pn) such that Pn - a.s. W > 0 , xr = xo+Jl bn(s,zr)ds+Jot an(s,x ~ ) d w ~ +JZJC~( S , X F - , z ) f i p (ds,dz). Construct a space On = D x W ox D, where D and W o are the totality of all RCLL real functions and all real continuous functions f ( t )with f (0) = 0, defined on [0,XI), respectively. Map (xn(.,w),wn(.,w ) , cn(., w ) ) into the O, where

> +

{z)

150

5. Stochastic Differential Equations with Non-Lipschitzian Coefficients

Ji hzI1l

rNkn(ds,dz), C = hZI<1zNkn (ds,dz) + and Nkn((O,t],U ) = Co<,st I O ~ A C E for U , t 2 0, U E B ( Z ) , Nkn (dt,dz) = Nkn(dt,dz) - ~ ( d z ) d t . From this map we get a probability law PPo on Rn. Now let R = X?!~R",5 = x & r , P = x?=',,P:, where = B D x !BW x !BD,and define Vw = ( w l , . - . ,wn,...) E 0, Zi(w) = xi(wl), Gi(w) = w:(w'),&w) = [Z(wl),

............

1

-7%

Z ~ ( W=)x?(wn), G?(w) = w;(wn), Ct ( w ) = <:(wn),

............

Then one finds that for each n , Z? satisfies the following SDE: P - a.s. an(s,Z:)dG," Z? = xo + bn(s,Z:)ds + J ~ J ~ C ( S , E : - , Z ) % ~ / - ( ~ S , ~ Z'& ) ,2 0, where Nktn((0,t ] ,U ) = Co<slt IoZnr:Eu, for t 2 0, U E B(z), ~ k t (dt, n dz) = N k ~(nd t , dz) - ~ ( d z ) d t ,

Ji

an%

+ Ji

4,1,1

Ji

Ct = zNktn(ds,dz) + z N k m ( d ~a). , -n Let us show that the following facts hold for q? = Ct ,G?,and Z? :

lim sup sup P(lqy1 > N ) = 0, N-+w n t5T limsup sup P(lqyl - qZ1, > E ) = 0, h4.0

n tl,tz<_T,ltl-tzlsh

1

In fact, as the proof of Lemma 173 one easily sees that condition: >N) limN-rn S U P , sup,,, = l i m ~ -S U+P ,~s u p t g ~Pz(lC:I > N ) = l i r n ~ -S + U P~t g - pio > N ) = 0,

~(ltq 1

Cy

(5.12) satisfies the

5.3 Weak Solutions. Continuous Coefficients

lim

sup P(sup IZPl n=l,2.-. t
> N ) = 0.

151

(5.13)

Furthermore, by Ito's formula to gm+l(Zy - Zp), s E (r,t], one finds that P - a.s. t gm+l(ZT - Z;) = 1 gk+l(ZF - Zp)bn(s, ZT)ds 2 J: g;+l(zF - ~ ) ' Y ( SZ;)~WI , f [I.&+l(% - $)g(s, Z;)II ds J,"lzgk+l (Zy- - Z;)C(S, Zz-, z ) N p (ds, dz) JrtJz[9m+l(ZT- - ZP C(S, ZT-, z)) - gm+l(ZT- - Z?) (ds, dz). -gk+l (ZT- - Z;)C(S, zy-, ~)]Nktn Thus one similarly has that VE > 0, 0 5 t - r 5 h, t 5 T,Vn, > E ) 5 supn P(suptST IZTl > N) P(lZ7 +&E(gm+l(ZT - zp) - l)ISUPtlT~zE"~<~

+ ST

+ + +

id,"

+

+

5 kb[J," cl(t)dt (t - r)]/(l - g,+l(~)) -,0, when h -+ 0. Therefore, (5.12) holds for @ = Zr. Hence Skorohod's theorem (Theorem 398) applies. By this and by Lemma 399 "the result of SDE from the Skorohod weak convergence technique" holds. (See Lemma 173 and Remark 174). For simplicity we still use the same notation as in Lemma 399 to denote the result. So we only need to show that (5.10) in Remark 174 holds. For this let us first show that lim~,, P(suptST I3$1 > N ) = 0. In fact, > N, P ( ~ ~ P , < T > N, = P(su~k=l,2,... 5 P(supk=l,2,... - ZpL > P ( S U P ~,... = ~IZ;,,"~ > $) 5 P ( S U P ~ = ~ ,lqk ~ , . .. Zp," > p(supk=l,2,... IZp," > $) = I1 where is the set of all rational numbers in [O,T]. However, for arbitrary given Z > 0 and for each r k we may take an nk large enough such

Iqk

{rk}El

I s)+ 1 i)+

Iek1

1

I

+ 12)

152

5. Stochastic Differential Equations with Non-Lipschitzian Coefficients

z. -

that ~ ( 1 %- 2I;: > I1 ) < &, k = 1,2. ... . Hence II 5 EEL& = On the other hand, by (5.13) there exists a 5 such that as N 2 N, I? < i. Therefore, limN,, P(sup,<, 1 3 1> N) = 0 holds true. Now let us prove the second limit in (5.10). Notice that by Remark 397 in the Appendix and from (5.13) and the result just proved we may assume that IETl 5 ko,Vt E [O,T],Vn= 0,1,2,... . Now for any given E > 0 ~ ( I J ,on(+ " 5:)diZ; - J,' O(S,z:)diZ:I > E)

I ($)2

EG I a n ( ~ , z ?-) o(s, %)I2 I

+P@ I = I;

~

+IT.

~2 ) d +~

s;~ I

~ z : ~ ~ ~ ~ I ~ ~ < ~ ~ ~ s ~~

~~2)dw:l (~ > ~ ~5 ) , ~

~

~

~

>

Notice that for any E > 0 as n ICg, F(lon(s,z:) - u ( s , 2 ) l 2 ~ ~ Z : ~ < k o ~ ~ Z> ~~
- ~ ( S$)I2 Y I

~

Z

~

21 > 7)

~

> &) ~

<

~

~

~

Z

~

~

<

~

~

+p(lfl(s, 2:) - o(s, 2 ) I ZI ~ Z ~ ~ < ~ J ~ Z ~ ~ <>~E). ~ I ~ Z : - ~ ~ < ~ Since o(s, x) is continuous in x, so it is uniformly continuous in 1x1 5 ko. Hence one can choose a small enough > 0 (which can depend on s) such that as lxl - xNI 7 and lxtl, lxtt) 5 b, Iu(s,xt) - a(s,xtt)I < E. This means that we can have the result that as n -+ m, F(Icn(s,z:) - o ( s l ~ ) lI 2~ E : ~ < ~ ~> I8)~5~F(Iz? ~ < ~-~ > 7) 0. So, by Lebesgue's dominated convergence theorem as n -+ oo, I? -+ 0. Now notice that a(t, x) is jointly continuous, so if we write um(t,x) as its smooth functions, then 2 limm,, Iom(t,x) - u(t, x)] = 0, Vt, x; and Iurn(t'x) - um(s,Y)I 5 km[It - sI IX - YIIi where k, 0 is a constant depending only on rn. Observe that

<

%I

>

I2 < 2 ( z ) E~ '~(,J+'J = IZ

+

I ~ ( s2, ) - om(s, %)I2 I ~ E ! ~ < ~ ~ ~ s

~ ~ Z ~ ~ < kz:)d~? ~ f f m( sJ;~

+ Igln.

2)d~:l > 5)

So for any given Z > 0 by Lebesgue's dominated convergence theorem we can choose a large enough m such that I,nl < Z/2. Then applying Lemma 401 in Appendix C we can have limn,, = 0. Thus we obtain that limn,, I2 = 0, and the second limit in (5.10) is established. The proof for the remaining results are similar.

(

~

5.4 Existence of Strong Solutions and Applications to ODE

153

5.4 Existence of Strong Solutions and Applications to ODE Applying the above results and using the Yamada-Watanabe type theorem (Theorem 137) we immediately obtain the following theorems on the existence of a pathwise unique strong solution to SDE (5.15).

Theorem 177 Under the assumption of Theorem 176 if, i n addition, the following condition for the pathwise uniqueness holds: (P WU1) for each N = 1,2,. - . , and each T < oo, 2 ( ( 2 1 - 2 2 1 , (W,$ 1 ) - b(t, ~ 2 ) ) + 144 x i ) - ~ ( $dl2 t , + JZ Ic(t, X I , 2 ) - ~ ( 2t 2,, 2)12~ ( d z ) 2 ~TN(t)PTN(lx1-x21 1, as lxil N , i = 1,2, t E [O,T];where $(t) > 0 such that cg(t)dt < oo; and p g ( u ) > 0,as u > 0 , is strictly increasing, continuous and concave such that So+ ~ U / P T N ( U ) = 00; then (5.11) has a pathwise unique strong solution.

<

<

f

Furthermore, by using Theorem 176 and Theorem 177 we immediately obtain a result on the ODE.

Theorem 178 1) If b(t,x) is jointly Bore1 measurable and continuous i n x such that I~W I c 1I( t ) ( l+ 1x1 944), where cl ( t ) and gk ( 2 ) have the same properties as i n Theorem 176, then the ODE

np=,

xt

= xo

+

I"

b(s,xs)ds,t

> 0,

(5.14)

has a solution. (It is not necessary unique). 2) In addition, i f b ( t , x ) is such the ( P W U l ) condition only for b i n Theorem 177, then ODE (5.14) has a unique solution. Proof. 1) is obtained by Theorem 176 by setting a follows from Theorem 177 by letting a = 0 , c = 0. H

=

0 , c = 0; and 2)

Example 179 Let b(t,z) = It#oIxZot-"lx (XI-' , where a1 < 1,O < P < 1. Then ODE (5.14) has a solution. Let b ( t , x ) = - 1 ~ ~ ~ 1 ~ ~ ~ t - ~ ~ ~ 1 x 1 - ~ where a1 < 1,O < 0 < 1. Then ODE (5.14) has a unique solution.

5.5 Weak Solutions. Measurable Coefficient Case In this section we will discusss the existence of weak solutions of SDEs with measurable coefficients. In this case we have to assume that the SDEs

154

5. Stochastic Differential Equations with Non-Lipschitzian Coefficients

are non-degenerate. In this case one sees that the Krylov type estimate is a very powerful tool for establishing the existence theorem for weak solutions of SDE with jumps under very weak conditions. Consider a d-dimensional SDE with jumps as follows: W 'q 0, Xt

= 50

+

1 t

+

b ( ~X ,S ) ~ S

1 t

a ( %X

+

S ) ~ W ~

C(S,

x8--,z)fik(ds,d z ) ,

(5.15) where wt and k k ( d t , d z ) have the same meaning as in (5.1) and all coefficients b, a and c are non-random. Theorem 180 Assume that l o Z = Rd - ( 0 1 , and n ( d z ) = d z l [zld+' ; Kt, x)l+ la(t,x)l+ J, Ic(t,x, 4124 d z ) ko, where ko > 0 is a constant, b, a and c are Bore1 measurable functions; 30 there exists a constant So > 0 such that for all p E Rd, (44 P ) 2 1p12S o Then there exists a weak solution for (5.15).

<

Before we prove this theorem let us establish the following lemma. Lemma 181 Under assumption of Theorem 180 there exist smooth functions: V n = 1,2, . . bn(t,x ) , ~ " (x 4) , (t,4 E [O, w ) x R d , which are the smoothness functions of b(t,x ) , a ( t ,x ) , on [0,co) x R d , respectively; and there exist smooth functions: V n = 1,2, - . . P ( t , x , z ) , ( t , x , z ) E [O,co) x Rd x { E , 6 IzI I E ; ' ) = A,, which are the smoothness functions of c(t,x , z ) , on A,, where E, J, 0 , such that set cn(t,x,z ) = P ( t , x , z)I{EnslZI<ER1}, then 1) lbn(t,x)l 6 ko; b n ( t , x ) l 6 ko,

<

Jz Icn(t,x , z)I n ( d z ) 2k0, V n = 1,2,. . . ; 2) V p II R d ,V ( t ,x ) E [0,co) x Rd, (an(& P ) 2 lp12 60; 3) V n = 1,2,... ,W 2 O,Vx,x' E R d , ' It - t'l], Ibn(t,x) - bn(t',x')l lan(t,x) - un(t',x')I < k n ~ [ l-x 11 2 12 J { E n j l z l s E RIcn(s,x,z) l~ - cn(s,xf,z)l 4 d z ) 5 4 k o [ l x - x I +It - t'121, 4) for each T < co, V N = 1 , 2 , - . . ,Vq 'q 1, as n + co, llbn - bllq,[O,T]X~~ + (Ian - a l l q , [ ~ , ~ J X ~ ~ and

+

+

+

l l ~ { r ~ d I z 1 6 I~cn ; ~-) r(dz)(lq,io,nlxsn< $il where S, = { x E Rd : 1x1 5 n ) .

Proof. Let us smooth out a to get an, i.e. let for all u E Rd+'

5.5 Weak Solutions. Measurable Coefficient Case

155

{

cd+l exp(-(1 - lu12)-I), for 1211 < 1, 0 , otherwise, such that the constant cd+l satisfies the condition JRd+l J d + ~ ( u ) d u= 1. and write for ( t ,x ) E [0,m) x Rd,n = 1,2, .. . x - n - ' ~J)( z ,~ ) d z & un(t,X ) = JRd+l ~ (- t = JRd JRl ( ~ ( tn-l?, x - n - l Z ) J(Z, Z)&&, where we define a ( t ,x ) = 0 , for t < 0. on are usually called the smoothness functions of a on [0,m ) x R d . For these a n , n = 1,2, . . . let us show that they satisfy 1) - 4). In fact, V p E Rd Ian(t,x)l 5 J R d f l Ia(t - n-lX,x - n - l ~ )J(T,Z)dZ& ] 5 kO, ( u n p ,p) = JRd+l ( a ( t - n-'Z, x - n-%)p, p ) J(z,Z)&& 2 lPl2 SO, Moreover, because the a n ( t ,x ) are the smoothness functions o f a ( t ,a ) , so an ( t ,x ) 4 a ( t,x ) , ax.( t ,x ) Hence for any q > 0 , for each T < m and N = 1 , 2 , . . . , as n --+ m, Jd+l(U) =

.[713[281

Furthermore, one easily sees that for each n = 1,2, . . . as x , x' E Rd ( a n ( t ,x ) - a n ( t ,x')1 knko I X - 2'1. Thus a n , n = 1 , 2 , . . . , satisfy 1) - 4). In the same way one can construct bn(t,x ) , n = 1,2, . . . , such that they satisfy I ) , 3) and 4). However, for the smothness o f c, t o meet our purpose we need more discussion. First we take a sequence E, 1 0. Set cEn = c I { ~ (~z ) ,~ and crn(t,2 , ~=) I { & n < l r l < E ~. P(t, } ( ~ )2,4, where P(t,x,z ) = JRl x R d X zcEn(t - m-I?, x - m - l Z , z - m P 1 Z J(T, ) Z , ~)d?&&, where we define c ( t ,x , z ) = 0 , as t < 0. That is, P(t,x , z ) is the smoothness ~ x { E , 5 IzI 5 E L ' ) . T h e n function o f c ( t , x , z ) on An = [O, m ) R~

<

Jz lcm12~ ( d z5) J{En51zl<E;l} JRd

JRl

~

J{En51tl<E;1} .

]c(t-rn-'f,z-m-'~,t.-m-~'i)I~

4

*Jzd

d+'

J(Z,Z, Z)dZ&cE

5 k~ & E n < I H I < EJRd n l }JRI 2J(zj5?Z)d'fi(tE< 2 k ~ > where w; have used the fact that for E , 5 121 < E ; ~ , and E , 5 lzl 5 E;' 5 ( 1 + p p + 1 5 2, i f we take m > &ek2, and EO > 0 is a constant such that ( 1 + < 2. Thus we have proved that Jz 1crnl2~ ( d z 5) 2k0, as m > '€E0- ' ." NOW for

q$g

each

E,

by assumption

2

ko 2 J{En
dt

2 J{c,,<~tl<E;l} 1'

d+ldz.

< m J ~ O , T l x S , x*{c-El t 1 2 E ; ~lc12 ) dtdxdz < m.

Thus by the property o f the smoothness f ~ n c t i o n s [ ~ ]as~ m [ ~-+~ co, ]~[~~]

~

~

~

5. Stochastic Differential Equations with Non-Lipschitzian Coefficients

156

~ [ O , T l x S N x { r n ~ l z lIc~ r-; cm12 l}

dtdxdz

- q o , ~ ~ x ~ N x { l rc ,- ~~ l ~~ dz t~d ~x d~ t 0. +

Hence as m 4 co, m 2 dtdxdz

& , T ] x ~ N x { E ~ ~ ~ 1'z- ~ < . ;I ' lZldfl ) m 2 dtdxdz -< J [ o , T ] ~ s ~ ~ { c . ~ l Icz ~-s ~ ;I ' F) j O l for each fixed n , N and T < co. This deduces that as m 4 co, ~{,n51zI~e;l} Ic-cm12 0, a.e-(t,x) X SN. (Otherwise, a contradiction is easily derived). Now applying Lebesgue's dominated convergence theorem one finds that for any q 2 1 as m + co, 1 - I 1.1 d z Lq([O,T]X S N )+ 0, for each n,N and T. Rom this for each n one easily choose a mn such that m, > $&i2 and

+

I

C-Cmn12

dz

-I

2

I

l l ~ { ~ ~ s ~1 ~ ~ s ~ p ; l } L ~ ( [ O , ~ ] X S<, ) For simplicity write cn for cmn. Since cn is smooth in t and x , so dz { n-< - < I n 1 -c n2 ,I 2 p

5 J { , s l z l s E ; }n

<

h.

+ It1 - t2121+

+

k;[lxl - x212 It1 - t212]. Finally, as n + co,

l Sz

lcn

- c 1 2 r ( d z ) 1 1 q , ( 0 , T ] X S-<~

lc12

"(dz)llq,[o,TlxSN

+ llJ{cns~z~sE;l} Icn - '12 ~ ( d z ) ~ ~ q , [ o4 , T0. lx~N The Proof is complete. Now let us prove Theorem 180 Proof. For bn, on and cn, which are constructed in the previous lemma, by Theorem 117 for each n = 1,2,. . . there exists a unique strong solution (x:) of the following SDE: t 1 0

Now applying Lemma 173 "the result of SDE from the Skorohod weak convergence technique" holds. So we only needs to show (5.10) in Remark 174 holds. As in the proof of Theorem 175 we may assume that IZyl 5 ko,Vt E [0,T ] , V n= 0 , 1 , 2 , . . . . Notice that for any given E > 0

P(lli(bn(s,Z:) - b(s,$))dsl

> E)

5 ;E'[J~ l(bn - bno)(s,Z:)l I\ql5kods + P ( l ~ i ( b " o ( sz:) , - bnO(s,$)dsl > ~ / 3 )

+$E'G I(bnO- b)(~,Z!31Ilsy5kodsl

5.5 Weak Solutions. Measurable Coefficient Case

157

= I y o + pn + I?. Obviously, by 4) in Lemma 181 and by the Krylov type estimate (Theorem 165) there exists a fi such that as n fi,no N, I:1no+ IT 2 . llbn - bnOI~~+I,[O,T]XS~, < 2r/4.

>

>

-

>

Now for each no fi, by (5.9) and by 3) in Lemma 181 as n -+ m , W E [O,TI, I p n-+ 0. Thus the first limit in (5.10) is proved. Now notice that for each no = 1,2, ... P(IJ," JZ cn(s,Zy-, z)P(ds,dz) - J,"Jzc(s,$-,z)iO(ds,dz)l

5 ($)'

> 8)

2

E J J,~ I(cn - c n O ) ( s , Z , z ) lI

~ ~ ~ ~ ~ k ~ ~ ( d z ) d s

+P(I J,"J, cnO(s,ZT-, z)Gn(ds,dz) -

s,"Sz

2-,z)2(ds, dd~rsup,,[o,TI ~ S ~ ~ 2

+ S U>~I) ~ ~ [ ~ , ~ ~ ~ ~

+ ( $ ) ' ~ ~ t ~ ~ l ( ~ ~ " - ~ ) ( sIli~l~kon(dz)ds , ~ : , t ) l

+

+

- I;"'0 I,""*" I,"". For an arbitrary given F > 0, as above (by using the Krylov estimate) one E n show tkat there exist a large enough fi such that for any fixed no N,asn> N

>

I y O+ q0< $E.

On the other hand, by using Lemma 400 in the Appendix one also finds that as n -+ m , Vno 13""~" 0. (5.18) -)

-

C3z = 1

pO,n 32

.

>

Notice that 'ds 0,Zy -+ 2 , in probability, as n Lebesgue's dominated convergence theorem,

1%

I<." < ~ , O E J , -

G SZ

- 2

-+

I l ~ ~ l ~ k o , l i ~+ i 50, k oasd ns 2

Now by E l c n O ( s , qi)l , r(dz)ds choose a small enough 6 > 0 such that I,":," < c/3.

m , so applying

+

m.

5 kot < m for any 5 > 0 one can

158

5. Stochastic Differential Equations with Non-Lipschitzian Coefficients

Let us show that for any 6 > 0 and no, as n

-t

m,

-

where knO,& > 0 is a constant only depending on no and 6;kno,a,ko> 0 is a constant only depending on no, 6 and Ico; and Z: is right continuous such that s j. si a xs 4 x s i . So by using Lebesgue's dominated convergence theorem, one finds that as X = maxi,o, ... , , ( s ~ + ~- si) + 0 nO,n p " , n no 13I331 33 5 ~ 6 ~ ~ 1 I~l ~ ~$ ' l s~k o6~( s-, x1~ - ,Z)

.

EJ,'

-C~=O ~ ~ Z ~ ~ < k ~ c ~ ~ ( ~ i , % ~ , ~ ) ~ ( s i , s i-t+ 0 ~](~)~~~(d~)ds Furthermore by Lemma 400 in the Appendix, for any given division, no n limn-too133; = 0. Therefore, (5.19) holds. Thus (5.18) is proved, and the third limit in (5.10) is also established. Finally, the second limit in (5.10) can be similarly proved. In fact, for arbitrary E > 0

- I;."' +

IT0.

Now the proof can be completed in the same way as that of the third limit in (5.10) by using Lemma 401. So the second limit in (5.10) is established. Thus we have proved that {5$}t, - - satisfies the following SDE on probability space

---

-

(R,5, P) for any T < m as t E [0,T]

5.5 Weak Solutions. Measurable Coefficient Case

1-20+

b(s,z ) d s

159

+ J,' o ( s ,z ) d z + fi JZc(s,Z f , z ) p ( d s ,d z ) ,

where WE) and p ( d t , dz) are a BM and a Poisson martingale with the compensator r ( d z ) d t , respectively. By using Theorem 180 and the Girsanov type thoerem we can obtain the existence of a weak solution to a BSDE with jumps under much weaker conditions.

Theorem 182 Assume that b, a and c are Bore1 measurable functions such that lo Z = R~ - ( 0 1 , and r ( d z ) = dz/ ~ z l ~ "; 2 , z)12 fl(dz) I ko, 144 x)l + J, where ko > 0 is a constant; 3" there exists a 60 > 0 such that for all p E Rd, ( d t ,X ) P , P ) 2 lp1260. 4 ( x ,b(t,x ) ) < c l ( t ) ( l f 1x1' nr=1 gk(x)), where c l ( t ) and gk(x) have the same properties as i n Theorem 176; furthermore, b(t,x ) is locally bounded for x , that is, for each r > 0 , as 1x1 < r , K t , x)l < k,, where k, > 0 is a constant only depending on r . Then there exists a weak solution for (5.15). Proof. The proof involves a combination of the results of Theorem 180 and Theorem 133. In fact, by Theorem 180 there exists a weak solution for the following SDE with jumps: V t 0 , -

>

2t

= xo

+ Jota(s,xS)dws+ ~ ~ ~ Z ~ ( ~ , x s - , z ) N k ( d ~ l d z ) .

Notice that by Skorohod theorem (Theorem 398 in the Appendix) we know that the above SDE holds in a probability space (Cl,z,P), where R = [O, 1 ] , 5= %([0,I ] ) , P =Lebesgue measure on [0,11. Since such ( 0 , g ) is a standard measurable space. So applying Theorem 133 the conclusion follows on W 2 0. rn In the above theorem we assume that a is bounded. Now we relax the coefficient a to be less than linear growth, (so, it can be unbounded). In this case we have to assume that a and c are jointly continuous.

Theorem 183 Assume that conditions l o ,3" and 4' i n the previous theorem hold, and assume that 5" Jz lc(t,x , z)12 n ( d z ) < cl ( t ) , 60 a ( t ,x ) is jointly continuous in ( t ,x ) ; and limh,h~-.+O JZIc(t + h', x + h, z ) - c ( t , x ,z)j 2 ~ ( d z=) 0 ; 7O there exists a 60 > 0 such that J a ( t x)l , > 60, and la(t,x)12< ~ l ( t ) (+l 1 x 1 ~ ) Then for any given constant xo E R~ (5.15) has a weak solution on t > 0. Example 184 If b(t,X ) = -Z lx12n1

+

nrZlg k ( x ) ,

160

5. Stochastic Differential Equations with Non-Lipschitzian Coefficients

where nl is any natural number, and g k ( x ) is defined in 9 of Theorem 176, then b(t, x) satisfies all conditions in Theorem 183. However, b ( t , x ) is very much greater than linear growth in x. Now let us prove Theorem 183. Proof. The proof can be completed by applying Theorem 175 and the Girsanov type theorem (Theorem 133). Since it is completely similar to the proof of the previous theorem. We do not repeate it. Finally, applying the above results and applying the Yamada-Watanabe type theorem (Theorem 137) we immediately obtain the following theorems on the existence of a pathwise unique strong solution to SDE (5.15).

Theorem 185 Under the assumption of Theorem 183 if, in addition, the (P W U I ) condition i n Theorem 177 holds, then (5.15) has a pathwise unique strong solution.

Part I1

Applications

How to Use the Stochastic Calculus to Solve SDE

To help the reader who wants to quickly know how to use the stochastic calculus to solve stochastic differential equations (SDE) we offer a short introductory chapter. Some instructive examples are also presented. Actually, some of the material only represent special cases of the general results in the first part of the book. However, these simpler cases can help explain the ideas more directly and clearly, and so may help the reader master the main ideas. A reader who is already familiar with Ito's formula, Girsanov's theorem and their applications may skip this chapter.

6.1 The Foundation of Applications: Ito's Formula and Girsanov's Theorem In solving ordinary differential equations (ODES) the following technique is frequently used: If we can guess the form of a solution, we will use the differentiation to check whether the guess is true solution, or to make some changes to make it true. Such idea can also be applied to the solution of SDE. However, the rule for differentiation in the Stochastic Calculus is different, and for this we need the Ito differential rule, that is, the Ito formula. Another frequently used technique is that if a tranformation can be made to simplify the ODE, then this will always be done first to make finding the solution easier. Such a technique is also applied to SDE. However, to use a trasformation in an SDE is much harder than in the case of an ODE. The

164

6. How to Use the Stochastic Calculus to Solve S D E

problem is that, after the transformation, is the new differential equation still an SDE that we can understand; that is, does the new stochastic integral still makes sense? To answer this question we need the Girsanov's transformation, or say, a Girsanov type theorem. As for the existence and uniqueness o f solutions o f an SDE we need a related theorem in SDE, and that is the theory in the first part o f the book. However, in this case the discussion o f the theory also needs the help o f Ito's formula. Let us look at the following examples. Example 186 Find the solution of the following 1-dimensional SDE dxt = axtdt bxtdwt, x0 = c, t 2 0, where a, b, c are constants, wt, t 2 0 , is a 1-dimensional BM.

+

I f b = 0 , then by the usual differential rule, one easily checks that xt = ceat satisfies the ODE dxt = axtdt, xo = c, t 2 0. However, setting xt = ceat+bwt,by Ito's differential rule (Ito's formula) this does not satisfy the above SDE, because it only satisfies the following SDE: dxt = cdeat+bwt= cdf (at bwt) = cf'(at bwt)d(at bwt) +;cf1'(at + bwt)d (bw), = ceat+bwt(adt + bdwt) + = xt(adt bdwt) jb2xtdt = axtdt bztdwt $b3xtdt. That is xt = ceat+bwtonly satisfies the SDE dxt = axtdt + bxtdwt + jb2xtdt, 5 0 = C , t 2 0. Here we have applied the following Ito's formula, which is a special case o f Thoerem 93 in the first part of the book, and we set yt = at+bwt, f ( y ) = eY t o get the above result. (Recall that by the notation before Lemma 62 ( M ) , = ( M ,M ) , is the characteristic process o f the locally square integrable martingale Mt such that ( M ) , comes from the Doob-Meyer decomposition: M2 =a local martingale + ( M ) ,. Now for a BM we have that (bw), = (bw,bw), = b2 (w,W ) = b2 ( w ) = b2t)

+

+

+

+

+

+

+

Theorem 187 (Ito's formula). Suppose that yt = Y O + A t Mt, where yo E T o , {At)tlo is a continuous finite variational ($t-adapted) process with A. = 0 , {Mt)t,O - E M2~10c~c. If f ( x ) E C 2( R ) ,then

+

Or, we can write it i n the differntial form: df ( y t ) = fl(yt)dAt + fl(yt)dMt + i f t l ( y t ) d( M I , . Recall that the differntial form is only a form, and its exact meaning is that the integral equality holds.

6.1 The Foundation of Applications: Ito's Formula and Girsanov's Theorem

165

Remark 188 To use the It0 formula one should know how to calculate the characteristic process ( M ) , of the given locally square integrable martingale Mt. For the convenience of the reader we recall the following fact t where w t , t E [O,T] is a BM, and here: If Mt = ~ ~ ~ a ( s , w ) d wE, ,[O,T], E J , (a(s,w)12ds < CO, then ( M ) , = l a ( s , ~ )ds,Vt 1 ~ E [O,T]. Now we can solve Example 186 by making a small change to the guess form as follows:

Solution 189 Set xt = ceat+bwt-?bat. Write yt = at + bwt - 4b2t, and f ( 9 ) = eY. Then by (6.1) dxt = cdf ( y t ) = cf1(yt)dyt+ & f " ( y t ) d (bw), = ceYt(adt bdwt - i b 2 d t ) ib2ceYtdt = ceYt(adt bdwt) = axtdt bxtdwt. SO xt = ~ e ~ ~ is+a solution ~ ~ ~of Example - 4 ~ 186. ~ Moreover, ~ it is the unique solution, because the coeflcient of the SDE is Lipschitz continuous and less than linear growth (Theorem 117).

+

+

+

+

Example 190 Find the solution of the following 1-dimensional SDE dxt = axtdt

+ f ( t ,xt)dt + bdwt, xo = c, t E [0,TI,

(6.2)

where b > 0 , a , c are constants, w t , t 2 0 , is a 1-dimensional BM, i n the case that 1) f ( t ,x ) is bounded and jointly measurable; 2) f ( t ,x ) is bounded and satisfies the Lipschitzian condition, that is, there exists a constant ko > 0 such that If(t,x)I 5 ko, V t E [O,T],vxE R 1 , E [O,T],&Y E R 1 . l kolx-yl, I f ( t , x ) - f ( t , ~ )5 In this example in case I ) , without the help of Girsanov's theorem we cannot even know that (6.2) will have a weak solution. Here, a "weak solution" means that the solution x t , t E [0,T I , with a BM wt, t E [O,T], on some probability space (a,5, {5t)tE[0,Tl ,P ) satisfies (6.2),but St # 5 y . So xt is zt-adapted, but is not necessary %-adapted. (See Definition 127). Intuitively we see that (6.2) equals dxt = axtdt f ( t ,xt)dt bdwt = axtdt b(b-'f ( t ,xt)dt d w t ) = axtdt bdGt, where we write dGt = dwt b-l f ( t ,xt)dt. However, the existence of a solution xt of (6.2) is not yet known. So we should solve (6.2) in an different way. First we solve the simpler SDE dxt = axtdt bdwt for a given BM wt to find a solution xt. Secondly, we let dGt = dwt - b-lf ( t ,xt)dt. If there is a theorem (we call it a transformation theorem) to guarantee that such a Gt is still 5 B M , but under some new probability measure pT, then we can arrive a t PT - a.s. dxt = axtdt f ( t , x t ) d t bdGt,x0 = c, t E [O,T].

+

+

+

+

+

+

+

+

+

166

6. How to Use the Stochastic Calculus to Solve SDE

That is, we have that x t , t E [O,T],with a B M Gt,t E [O,T],o n the probability space (Q5,{5t)tE[0,T1 , satisfies (6.2). So (6.2) has a weak solution. Fortunately, such a useful transformation theorem exists, and it is the so-called Girsanov type theorem that can be stated as follows, which is a special case o f Theorem 124 in the First Part o f this book.

FT)

Theorem 191 If on a given probability space (0,5,{St)tEIO,Tl,P ) a 1dimensional process 0: is St - adapted such that

where c(t) 2 0 is non-random such that S: c(t)dt < co,then defining as dkT = e x P [ cBydw, 10y12d s l d ~ , where wt, t E [0,TI is a BM on this probability space, PT is a new probability measure, and wi

= wt

-J(;

t

6:ds.t E [O,T],

is a new BM under the probability pT. Now let us use this theorem to solve SDE (6.2) by using this approach. Solution 192 For a given probability space ( O , g , { ~ t ) t E ~ o,,PT )l and a given BM wt, t E [0,T ] defined on it, we can solve the simpler SDE dxt = axtdt bdwt, xo = c, t E [0,TI, to get a unique 520-adapted solution P - a s . t xt = eatc + b So ea(t-s)dw,, Vt E [0,TI. (In fact, by Ito's formula one can easily checks that it satisfies the simpler SDE, and by Theorem 117 it is the unique 520-adapted solution). Now applying the above Girsanov type thoerem (Theorem 191) Gt = wt - bF1f ( s ,xs)ds, t E [0,TI, is a new BM under the new probability measure PT, where T -1 ~ P T= exp[Jo b f ( 3 , xs)dws - SoT Ib -1 f ( s ,x s ) 2 dsldp. So we have that PT - a s .

+

Jot

1

dxt

= axtdt

+ f ( t ,xt)dt + bdGt, xo = c, t E [0,TI.

(6.3)

where xt = eatc b ea(t-s)dws, So ( ~ t , G t ) ~ is~ only [ ~ , a~ weak ] solution of xt E 520 c 5t, but xt $ (6.2) in case 1). Next we discuss case 2). In this case the pathwise uniqueness of solutions of (6.3) holds. Hence one can apply the Yamada- Watanabe theorem (Theorem 137) to get the result that (xt,i&)tEIO,Tl-is actually a strong solution of SDE (6.3); that is, xt E 5f (xt is 520-adapted). So there

+

SF.

6.2 More Useful Examples

167

exists a Baire function F such that xt = F(G,,s L. t). Therefore, let ~ [ (6.2) ~ , on ~ the ~ original probZt = F(w,,s L. t ) , then (Zt,w ~ ) ~satisfies ability space (Q, 5, { ~ t ) t E I O ,P T l) with the original BM wt, t E [0,TI, and (Zt)tE[o,Tl is the pathwise unique strong solution, that is it is unique and

st E 59.

6.2 More Useful Examples In the later Chapter we will meet a Stock price SDE as follows: dP: = P,"[rtdt+ atdGt)],PJ = PJ; Vt E [0,TI, F - a.s., where for simplicity we assume that all of the processes that occur here are real-valued, and Gt,t E [0,TI, is a BM under the probability measure F. Example 193 Under assumption that rt, at are non-random and .f[lrtl + lat121dt < co the solution of the stock price SDE is: F - a.s.Vt E [O,T] t P,1 = PJ exp[$ r,ds So a,dGs - $ J: la,12 ds].

+

Proof. Write yt = So rsds + So o,dGs - $ J: la,12 ds, and f ( y ) = eY. Applying Ito's formula (Theorem 187) to P: = PJeYt we find that fj - a.s. dP: = PJ@tdyt + $ ~ J e y(at12dt t 2 = P;eYt[rtdt + atdGt - $ latl dt] $pteYt lat12dt = P,"[rtdt + atdGt]. So P> = PJeYt solves the stock price SDE. Moreover, by Theorem 117 it is the unique strong solution of the stock price SDE. In the later Chapter we will also meet a wealth process SDE as follows: dxt = rtxtdt .lrtatdGt,xo = X O , P - as., where for simplicity we assume that all of the processes that occur here_are real-valued, and Gt,t E [0,TI, is a BM under the probability measure P. t

t

+

+

Example 194 Under assumption that rt is non-random and Iatl 5 ko :J IrtJdt < co, and E lat12dt < co, the solution of the wealth process SDE is: F - a.s.W E [O,T] t t xt = exp[$ r,ds]xo So exp[Ssr,du].lr,a,dG,.

+

Proof. Applying Ito's formula (Theorem 187) to the above xt we find that F-a.s. dxt = zod(exp[$ r,ds]) d exp[Jstr,dul.lrsasd61 t t = rtxo exp[$ r,ds]dt atotdGt rtdt So exp[S, r,dul.lrsasdG = rt [ e x p [ r,ds]xo ~i S' e x p [ ~r,du].lr,~,dG,]dt ,~ + .lrtatdGt = rtxtdt ntatdGt, where we have used the following result: ,~ dJ : e x p [ ~r,du].lr,a,dG,] st = atatdGt rtdt e x p [ ~rrdu].lrsa,d~,. Indeed, let yt = J: expi- S{ r,du].lr,a,dG,]. Then by Ito's formula

+

+ +

+

+

+

168

6. How to Use the Stochastic Calculus to Solve SDE

d Jot exp[Jst rudu].lr,a,dG,] = dej: ruduyt = rteSOt ?uduytdt+ el: rud"dyt e- So"rudu.lrsgsdGs+ & ?udue- Sot r~du.lrtatdGt = rtdteS: ?udu = el: rudu.lrsa,dG, .lrtatdii&. So the result is true. Thus t t xt = exp[Ji r,ds]xo Soexp[J, rudu].lr,a,dGs solves the wealth process SDE. Moreover, by Theorem 117 it is the unique strong solution of the wealth process SDE. rn

Jot

+

+

Linear and Non-linear Filtering

In many cases the signal process which we want to examine cannot be observed directly, and we can only examine a different observable process which is related to the signal process. This poses the question of how we estimate the true signal process by using the information obtained from the observable process? Such an estimate of the present signal obtained by using the information from an observable process up to the present time is called a filter. In this chapter we will discuss the fitering equation for both the non-linear case and the linear case. When the linear case is without jumps, we will derive and solve the continuous linear filtering equation the famous Kalman-Bucy equation, and we will also consider the non-linear case when we will derive the Zakai equation and discuss its solutions.

7.1 Solutions of SDE with Functional Coefficients and Girsanov Theorems In discussing filtering problems we need to consider a SDE with functional coefficients. Let us begin by introducing some notation. Let D = D([O,co);Rd) = The totality of RCLL maps from [O,co) to R ~ , with the Skorohod metric (see Lemma 388 in the Appendix), DT = D([o,T];Rd), C = C([O,00);R ~ =) the totality of continuous maps from [0,m) to Rd, CT = C([O,TI);Rd), B(D)= The topological a-field on D,

170

7. Linear and Non-linear Filtering

(that is, the a-field is generated by the totality of open sets), %(D) = B(Dt). Consider the following SDE with jumps and functional coefficients in d-dimensional space : Vt 0,

>

<

where w: = (w:, . . . ,w,dl),0 t, is a dl-dimensional &-adapted standard Brownian motion (BM), wF is the transpose of wt;k* = (kl,. . .,kdz) is a d2-dimensional st-adapted stationary Poisson point process with independent components, and I?ki (ds, dz) is the Poisson martingale measure generated by ki satisfying

Nki(ds, dz) = Nki(ds, dz) - n(dz)ds,

i = 1 , . . . ,d2,

where n(.) is. a a-finite measure on a measurable space (2,%(Z)), and Nki(ds,dz) is the Poisson counting measure generated by ki,where for simplicity we always assume that Nki({t), U)Nk,({t) ,U) = 0, as i # j , for all U E !B(Z), n(U) < m. For simplicity we also write b(s,x(.),w) = b(s,xr,r s,w) = b(s, {xr),<, ,w), , r S, W) = ~ ( s{, ~ r )- i ,w), <~ ~ ( sx(.), , W) = ( ~ ( 3xri and C(S-7 x(.), z, W) = C(S,xr, r < S, z, W)= c(s, { ~ r ) ,~~1 w). < ~ The definitions of a solution and the pathwise uniqueness of solutions to (7.1) are completely the same as those of (3.1). Furthermore, by using almost completely the same proof as that in Lemma 195 we have an a priori estimate of the solution to such an SDE.

< <

Lemma 195 Assume that xt is a solution of (7.l), and assume that 1" E lxo12 < co, and

where Ilxllt = supslt Ixsl, and 0

for any 0 < T < co.

< c(t) is non-random, such that

7.1 Solutions of SDE with Functional Coefficients and Girsanov Theorems

171

where kT 2 0 is a constant depending only on CT and E lxo12 For the uniqueness of the solution we have the following similar lemma.

Lemma 196 Under the assumption in Lemma 195 i f for each N = 1,2,. . . there exist non-random functions c N ( t ) and p N ( u ) such that as Ilxl lit and 11x211, 6 N , Vx1(.),.2(.) E D

where c N ( t ) 2 0 satisfies the condition c N ( t ) d t < co for each T < oo, and p(u) 2 0 , as u 2 0 , is increasing, continuous and concave such that So+ d u / p ( u ) = oo, then the solution of (7.1) is pathwise unique. Since the conditions above are a little different from Lemma 115, we give a proof here. Proof. Asume that { ~ f ) i ~= ~1 , 2~ are , two solutions of (7.1) with the same BM

and

and Poisson martingale measure fik(dt, d z ) . Let

+

7 N = inf { t 2 0 : llxlllt llx211t > N ) . Then by Ito's formula and by the martingale inequality one sees that

Hence by Lemma 116 for any T < CO, P - a.s. Z,,,, Letting N -+ co one finds that P - a.s.

=

0,Vt E [O,T].

By the RCLL property of { x ~ ) , , ~ i= , 1 , 2 the conclusion now follows. The proof of the following theorem is also completely the same as that of Theorem 117. So we will not repeat it.

172

7. Linear and Non-linear Filtering

Theorem 197 Assume that lo b(t,x(.),w) : [O,co) x D x R H

R~

is a St- adapted and measurable process such that P - a s .

where c ( t ) is non-negative and non-random such that for each T

< CQ

where c(t) satisfies the same conditions as in 1'; 30 $0 E 80,E 1 x 0 1< ~ co. Then (7.1) has a pathwise unique St- adapted solution. In the case that

-

-

b(t,x , w ) and a ( t ,x , w ) are 8ylNk- adapted, and c(t,s,z , w ) is $ytNk- predictable, then the solution is also SytNk- adapted, i.e. it is a strong solution. Similar to Theorem 120 we also have the following theorem. Theorem 198 If the condition 30 i n -Theorem 197 is weakened to be 30' there is a given process E such that E l[,l2 - < CQ; and assume that b(t, - x(.),w) and a ( t ,x(.),w) are adapted, and - predictable, then there exists a pathwise unique c(t,x(.), z , w ) is strong solution x t , t 2 0 , satisfying the following SDE:

c, zylNk

z?lNk-

7.1 Solutions of SDE with Functional Coefficients and Girsanov Theorems

173

Later, when deriving a non-linear filtering equation, we will need a martingale representation theorem for a SDE with functional coefficients. However, such a representation theorem depends on the Girsanov type theorems discussed below. So we discuss them here. Suppose that xt solves the following SDE with jumps and with functional coefficients on ( Q S ,(St),P):

where we make the following assumption that (H)2 b(s,x ) : [O, co) x D -,Rd is jointly measurable and St-adapted. Thus we have

Theorem 199 Assume that b'x = x(.) E D, l0 lb(t,x)12 b ( t , x ) l 2+ JZ Ic(t,x, z)124 d 4 I ~ g ( t ) ( l llxll;), where c o ( t ) 2 0 is non-random such that co(t)dt < co, for each T and write I141t = SUP& Ix(s)I , 1?" a - ' ( t , x ) exists and is bounded; that is, la-'(t, x)l I ko, 8 E 1x01~ . . < oo. Let dPt = zt(a-'b)dP, where

+

zt(-.-'b)

+

= exp[-

I"

< co,

t

a-'b(s, x ) . dw, -

lo-'b(s, x)12 ds],

(7.4)

and xt is the solution of (7.3). Then 1) E z t ( a P 1 b )= 1, W 2 0 , 2) there exists a probability measure P defined on such that ( d P / d ~ ) = zt(-a-lb) 3) let Gt = wt a-lb(s, x ) d s , then Gt is a P -BM, 4) fik(dt, d z ) is still a P -Poisson martingale measure with the same compensator .rr(dz)dt, 5) xt also solves the following SDE on (Cl,S, (St),P ) : V t 0

(0,s)

+ Ji

>

The proof of this theorem is almost completely the same as that of Theorem 132. So we still omit it. Now we discuss some other conditions on b(t,x ) to get the Girsanov type theorem.

7. Linear and Non-linear Filtering

174

Theorem 200 If we replace the condition 1' i n Theorem 199 by 1"' la(t,x)12 JZ Ic(t,x, z)12 ~ ( d z5) c ( t ) ( l + I I X ( ~ ~ ) , VEX D, where c ( t ) 1 0 is non-random such that J: c(t)dt < oo, for each T < oo; and Ib(t, x(.))12dt < k~ < c q V x E D, where l c > ~ 0 is a constant only depending on T ; then the conclusion of Theorem 199 still holds.

+

fl

Remark 201 If we introduce Novilcov's theorem as follow^:[^^ ~ e ~ 4~ I ' ( ~ "<4 l ~=+ ~ ~E Z T f i ) = 1, where 2 ZT = exp[ T(s,w ) dw, l;(s, W ) ds], then the conclusion of Theorem 200 becomes trivial. However, to prove Novilcov's theorem needs more pages of detailed analysis and also some properties of special functions, so we will give a direct proof of Theorem 200.

,

Ji

fi

6)

1

Proof. We only need to show that Ezt(-a-lb) = 1, W 2 0. For any given T < ca let bN = bIlbllN.Then by the previous theorem Ezt(-a-'bN)(x) = 1, b't 2 0 , where t 2 ~t(-a-lb~)= ( ~e)x p [ - ~ i a - ' b ~ ( s , s.dw, ) - $So ~ o - ' ~ N ( s , x ) ~d s ] , and xt is the solution of (7.3). However, by Lebesgue's dominated convergence theorem as N -+ m, T E J o l a - l b N ( s , z ) - a-'b(s,z)12 ds -+ 0. Hence as N oo, -+

2

~ - ' b ~ ( xs), - dw, - f ' o - ' b ( s , r ) . dw,l -+ 0 . Esupt<, Take a subsequence denoted by {N) again, such that as N -+ oo, I ~ ; o - l b ~ ( s , x.dwS ) - J,t a -1 ~ ( s , x dwsl ) . -+0 , and l 0, ~ a - ' b ~ ( s , x ) l ~-d s 1 a - ' b ( s , x ) 1 ~ d s -+ uniformly w.r.t. t E [O,T],P - a.s. Notice that sup,E ~ w ( - a - ' b N ) - E ~ - 20-~b~(s,x)~dw.-f~ I U - ~ ~ N ( S , X ) ~ ~ .~ eS / ~ l ~ - 1 b ~ ( s . ~ ) 1 2 d d 20-'brv(s,~).dwa-4Jot 4 1 a - ' b ~ ( s ~ )dls < e k o l c ~ . < elcokTEe-

If,'

l2

-

-

Hence { Z ~ ( - O - ' ~ ~ ) ) : is = uniformly ~ integrable. Therefore as N -+ oo, EzT(-a-' b ~ --+ ) E z ~ ( - a - ' b ) . So ~ z ~ ( - a - l b=) 1. Later, when deriving the non-linear filtering equation we also need some martingale representation theorems for the SDE with functional coefficients

7.1 Solutionsof SDE with Functional Coefficients and Girsanov Theorems

175

as follows:

where for simplicity assume that fit is a 1-dimensio~al centralized Poisson process with constant density A such that ENt = ENt +At. (In this case we also say that Et has a compensator At). This SDE can be seen as a special case of (7.1). In fact, in (7.1) if we assume that ~ ( 2=)A, c(s-,x(.)) = c(s-,x(.), z ) does not depend on z, then from (7.1) we get (7.6). For such a SDE we can relax condition 10' in Theorem 200 to get a better Girsanov type theorem as follows:

Theorem 202 If we relax the condition 1'' in Theorem 200 by lo'' la(t,x)12 Ic(t,2,z)12 L: c ( t ) ( l I I X I I ; ) , V X E D, T where c(t) 2 0 is non-random such that So c(t)dt < m, for each T < m; and 1b(t,x(.))l2d t < m) = 1,Vx E D; and assume that the condition P in Theorem 197 holds; then the conclusion of Theorem 199 still holds.

+

+

p(C

Remark 203 This theorem cannot be derived from Novikov's theorem. (See remark 201). Proof. By assumptions there exists a pathwise unique strong solution yt satisfying ~t = so Ji u(s,y(.))dwS+ J : c(s-,y(.))dfiS,t1 0. For each T < m and z E C let r , ( z ) = T ~ i n f { t>O:Ji~(o-'b)(s,z(.))l~dt> n } , Then by Theorem 198 there exists a pathwise unique strong solution x? of the following SDE: Vt E [0,T ]

+

X? =

xll ~ ~ ~ , , , , , d sf

t

- l S < T-n ( Z ) ) a ( s ,

xn(*))dwS

+ So"(1- Is T,(x) dx? = a ( t ,xn(.))dwt c(t-, xn(.))dfit. However, if rn(xn) > T,(x), then by the definition of T,(x") we must have b(s,xy) = 0, as s E [rn(x),rn(xn)]. Hence in any case one has that dx? = b(t,xn(.))It<,,,(,n)dt a(t,xn(.))dwt ~ ( t - , x n ( . ) ) d N t . Now since VT < m 2 I(a-lb)(t,xn(.))It
<

+

+

SoT

+

176

7. Linear and Non-linear Filtering

- j?~n(%")u-'b(s,xn).dws-8~OTATn(s"'~u-'b(~,xn)~ads =e TAT,,(+")-1 = exp[a b(s,xn) . ( s ,xn)dxy

+;

So

xn)12

x") A X: rTn(+") d s + C0
~

-

~

~

(

~

)

-

-

= Jrn(ZECT:Tn(Z)=T} Z~~T,(=)(-u-'b)(z)d~xn(z).

So &(z) = Z T A T n ( , ) ( - ~ - ' b ) (on ~ ) ,r E { r E CT : ~ ~ (= 1 T )). hrthermore, VI' E %(CT), px (r)= limn,, p, (I'n { z E CT : r n ( z ) = T ) ) = limn,, pxn(I'n { z E CT : r n ( z ) = T ) ) = limn+, Jrn{ZeCT:Tn( Z ) = T ) ) ( Z ~ ~ T n ( r ) (b-)u( z- )l) - l ~ y ( d z ) = limn+, S ~ ~ { Z E C T : T , , ((~Z) T =(T-}~) - ' ~ ) ( Z ) ) -(dz) 'P~ = Jr(z~(-a-lb)(z))-l~y(d~). This shows that &(y) = ( ~ ~ ( - c r - l b ) ( ~ ) where )-', .t4

ZT(-a-'b)(y)

+

= exp[-

0-'b(S, Y ) ~ a - l ( Sy)dy,+i , 10-'b(s, y)12 ds Y ) a 9.3 - A ~ ~ A T n ' x n ' ( u - l c ) ( sYl ) ~ ' ] '

~o<s5TAr,,(xn) Notice that EZT(-a-lb)(x) = J, ZT(-a-lb)(x)dP = JcT Z T ( - ~ - ~ b ) ( r ) p , ( d z )

= JcT % d p ,( 4 ~ + ( d 4 = JCT p y ( d 4 = 1. Hence, let for Vt 2 0 (dF/dp)lzt= zt(-aP1b)(x),and s)ds, Gt = wt + J; aW1b(s, then Gt,t 2 0 is a P -BM, and fit is a I-centralized Poisson process still with the same compensator At. Therefore, SDE xt = xo S," b(s,x(.))ds + a ( s ,x(.))dws + J ~ ~ C ( S - , Z ( . ) ) ~ $ ~2, ~0; P - a.s.

+

Si

7.2 Martingale Representation Theorems (Functional Coefficient Case)

177

can also be rewritten as

xt = 20

+ J: a ( s ,x(.))dGSJot c(s-, z(.))dfi,, t > 0; P - a s .

7.2 Martingale Representation Theorems (Functional Coefficient Case) Consider SDE (7.6). We have the

Theorem 204 Assume that lo lla(t,x)1l2+ lc(t,x)12I co(t)(l+ 11x11;), v x = 4.)E Dl where co(t) > 0 is non-random such that :J ~ ( t ) d
where k~ > 0 is a constant depending on T only; P a-'exists such that ko, [la-' 9 Ila(t,x>- a(t,y)112+ Ic(t,x) - ~ ( y)12 t , 5 ko(llx - yll3, for all x , y E D , t 1 0, Now assume that xt satisfies (7.6). If I , is a RCLL Rdl-valued square integrable 5; -martingale, where 5: = a(x,, s I t ) , then there exzsts an f (t,w): [O, oo) x R -+ R ~ ~,@ g(t,w) ' ~ .: [0,oo) x R -+ R d l , which is 5;-adapted, 5;-predictable, respectively, satisfying for any 0 5

11 <

Proof. Let t zZt = z$,(-a-'b) = exp(o-'b(s, x ) . dw, - So la-'b(s, x ) ds), d& = z&dP, where xt is the solution of (7.6) given in the assumption. Then by Theorem 202 above there exists a probability measure jS on ( R , 5 ) such that

Sot

and

+

4

112

a-'b(s, x)ds, 0 5 t Gt = wt is a BM under the probability measure P, moreover, fit is still a centralized Poisson process under P with the same compensator At. Hence xt also

7. Linear and Non-linear Filtering

178

satisfies that

F - a.s.Vt > 0

Since by Theorem 197 the solution of (7.8) is a (pathwise unique) strong solution. Heace for all t 0. 5: c On the other hand, by (7.8) Nt = C o < s < tc s - ( ~ ) - ~ xs -

>

a

is $:-adapted, Hence 3: = @I".

and so is %t

==

Nt-At. However, iijt is obviously g?lN-adapted.

-

Let us show that tt . (z&)-' is a 5FN- adapted locally square integrable s5t martingale under probability F. Indeed, for all A E

el',

JA

tt . (zXt)-'dF

=

L.a

I ~ t.tE(z j

( dJ) dP = JA ( I . d P

es . (z&-'dF. This shows that {& (zXt)-l, 51"''}tL0 is a F= JA

.

martingale. On the other

.

hand,

I

let . ( z ~ ~ ) - l lsuptST = E letl = suptsT E IqtT111') I EItTI < m , since Jt is a (591N, P)-martingale, and we have applied the Jensen SUP~STEF

inequality, and = 1, for all 0 5 t 5 T. E ( z k 13: >') = E [ E ( z b 13;) Moreover, let U N = inf { t E [O,T] : ((z&-'1 IJtI > N) . Then 0 N f m , as N f m, since (z$')-l is a continuous process, and Et is a RCLL (right continuous with left hmit) process. Now since tt is a square integrable (3:lN, P)-martingale, we also have

]":$I

+

1

1

1 2

S U P ~ S T I~< tP/ \ U N ' (zctk7N)2 =N ~ ~ P ~ s T E 5 N s u p t < ~ EI E

Ic~A,, I

This shows that

tt . (z&-'

is locally

tt . (z&)-'

2 '

(zctr\UN)-ll

~
F-

square integrable. Therefore

-

is a locally square integrable (8FtN,F)-martingale. Now by T < m there the martingale representation theorem for each given 0 I exists an fT(t,w) : [O, T ] X 0 -+ RdlBd,

which is @"-adapted, satisfying there exists a g&w) : [O,T] x 0 + Rdl,

F(J: 11 fT(t, w)l12dt <

m ) = 1, and

7.2 Martingale Representation Theorems (Functional Coefficient Case)

179

where - (denote A* = the transpose of A) ~ T ( s , w= ) ~&~T(s,w -)E, . (o-lb(s,x))*, ~ T ( sW) , = Z$SST(S,w). Moreover, it is evident that TT(t,W)is 5;-adapted,

2 P(SoT IlfT(t,w)II dt < m, =

and gT(t,W)is 8:-predictable, P ( J ~1gT(t,w)l2dt < 00) = 1. However, by assumption, Et is R~I-valued and square integrable. Hence applying the representation (7.9) yields

where

J, l ~ f T ( s , w ) l+/ ~~ f T ( s , w ) ~ ~>l dNs ) fir inf{@) . Now as N + m , T N --+ T. Hence by Fatou's lemma TN

= inf {t E [(),TI :

,

T N = T,

Recall that, by the definition of a stochastic integral, if fi(t, w), g"t, z, w), i = 1,2, are such that

7. Linear and Non-linear Filtering

180

and f l , g1 are not distinguished from f 2, g2 in the above integrals, and so we denoted by fl f 2 , gl 92. Now if there exist f$,g$, i = 1,2, for each i, with both satisfying (7.9) and (7.10), then (7.11) holds. Hence f l I f 2 1 2 2 T - T, QT -ST' Now define f l ( t , ~ ) as , t E [0,111 f ( t ' W ) = { f n ( t , w ) , a s t ~ ( n , n + l ] , n = 1 ,,...., 2 91(t,w), as t E [O, 11, g ( t ' W ) = { b ( t , w ) , a s t ~ ( n , n + l ] , n = 1 , ,.... 2 Clearly, f (t, w) is 3:-adapted, and g(t, w) is 8:-predictable such that P - a.s for all t 2 0 f (s,w) . dws g(s, w)dGs. Et = EO Moreover, for any 0 5 T < ca llf (t14112dt + lg(t,w)12dtl < 00.

--

+

+

E[S,T

SOT

.

7.3 Non-linear Filtering Equation After the above preparation now we can discuss the filtering problems. Let us consider the following partially observed Stochastic Differential System:

where yt is a d-dimensional RCLL (right continuous with left limit) square integrable martingale with yo = 0, and we call ht a signal process, Et an observable process, and they are all d- dimensional processes. In this section we are going to derive the SDE satisfied by the non= o(Es; s 5 t). Naturally, in linear filtering processes E(ht I g ) , where (7.12) we assume that h, H, y, <,A and C are d-dimensional vectors, B is a d xd matrix, and on a given probability space (R,3, (St), P ) wt is a d dimensional standard St-Brownian Motion process, and for simplicity we assume that Nt is a 1 dimensional St-centralized Poisson process with the inte_nsity Xdt such that Nt = Nt - Xdt, where X > 0 is a constant, Nt is a Poisson process with ENt = Xdt. We always make the following assumption (HI) At(w) : [0, ca) x R -+ Rd is jointly measurable and -adapted, Bt(x) = B(t, x(r), r 5 t) : [O, ca) x D -+ RdBd is %([O,ca)) x %(~)/!B(R~@~)-measurable, and for each t 2 0 Bt(x) is (D)/%(Rd)-measurable;

g

zt

7.3 Non-linear Filtering Equation

181

C t ( x ) = C ( t , x ( r ) , r 5 t ) : [O,m)x D + R ~ , has the same properties as those of B t ( x ) ; (Hence. if we set Ct- ( E ) = C ( t ,( ( r ) ,r < t ) , where E(r) is a gt-adapted RCLL process, then Ct- ( E ) is 3t -predictable. Similar notation is also made for B t - ( 0 ) . We have the following Theorem 205 Assume that ( h t ,E,) is a 3t-adapted solution of (7.12) and for all T > 0 p(S,T IAs(w)(ds < 03) = 11 [Bs(F)12ds < m) = 11 l ~ s ( F )Xds l ~ < m) = 1, and B-I exists such that

~ ( g

ws,T IB-I1

< ,to.

Then 1) Mt = J: B ~ ( E ) [-~E(A, E , I8$)ds], is a locally square integrable martingale on the probability space ( R , 3 , ( ~ $ 1P, ) . (Recall that = a([,, s I.t ) ) . The 2nd equation of (7.12) can be rewritten on the probability space ( R ,5, P ) as

~

(zf),

2) Furthermore, i f

-

M~ = where

+ +S,t

(t)exists, then

B;.(J)C,-(E)~~~~,

+

a:,

Zt = Bl1(F)( A s - E ( A , 132))d.s wt = the continuous martingale part of Mt, is a BM on ( R ,5, P ) ; and & = jS,tc;:(o~.-(od~;, -d . where M t zs the pure discontinuous martingale part of and fit is still a centralized Poisson random process with intensity function At on (%5, (39,P ) such that Nt = Nt - Xdt, that is, Nt is-a Poisson random process such that E N t = Xdt. Usually, we call Tiit and Nt the innovation processes for wt and fit. The 2nd equation of (7.12) can be rewritten on the probability space ( R , s , ( ~ $ 1P, ) as

(sf),

at,

First we notice that E IBS(F) - B S - ( a 2 ds = E S T IB(s,F(r),r s ) - B ( s , t ( r ) , r < s)12ds = 0 , since J t is a RCLL function only with countable discontinuous points in t for a.s. w. Secondly, we also notice that the coefficient E ( A s 15:) is always

s,T

<

182

7. Linear and Non-linear Filtering

a functional of J,, s 5 t. That is why we need to consider the SDE with functional coefficients in the previous sections. Now let us prove Theorem 205. From now on let us write r t ( A ) = E(Atl$), etc. Proof. 1): First, we make a stronger assumption that

Let us show that

d& = r t ( A ) d t

+ dmt ,

where (mt,&)t?o E M21loc.In fact, comparing this equation with the 2nd equation of (7.12), we have

7.3 Non-linear Filtering Equation

183

~[m; m i l g ] = 0. So, for any t <2 u 5 T < m E[m, - mt = 0. Hence mt, t E [O,T] is a ~- martingale. Since T < m can be chosen arbitrarily, we have that (mt,$$)t20 is a martingale. The result E lmtl2 < m can be derived by working the stronger assumption (7.15). Now let

1261

I

<

Since B,' (J) ( to, we have ( z t ,g ) t 2 0 E M 2 . Therefore d(, = =t(A)dt+ Bt-(()dat, and 1) is proved under the stronger assumption (7.15). However, in the general case for all T > 0 P(S,T IAs(w)l ds < m ) = 1, l ~ s ( ~ ) l< ~m d )s = 1, p ( f l l c S ( t ) 1 2 ~ s< m ) = 1, 1) is also true by taking the limit through a series of stopping times a, T m . 2): Without losing any generality we may assume that the stronger assumption (7.15) holds. Notice that

p(g

at, = G- ((1 N. Hence Nt = C,<,$t Cs- (0-l A t s is 26-adapted, and so is fit. However, we already know that (Gt,8t)tLois a centralized Poisson process with intensity At, and so is (fit,$)t20 with the same intensity. In fact, Vs < t, E[(Nt - At1821 = E[E[(Nt - ~tl8SlI821= E[Ns - As)l821 = Ns -AS, where we have used the fact that Et it 5: c and Et = Nt - At. So Nt = fit +At is still a &-~oisson process with the same density A, (or say the same intensity At). Now write a = ~,-_l(t)[dt,- E(A, 182) ds - cs-(wfiSl = S,"Bl1(0(As - E(As 182))ds wt. Obviously, Vt is 8:- adapted by the first equality. By Ito's formula V? E

g,

s,"

Rd,

zt,

+

-

e'(*lztpzs) -1 = i (?, SSte'("."ze)

[By1(() (A,. - E(A, 18:))dr

+ dw,], )

e'(i>zT-z8) d,.

2-

where M t and fit are both square integrable %$-martingales. Hence for any G E 82,s <_ t, E I ~ ~ ~ satisfies ( ~ ~the~following ~ - ~ordinary ~ ) differential eauation: Solving it gives

7. Linear and Non-linear Filtering

184

P ( ~ ) e - I ~ I-

2

(t-s)P, that is, Et is a 5:-BM. By the definitions of Et and hit, and also from (7.16) one immediately sees that M t = ~t ~;'(t)~,(t)dfi,. Now (7.14) is easily derived. rn In the following we are going to derive the non-linear filtering equation for the partially observed SDE system (7.12). More precisely, we are going to show the following theorem. (Recall that 11x11: = - Ix,~). E I G e i ( X % - ~= ~)

+ Ji

Theorem 206 . Assume that 10 I B ~ ( X ) I ~~ ( x)12 t , I &)(I vx = x(.) E D , where co(t) 2 0 is non-random such that q,(t)dt < co, for each T < co; and C - ' ( t , x ) exist and we still denote the norm of matrix a ( t , x ) by lo(t,x J ; and for any T < m, E J o l ~ t ( 4 dt1 < ~ w; 5' B-l exists such that 5 ko; 9 K t , x ) - ~ ( y)12 t , + IC(t, 2 ) y)12 I ko Ilx - Y I I ;, for all x , y E D , t 2 0; 4' E lho12 < 00, E lto12 < w , dt < co, s u p t E [ o , ~lht12 ~ E < co, V T < co. E J: Then the optimal filter ~ t ( h = ) ~ [ h t l g ]or, say, the best estimate i n L 2 ( R , 5 ,( $ ) t 2 0 r P ) , for estimating ht based on the information 3: (that is, measurable), will satisfy the following non-linear filtering equation it is (P -as.)

+

+ 11~113,

p-'l

w,

&

where A* means the transpose of A , Et is the innovation process defined i n the previous Theorem 205, and (Y?, ~ l ) t' ' ' (Y:I ~ d ) t Dgdt = d ( y Cw , ), =d ... ... * , ( Y : I W ~ ) ~' ' ' (y:,wd)t

[

[

]

]

D:dt = d [Y?, fi] . . . [Y:, fi] * , (yc,w ) , is the (predictable) quadratic variational 5t - adapted process corresponding to the martingale yc and w , and [ y f , f i ] = CsstAyt A fis. Moreover, the adapted Radon-Nikodym derivatives Dg and Df exist. Let us first show that E [ ~ ~ Jiss the : ] best estimateof ht in L 2 ( R ,5, P ) . Suppose that hlo,tl E is any given estimate.

(g)ttO,

7.3 Non-linear Filtering Equation

So, if x p t 1 = E(htl$).

then the estimate error E 2

minimum E lht12 - E IE(htlg)l

l&o,tl

185

- ht12 attains its

. Therefore E[htl$] is the best estimate

of ht in L2(R,5,(39,,,, P). However, to derive the non-linear filtering equation (7.19) we need to establish a series of lemmas. First we see that by (7.12)

Let us discuss (7.20) term by term.

Proof. By Jensen's inequality it is obviously square integrable. Moreover,

VA ~ 3 $ ,<st , E[IAE(Y~I$)I = E [ E ( I A Y ~ I = ~ )E(IAYt) ] = E(IAY,) = E(IAE(Y,I~$)), since yt is a St-martingale. This shows that as s E ( E ( Y ~ I s E ) I ~=$ E(Y,I%). ) Hence { E ( ~ ~ I ~ is~ a) $} ~ martingale. , ~ rn

< t,

186

7. Linear and Non-linear Filtering

Proof. Since by Theorem 205 (7.14) holds. Hence, applying Lemma 207, 208, and Theorem 204, the conclusion is derived. rn Now we are in a position to prove Theorem 206. We see that to derive the non-linear filtering equation (7.19) we only need to show that f(.,w) = K(DC)+ [ W h A * ) - ITs(h)n,(A*)lB,l(t)*, g(s, w) = E ~ , - ( D ~ ) . Proof. Let zt = X:(t)fi, ~ : ( t ) d E , , where X;(E) is any given adapted random process, and A:([) is d x d matrix - valued bounded any given d- dimensional bounded predictable random process. Then {zt, ; I $ }E~ M~ 2 . Moreover, ~

Jot

+ Jot g$2-

On the other hand, since gt = nt(h) - no(h) -

Jot n, (H)ds

E 52. Hence

7.3 Non-linear Filtering Equation

187

Now one finds that by (7.22)

However, Zt is a st-adapted square integrable martingale with respect to the probability measure P. Hence EZt . ho = E[ho. E(Zt [8o>] = E[ho. ZO]= O., and E J i Zs . Hsds = E J ~ [ E (18s) Z ~ . Hs]ds= E(Zt . Hsds). Therefore E[Zt . ht - J; Zs . Hsds] = E[Zt - (ht - ho - S: Hsds)] = E[Zt . yt] = E ( 2 , y c ) , E[Zd,ydIt. Hence

+

where C

- d(~c,w = )dt

Dt -

dt

[

(~;,wl)t" '

... (Y% w1)t . . . ..

(~;,wd)t

,

(Y& ~

d ) ,

@ = d [ [y?,zIt [ ~ ; , f i ]]*~l d t , are some versions of the Radon- Nikodym derivatives, which exist and are st-adapted by Lemma 210 and 211 below. Now computing the second item on the right side of (7.23),one obtains Eht . ~f ( t ) ~ , ' ( t ) ( ~ s- rs(A))ds = E J ; ~ B , ~ ( J ) ( A S - r s ( A ) ). hsds + E J ~~ f ( t ) ~ , ~ ( rt s)( (A )& ) ). (ht - hs)ds. Hence

Ji

7. Linear and Non-linear Filtering

188

Hence by (7.23), (7.24) and (7.25) we get

E F ~. zt = E J; A:([) . D ~ ( s ) ~ s +E J: t r . ~(0 ; [DC(s)* + B y 1(6)(r,(Ah*)- ( K , ( A ) r ,@)*))Ids =E A: ( E ) . X , ( D ~ ) ~ S +E J: tr.X:(C)[rS(DC*) + B;l(E)(rs(Ah*) - (.rr,(A)r,(h)*))lds Comparing (7.21) and the above equation, one finds that Is(()* =p.(Dc*) + B;'(o(r.(Ah*) - ( r S ( A ) n s ( h ) * ) ) , d r )= X r s - ( ~ d ) .

J,t

The conclusion now follows. I The following two lemmas show the existence of Radon-Nikodym derivatives.

Lemma 210 Consider the 1- dimensional random processes. If { x t ,zt)t>o - E is a BM, then there exists a {at(w), E L;'"(R) such that (P - as:) t ( x , w )= ~ So a,(w)ds, Vt 2 0. Naturally, we write d ( : y ) k = at(w).

M 2 , and {wt,

Proof. Fix an arbitrary T < m. For V S x A E %([O,T ] )x TT let Q ( S x A ) = E V A Js d ( x ,w),l Then dQ << dt x dP. (That is, dQ is absolutely continuous with respect to dt x dP, which is equivalent to requiring that VS x A E B([O,T])x ZT J: ~ ~ ~x d ~P ) =( 0 d+t J: I ~ = Q~( S x A) ~ = 0 )~ . Q t In fact, if S E %([O,T])is such that Isds = 0, let yt = So Isdw,, then E

lyt12

0 = E ( x ,y),

=

E

=E

1~:

2

t

lsdwSl = E J o Isds

=

0. yt = 0, P - ass. Thus

J , Isd ( x ,w ) , . So dQ << dt x d P easily follows. Hence by

7.3 Non-linear Filtering Equation

189

the Radon-Nikodym theorem[28]there exists a %([O,T I ) x ST-measurable function f (t,w ) such that Q(S x 4 = E V A J, f (t,w)dtl. So E [ I A( x ,w),] = E [ I A f ( s ,w)ds],and furthermore, P - a.s. ( x ,4,= f ( s ,wW,Qt e 10, TI. Now let the smoothness o f f be fn(t, W ) = n J; e-n(t-s) f ( s ,UPS. T h e n it is easily checked that there exists a subsequence { n k ) o f { n ) , denoted b y { n ) again, such that as n -+ m J,T Ifn(t,w) - f (t,w)l dt + 0 , P - a.s. I fn(t, w ) - f ( t ,w)l --+ 0 , dt x d P - a.e. Now let us show that the solution x r o f the following O.D.E. for each fixed w e a dxn -d= t - n ~ ~ + n ( x , w ) ~ , x ~ t=EO[O,T]. , satisfies = fn(t,w). In fact, x r = n ~e-n(t-s) ; ( x ,w ) , ds. Hence we have that f ( s ,w ) ( nJSt e-n(t-u)du)ds] = n [ ( x ,w ) , - $1 = n [ cf ( s ,w)ds - n e-n(t-s) f (s,w)ds = fn(4 w). Since x; is Tt-adapted, it follows that fn(t, w ) is also St-adapted, and so isitslimit T ( t , w ) =limn,, fn(t,w). Hence f ( t , w ) = f ( t , w ) , d t x d P - a . e . on [O,T]x 0. By the continuity o f ( x ,w ) , one has that P - a.s. t ( x , w j t = JO f ( s , w ) ~ s Qt , e E0,Tl. Finally, let yn(t) = J: T ( s ,w ) ~ ~ ~ ( ~ , ~ ,Notice l < ~ that dw~.

Jgt

~gt

-

0 5 E(XT- Y ~ ( T ) ) ~

,Tl

w) I

E J: h s , This gives that

,
2

~ l f i ~ , ~EX$ ) ~< ~ m.~ d ~ 2

ds < m. Since T m is arbitrary, one easily obtains the result. H Let us establish a similar result for a pure discontinuous martingale and a centralized Poisson process.

E J:

Lemma 211 Consider a 1- dimensional random process. If { ~ f , ~ ~ ) , , ,E

M 2 is a pure dicontinuous square integrable martingale, and N t , St

-

Z,,

is a centralized Poisson process with the compensator intensity At, where A > 0 is a constant, that is, Nt = Nt At is a Poisson process such that

+

190

7. Linear and Non-linear Filtering

E N t = At, then there exists a {bt(w),&}t,o- with E V T < co, such that (P - a.s.) [ x d ,f i ] = $ b.(w)ds, V t 2 0. d[md,fi] dt = bt(w). Naturally, we write Proof. Fix an arbitrary T

< m. For V S x A E B([O,TI) x & let

Q ( S x A ) = E [ I ASs d [ x d ,G] 1

<< dt x dP. In fact, i f S

Then dQ let yt

lsd@,, then E $12

=

r lbs(w)12ds < co,

.fi

B([O,T]) is such that - Isds = 0 , 2 = E I ~ , ' ~ ~ d f=i ~ EX/; l Isds = 0. yt = 0, E

-

P -ass. Thus O = E [ x d ,y ] t = E J,' Isd [ x d ,fi] . So dQ << dt x d P easily s follows. Hence by the Radon-Nikodym theorem there exists a B([O,T]) x &-measurable function f (t,w) such that & ( S x A ) = E [ I ASs f(t,w)dt]. So E [ I A [ x d ,fi] t ] = E [ I ASot f ( s ,w)ds],and furthermore, P - a.s. [ x d , f i ] = S,' f(s,w)ds, 4 E [O,T]. t Now the proof can be completed in exactly the same way as the proof in the previous lemma. T h e only thing we need t o do is t o replace ( x ,wjt there by x d , N here. -It Here we should mention that even in the filtering equation (7.19) the jump term coefficient C ( t ,xs(.),s 5 t ) o f the observable equation does not appear directly. However, equation (7.19) does depend on C(t,x,(.), s t ) , because the solution tt o f the observable equation does.

[

<

Corollary 212 If bs(w)dws Sot cf (w)d@i ~ ~ ~ c ~ ( w ) d @ ~ , yt = bt(w)dwi where { w j , is another d- dimensional BM independent of {wt,St)t>O, -

+

+

+

-

{f i ,S t )t>o is another 1- dimensional centralized Poisson process with the -

intensity EN: = A1t, and &: = N: - A1t such that it has no common jump times with fit; that is, A@; A fit = 0,Vt 2 0 , where A1 > 0 is a constant; and for simplicity we assume that for all T < oo, ~ b f ( w ) l ~ d s + ~ : l b ~ ( ~ ) ~( C~~d( Ws )+~ ~~~ S: + J : ~ C , ( W ) I ~ ~ S ] < m, then the non-linear filtering equation (7.19) becomes

E[C

7.4 Optimal Linear Filtering

191

Corollary 213 1) In the case that Ayt 0, that is, for yt without jumps, then the non-linear filtering equation (7.19) becomes

2) In the case that Ct(J, z ) z 0, that is, the observation process Jt has no jumps, one also finds that (7.27) is the filtering equation.

7.4 Optimal Linear Filtering Let us apply the previous results to a linear system. For simplicity suppose that we have an 1-dimensional signal process satisfying the following SDE:

which cannot be observed, and we have a 1-dimensional observable process related to the signal process satisfying the SDE as follows: {Jt,

- and {wt, St}tloare two independent BMs, and

where {w:, and Nt, St

-{ -

}+,n

are two centralized Poisson processes such that

-.

N,1 = N,1 - ~ ' t Et , = Nt - Xi! and A', X > 0 are constants, moreover, AS: A fit = 0,Vt 0; that is, E; and fit have no common jump times. Let us assume that all coefficients are non-random such that B-'(t)l 5 kO, I ~ - l ( t ) l5 ko, and

>

I

+

s:[la(t)l+ IA(t)l+ lbM2 + lb1(t)I2 lB(t)12 lc(t)12 1c'(t)l2 lC(t)12]dt < co. By theorem 117 (7.28) has a pathwise unique strong solution Ot = erot a ( u ) d u [ O o + ~ ~e-1; a(u)du(bl(s)dw~+b(s)d~s+cl (s)dfi;+c(s)dfis]. Applying Corollary 212 one immediately finds that

+

+

+

192

7. Linear and Non-linear Filtering

However, (7.30)is not very convenient, because there is another term 7rs(e2) in the equation. Notice that ~ s ( 6 ' )= ~ [ 6 : 1 3 $=] E[(6s - ns(6) ~ s ( 6 ) ) ~ 1 3 $ ] = E[(6, - ~ , ( 6 ) ) ~ 1 3 $~] ~ (- 2ns(6)E[Qs 6 ) ~ -~s(6)13$] = E[(BS- ns(6))'13$1 ~ ~ ( 6 ) ~ . So if we write 7 , = E[(6, - nS(6))'1$$],then (7.30)can be rewritten as

+

+ +

Notice that yt = 7rt(e2)-nt(6)'. Applying Ito's formula to (6t)2and ~ respectively, one has that

~(e)~,

and

respectively. Now applying the non-linear filtering equation (7.26) to (6t)2 one has that

+

l

t

+

[2b(s)rs(6) B-1(s)[A(s)ns(63) - A(~)a,(6~)n,(B)]&, t

+l

( ~ C ( S ) K , - (6)

+~ ( ~ ) ~ ) d f i , .

Therefore, substracting (7.34) by (7.33) one obtains

(7.34)

7.4 Optimal Linear Filtering

193

Equations (7.31) and (7.35)are the equations for the (optimal)filtering and the conditional mean square error of filtering, respectively. The interesting thing is that there is no jumps in (7.35). Thus we arrive at the following theorem. Theorem 214 Suppose that a 1-dimensional signal process {6t,&}t20 and an 1-dimensional observable process {[t,5t)tlo are given by (7.28) and (7.13). Assume that BP'(t)l I ko, IC-'(t)l I ko, and

I

&%(t)l+ IWl+lb(t)I2+ lb1(t)I2+ IW)I2 lc(t)12 lc1(t)12 1c(t)l2]dt< 03. Write nt(e) = = E[(& - ~ ~ ( 6 ) ) ~ 1Then @ ] . they satisfy the following filtering equation and equation for conditional mean square error:

+

where

+

+ ~[e~lg],?~

$)tto-

a BM, and

is still a centralized Poisson

process such that Nt = Nt - At, (where Nt is a Poisson process with the intensity ENt = At), both under the original probability P such that P-a.s. dEt = A ( t ) ~ t ( B ) d t B ( t ) f i t c ( t ) d N t .

+

+

The interesting thing here is even the filter ~ ~ (is6RCLL ) (right continuous with left limit), that is, it can be discontinuous; however, the conditional mean square error yt of the filter is continuous, and its SDE is without jump terms.

7. Linear and Non-linear Filtering

194

7.5 Continuous Linear Filtering. Kalman-Bucy Equation In the case that b(t) = c(t) = c l ( t ) = C ( t ) = 0 , the signal process and the observable process (7.28) and (7.29) become

+

d6t = a(t)6tdt bl(t)dwf, qt = A(t)Btdt B(t)dwt,

+

(7.38)

where {wf ,S t )t,o and {wt ,5t)t.,0 are two independent BMs and (60,to)is ~o-measurable.-Moreover, the filtering equation and the conditional mean square error equation in Theorem 214 become

~} t 20 is a BM. Recall

where Et = B B 1 ( t ) ( q t- A(t)rt(6)dt),and { E t , that

y,

= E [ ( 6 , - ~ , ( 6 ) ) ~ 1 3 We $ ] . have the following theorem.

Theorem 215 Under the assumption i n Theorem 214 and the assumption that (et,l t ) t > 0 is a jointly Gaussian process, (60,to)is &-measurable, { w ~ , 5 t ) t t oand { ~ t , $ t } ~ > ,are , two independent BMs, and b(t) = c ( t ) = c l ( t ) = C ( t ) = 0 one has that 1) 7, = E[(6, - %(6))21. 2) yt satisfies the following ODE (ordinary difserential equation)

Equations (7.39) and (7.41) are called the Kalman-Bucyfiltering equations. Equation (7.41) is also a Riccati equation. To establish this theorem we need the following lemma.

Lemma 216 If then

(E, 6 ) are jointly E(W3 =

Guassian (both can be multi-dimensional),

+ DecD,f,(E - BE),

(7.42)

c o v ( 6 , 6 l l ) ~ E [ (-e E(elE))(e- E(6lE))*IEl = Dee - DeCD&D&, (7.43) where Dee = E[(6- E6)(J- E l ) * ] ,and Dee, DEEare similarly defined and, moreover, we write D& = D&', i f DEE> 0; and D& = 0 , otherwise.

7.5 Continuous Linear Filtering. Kalman-Bucy Equation

195

+

Proof. Notice that if Q = (6 - EO) C(J - EJ), then C =-D~~D& ===+ Eq(J - EJ)* = 0. In fact, if DEE > 0, then D& = DG'. Thus the above statements are obvious. If DEE= 0, then J = EJ. In this case any constant C will make EQ(J- EJ)* = 0. For C = - D ~ D &one has that {Q, J) is an independent system, since Q and J are jointly Gaussian and not linearly correlated. So,

0 = E(Q) = E(v1J) = E(0lJ) - Eo - DqD&(J - EJ). Thus (7.42) is obtained. On the other hand, substracting (7.42) from 0, one finds that Q = 0 - E(0IJ). By the independence of {Q,J) E[(o - E(oIJ))(o - E(oIJ))*IEl = EQV*. Using (7.44) one has that as Dee > 0 EGQ*= Dee DeeDG' DEED;F1 D& - DoSD:c1 D& - Dee DG' D& = Dee - DeeD:;' D ; ~ . In the case Dee = 0, so then J = E J , D& = 0, and EQQ* = Doe = Dee - DsED&D&. The proof is complete. w Now we are in a position to prove Theorem 215. Proof. Let us show that 7, = E[(Ot - T ~ ( O ) ) ~For ] . this let us make a subdivision on [O, t] by <...
+

;=g

le,nl,

my = ~[otl8f,n],$ = E[lot - m?12 and mt = = nt(O),yt = E[lot - mt12 18f1 = E[Pt - rt(0)l2 [ e l . Since (Ot, (Jts, . . . ,JtTn))is Gaussian by Lemma 216

~[otlel

Y; = E[IOt - m a 2 ~$,n] = Dee - DeenD:nC-D&n 2 = E[Dee- Detn D&,, DZ;~,] = E (06 - my1 = 7:) where Jn = (Jton,... ,Jtgn).If we can show that as n 7;

+

co,

Yt = E[IOt - mt12 18f1,

7t = E lot - mtl 2 , then yt = 7,) and taking the expectation

7:

-+

+

in both sides of (7.40) we obtain (7.41). However, by Levi's theorem as n co, rZ"= E[let - m?12 18f,,1 = ~ [ l o t18$,n] l ~ - 2~[etl@,n]m?* + lm?12 = ~ [ l o t18f,n] l ~ - lm?l2 18f1 - lmtl2 = E [ I-~mt12 ~ 1821 = rt. On the other hand, 72"= E 10, - m?12 = E I(0t - mt) (mt - m?)(2 = 7, E Imt - m?l2 2E(Ot - mt)(mt - my)*. By this we easily see that as n -+ co, so 17; - Ttl -+ 0. Hence the proof is complete. -+

-+

lo^^^ +

+

+

196

7. Linear and Non-linear Filtering

Corollary 217 Under the assumptions i n Theorem 215, the Riccati equation (7.41) has a unique solution. Hence (7.39) also has a unique solution. Furthermore, i f yo > 0, then we also have 7, > 0,Vt > 0. Proof. By Theorem 215 it is already known that (7.41) has a solution = E [ ( & - ~ ~ ( 8 ) Let ) ~ us ] . show the uniqueness. First, by Ito's formula one easily sees that the solution ( O t , f,) of SDE (7.38) has the following property: for any T < co < col E [ s u ~ t E [ O , ~ l loti2 + SuPtt[o,~] lftl2] 5 where kT is a constant depending on T only. Hence as t 5 T yt = E [ ( B-~~ ~ ( e 5) 41c~ ) ~ <] 00. Now suppose that ?Eli = 1,2 are two solution of (7.41). Set yt = 17; . By (7.41) one has that 0 5 yt 5 Sot[2a(s)yS ~ ~ T ~ : A ( s ) ~5]8Y ks Sot ~[.(s.) + J$A(s)2]~sds. So applying Gronwall's inequality one finds that yt = 0 , i.e. the solution of the R'iccati equation (7.41) is unique. By the Lipschitzian condition, (7.39) also has a unique solution. Finally, let us show that 7, > 0 ,V t > 0. In fact, let St = Bt - ~ ~ ( 0Then ) . by (7.38) and (7.39) it satisfies the following SDE d& = a(t)dtdt + b(t)dwf - A ( t ) y t B - 2 ( t ) ( d f s- A ( t ) n t ( e ) d t ) = a(t)btdt b(t)dwt - ytB-2(t)A2(t)&dt - ytB-l(t)A(t)dwZ. Applying Ito's formula one finds that 1&12 = 1 6 0 1 ~ 2 $ Xs 16S12d~ c [ b 2 ( s ) ~ ~ ~ B - ~ ( s ) ~ ( s ) l ~ ] d s +2 J," Ssb(s)dwi - 2 6 s B - 1 ( s ) A ( s ) y s d w ~ , where & = a ( s ) - y , B - 2 ( s ) A 2 ( s ) . Hence as yo > 0, -e2~Asds+~~eS~2;i,dr[b(~)2+~-2(~)~(~)2y~]d T t = El&l = yo

yt

-?;I

+

+

+

+

+

7.6 Kalman-Bucy Equation in Multi-Dimensional Case For the continuous linear filtering problem in multi-dimensional case we have the following corresponding theorem.

Theorem 218 Suppose that a k-dimensional signal process {8t,8t)t,0 and an 1-dimensional observable process {(tlzt)t,o are given by the f&lowing SDE: dot = a(t)&dt bl(t)dw:, dEt = A(t)Btdt + B ( t ) d w t , where {w:, 5t)t,o and { w t ,zt}tlo are two independent BMs, the first one is k - dimension~l,and the second one is 1-dimensional; moreover, ( B t ,Et)t2o is a jointly Gaussian process, ( B o , f o ) is 80-measurable, and a ( t ) ,b l ( t ) E ~ ( t ~) , ( tE )R @ ~ .

{

+

7.7 More General Continuous Linear Filtering

197

<

Assume that J ~ - l ( t ) l ko, and d tco. S,T[la(t)l + 1 4 1 1 + lb1(t)I2+ I ~ ( t ) l ~ l < Denote ~ ~ ( =0 ) yt = E[(&- T ~ ( O ) ) ~Then ] . they are the unique solutions of the following filtering equation and equation for mean square error: a(s)~,(O)ds ~,~*(s)B*-~(s)dG,, nt(O) = TO(O) 7, = yo ~:[a(s)y, ?,a* (s) bl(s)bl*(s)]ds

~[O~lzf],

+2

+

+ Sot +

+

- So"r,~*(~)(~(s)B*(s))-~~(s)y,ds,

{ "),>,,

where Vt = BB-t)(dt, - A(t)nt(0)dt), and Etl thermore, if

is a P-BM. Fur-

yo > 0, that is, it is positive definite, then yt > 0, W 2 0.

Since the proof is almost completely the same as in the 1-dimensional case, we will not repeat it. This theorem actually gives us a practically closed form method for solving the filtering equation. In fact, we can first solve the second ODE (the so-called Riccati equation) to get the mean square error for the filter, and then put it into the first equation and solve the first linear SDE to obtain the filter ~ ~ ( 0 ) .

rt

7.7 More General Continuous Linear Filtering In this section we will consider the filtering problem on a more general continuous partially observed system:

where {w:, zt),>o and { ~ : , z ~ ) , > are~ k-dimensional and I-dimensional and BMs, respectiveiy, and they areindependent; moreover, {Ot, {[t,zt}t20 are k-dimensional and I-dimensional random processes, respectively. Naturally, we assume that aO(t)E Rk@l;al(t), bl(t) E RkBk;a2(t),b2(t) E Rk@'; AO(t)E R I @ ~ ; A1 (t), B1(t) E R1@k;A2(t),B2(t) E R1@', 0 1 2 A0 A1 ~ 2 . g-lh(t)ldt
+ +

+

~

7. Linear and Non-linear Filtering

198

b1( t ) b2(t) ( B 1 ( t ) B 2 ( t ) ) ( 3) . So that we can rewrite the partially observed system as the simpler one: dot (aO(t)+ al(t)Ot + a 2 ( t ) f t ) d t+ d l ( t ) @ + dz(t)&$, d t t ( A O ( t )+ A1(t)Ot + A2(t)Et)dt+ D2(t)&:. { 2) Suppose 1) is done. If we write the second SDE as =

=

=

and let vt = O t - 8t, then we should have d f t = d t t - ( A O ( t ) A1(t)vt A2(t)Jt)dt, and dfi~:= D z l ( t ) [ d & - ( A O ( t ) A1(t)Ot A2(t)Et)dt]. Moreover, substituting this expression for diii:, and Ot = Jt v t , into the SDE for O t , we find that d8t dvt = (ao(t) a1(t)(8t v t ) a2(t)&)dt dl (t)diiii + d z ( t ) ~ ; l ( t ) [ & t - ( A O ( t ) A1(t)(Bt ~ t )A2(t)&)dt] = [a1( t ) - dz(t)D;' ( t ) A 1(t)]&dt dl (t)&i +(ao(t) al(t)vt a2(t)&)dt d z ( t ) D ; l ( t ) [ d ~-, ( A O ( t ) A' ( t ) v t ) +A2(t)&]dt. So that if we set dvt = [(ao(t)- d z ( t ) D ; ' ( t ) ~ O ( t ) ) ( a l ( t ) - d 2 ( t ) D z 1 ( t ) A 1(t))vt (a2( t ) - d2 ( t ) ~ ; '@)A2( t ) ) t t ] d t d2 ( t ) ~l (, t ) d t t , (this linear SDE, when Et is given, always has a solution v t , which is &-adapted), then

+

+

+

+

+ +

+

+

+ + + + + + + +

+

+

+ +

+

Now the partially observed SDE system (7.46) and (7.47) is similar to the = $ , ~ t2 0 , then type considered-in Theorem 218. If we can show that E [ . I ~= ; ] E [ . I ~Thus ] . by means of OL= vt Ot we have mt = vt + K t , where mt = and Et = E [ & l g ] . Furthermore, we also have 7, = since E[(Ot - mt)21$] = E[(& SOthe filtering equation can be derived by using Theorem 218. Let us show the following lemmas.

yt,

+

~[Otlg], -

Lemma 219

g = sf,w 2 0. -

e,

ct

Proof. Obviously, 3; C since vt is 3;-adapted, and is expressed by a function of ([,,us), s 5 t. However, by the above discussion we can also write

7.7 More General Continuous Linear Filtering

199

Hence, by this linear matrix equation when I,, s 5 t is given, the solution (Et,vt) is ?$-adapted. (Actually, it can be expressed as some exponential function of ,s 5 t ) . So 3: > Applying Theorem 218 to the partially observed SDE system (7.46) and (7.47) we obtain the following lemma. -

<,

g.

Lemma 220 The filtering equation for (7.46) and (7.47) is d E t = [a1( t ) - d z ( t ) ~ ; '( t ) A l ( t ) ] ~ ~ ~ d t + - T t ~ l *( t ) (( t~ )D;~(t))-l (d& - ~l ( t ) m d t ) , @A = [a1( t )- d z ( t ) ~ , l ( t ) A 1(t2]Tt [a1( t )- d2 ( t )D;' (t)A1(t)]* -7,A1* ( t )(D2(t)D;(t))-1A1(t)yt dl (t)d;(t).

+ Tt

+

However, to obtain the filtering equation for the original partially observed SDE system (7.45) we need to find out the concrete form of Dt; that is, we should complete step 1). For this we see that if step 1) is true, then E jot D,D,*~s= E jot B,B,*~s, w 2 0, where Bt =

(

z{("j),"$(":) )

. So we may use the following approach to

find Dt. Let

Then we have the following matrix equations di(t)dT(t) &(t)d;(t) = bi(t)b;(t) bz(t)b;(t)= (b 0 b)(t), dz(t)D,*(t)= bi ( t ) B ; ( t ) bz(t)B;(t)= (b 0 B ) ( t ) , Dz(t)D;(t)= B l ( t ) B i ( t ) Bz(t)B;(t) = ( B 0 B ) ( t ) . Thus, by using such definitions for (b o b)(t),(b o B ) ( t ) and ( B o B ) ( t ) ,we obtain the following lemma.

+

+ +

+

I(B

Lemma 221 Assume that ( B O B ) - ' ( t ) exists and is bounded o ~ ) - l ( t )5 l ko,W 2 0. Then 1) the sysmetric matrix ( b o b)(t)- ( b o B ) ( t ) ( B0 B ) - l ( t ) ( b o B ) * ( t )2 0, that is, it is non-negative; 2) if we set D2(t)= D; ( t ) = ( B o ~ ) l / ~ ( t ) , d2(t)= (b o B ) ( B o ~ ) - l / ~ ( t ) , d l ( t ) = dT(t) = [(b0 b)(t) - (b 0 B ) ( t ) ( B0 B ) - l (t)(b0 ~ ) * ] l / ~ ( t ) ,

then Dt =

( db(t) $2yj) ) solves (7.50); Sot

3) let zt = D;lB,dw,, (since DtD; = BtB; > 0 , this definition makes sense), where wt = (w:, w:) is the original Ic 1-dimensional BM given in 5t)t,o - is a BM, and $'D s m s = B,dw,. (7.45), then {zt,

+

Jot

200

7. Linear and Non-linear Filtering

Proof. Recall that a conditional mean square error is always non-negative. So we will use Lemma 216 to prove result 1). For this we introduce k I-dimensional Gaussian vectors (a1,a2)* = ((a:, - . - , (a:, . - . ,E:)) with independent components and E E = ~ 0 , Daj = E(E$E;)= 1. Set 0 = blal b2a2, ,$ = B1sl B2c2. Applying Lemma 216 to these two random vectors, we have (b o b)(t) - (b o B ) ( t ) ( Bo B)-l (t)(bo B ) * ( t ) = ~ ~ ( 0 , 0 1 0 = E [ ( o- E ( o I t ) ) ( o- E(olE))*l 2 0 , where (b 0 b)(t) = bl (t)b?(t) b2(t)b;(t), (b 0 B ) ( t )= bl ( t ) B ; ( t ) b2(t)B,*(t), ( B 0 B )( t )= B l ( t )B ; (t) B2 ( t )B," (t). Thus 1) is proved. 2) is obviously true. Let us prove 3). Notice hat W > u E[(mt - T&)(TBt - mu)*lzu]= E[J: DL' BsB,*(D,l)*d~lzs] = E[J: D ; ~ D , D , * ( D ; ~ ) * ~ s=~t~-, ]u. Applying Theorem 97 we find that { ~ t , 5 t } is ~ ,a ~BM. By definition Jot Ds&, = Jot Bsdw,. The proof is complete. Finally, we can deduce the filtering equation for the original partially observed SDE system (7.45). Let us write out the final result as follows.

+

k),

+ +

+ +

+

Theorem 222 Suppose that a k+l-dimensional partially observed processes { O t , ~ t } t 2 0and { t t , 5 t ) t 2 0 are given by the SDE (7.45), and suppose that the assumption made i n the beginning of this section holds. Moreover, asko,V t 0 , sume that ( B o B)-' (t) exists and is bounded ( B 0 B)-l(t)l and (&, tt)tlo is a jointly Gaussian process, (00,to)is 30-measurable. ] , = E[(Ot - ~ t ( e ) ) then ~ ] , they are the unique Write nt(0) = ~ [ 0 t 1 3 !yt solutions of the following filtering equation and equation for mean square error.. [aO(s) a' (s)wS( 0 ) a2(s)t,)ds nt ( 0 ) = no(0) Jot [(b0 B ) ( s ) T,A1* ( s ) ]( B 0 ~ ) - l ( s ) .[dts - ( A O ( s ) A1(s)ns(0) A2(s)t,)ds], ;jit = yo Jot[al(s)T, Tsal*(s) b 0 b(s)]ds - ~ o t [ (obB ) ( s ) T s A 1 * ( s ) ] ( oB B)-'(s)[(b o B ) ( s ) T,A1*(s)]*ds. hrthermore, if To > 0, that is, i f it is positive definite, then Tt > 0,Vt 2 0.

I

+

+

+ + + +

+

+

<

>

+

+

+

+

-

Proof. Notice that nt(0) = mt = vt + W i t , and 7, = 7,. Hence by using Lemma 220 and by using 2 ) of Lemma 221, and (7.48), (7.49) we easily derive the final result.

7.8 Zakai Equation

201

7.8 Zakai Equation In this section we are going to derive the Zakai equation for some concrete partially observed system. Suppose that we are given a signal process xt satisfying the following SDE :

where wl, t 1 0, is a d-dimensional BM, fikt(ds,dz) is a Poisson martingale measure with the compensator rf(dz)dssuch that Nkr (ds,dz) = Nk,(ds,dz) - r f(dz)ds, where r f ( d z )is a a-finite measure on the measurable space (Z,!B(Z)), Nkt(ds,dz) is the counting measure generated by the Poisson point process Ic; and xo is supposed to be independent of and suppose that the observation tt is given as follows:

-

~r"~',

where At is assumed to be a bounded function, B , C satisfy the same conditions stated in Theorem 206, and fit is a centralized Poisson process like that given in the same Theorem. Write ( w f i , w j )= Jot p y d ~1, 5 i ,j 5 d .

Ig), etc.

For the random process At(%(.))write r t ( A ) = E(At(x(.)) Then we have

Theorem 223 . Suppose f E C;([O,OO); R1). Then by Ito's formula f ( x t ) = f ( 5 0 ) L f ( x ,S ) ~ + S ~f (xa)' 4 3 , X ( . ) ) ~ W ; ji jz L ( ~ ) ~s-)fikt(ds, (x, dz), where for each x(.) E D([o,oo), R ~ ) , L f ( x ,s ) = b(s,x(.)) . ( x s )+ i t r . ( a ( ~ , x ( . ) ) * ( ax~sf) l ~ x 2 x(.))) )~(~, J Z ( f ( x s - + c ( s - , x ( . ) , ~ )) f (xs-) - ~f (xs) . C ( S - , X ( . ) , Z ) ) ~ ' ( ~ Z ) , L ( l ) f ( x , s - ) = f(xs- +c(s-,x(.),z)) - f(xs-). Furthermore, if (w:,w ~= ) &ds. ~

+

+ Ji

Ji

vf

+

Ji

7. Linear and Non-linear Filtering

202

then

.(ns(fA*) - n . ( f ) n , ( A * ) ) ~ ; ~ ( e ) * ] f i ~ 11

+

-4

where Dt dt

= d[yd,@ ] t , and

~,= d J:

JZ

/d

n s - ( ~ d ) d G s , (7.53)

L(')f ( x ,s - ) f i k ~(ds,dz).

Theorem 223 is a direct corollary o f Theorem 206. One only needs t o see f ( x t ) as the signal process ht in (7.12), then applying Theorem 206, the conclusion is derived immediately. Corollary 224 Assume that i n (7.51) and (7.52) the signal and observation noise are independent, and the jump noise in the signal and observation have no common jump time, i.e. p = 0, Ayt . A N t = 0; and P [ x t 5 x

-

/g],

sE,

(which is the conditional distribution of xt under given and {xt 5 x ) is the set { x : 5 x l , . . . ,xf 5 x d } ) has a density at,r ) = d P [ x L5 x 1$]/dx, which satisfies suitable differential hypothesis, and x ( x t ) = At(x(.)) only depends on x t , and it is a bounded function. Then one has the following Zakai equation satisfied by the conditional density p^(t,x)

where L* is the adjoint operator of L. Proof. T o show Corollary 224 on_e only needs-to notice that nt (2) = J,, 3 (x)p^(t,x)dx = (A*p^(t,.)) = ( A * ,ph), and t o apply integration by parts. Indeed, by assumption p = 0 , A y t . A N t = 0, hence (7.53) becomes Vf E C z ([o,co);R' ) (where C: ([o,co);R') is the totality o f functions f : [0,co) -t R 1 , with continuous derivatives up t o the second order and with compact support), n t ( f )= n o ( f ) r S ( L f)ds + $ ( n S ( f ~* n) s ( f ) ~ s ( A * ) ) B ~ l ( J ) * f i s . Or, V f E Ci([O,co);R1) ( f , dtp^(t, = dt(f,p^(t,.)) = ( f ,~ * p ^ .))dt ( t , ( f ,gt, .)(A* - n t ( ~ * ) ) ~ , - l * f i t . Hence (7.54) now follows. T h e advantage o f Zakai's equation is that it is a linear partial stochastic differential equation (PSDE) for g t , x ) , and usually a linear PSDE is much easier t o handle. As soon as the solution p^ is obtained, then the non-linear filter r t ( z )= ~ ( x t l g=) JRd xp^(t,x)dx is also obtained.

-

+ Jot

+

7.9 Examples on Linear Filtering

203

7.9 Examples on Linear Filtering In many cases, or in an ideal case, we will consider the original signal system to be non-random, that is, the coefficients of the signal dynamics are non-random. However, the initial value of the signal process may be random. For example, the initial value of the population of fish in a large pond actually is random. Moreover, since the signal process itself usually cannot be observed directly, we can only estimate it and understand it through an observable process that is related to it. Obviously, the observed results will usually be disturbed by many stochastic perturbations. So the appropriate assumption is that we have a pair of a signal process Ot and an observable process I, as follows, where for simplicity we consider them both in 1-dimensional space: dOt(w) = a(t)Ot(w)dt,Oo(w) = OO(W); dt, = A(t)Otdt B(t)dwt, l o = l o ,

+

(7.55)

where all coefficients a(t), A(t) and B(t) are non-random. By Theorem 215 we have that

where rt(0) = EIOt13!] is the estimate of Ot based on the information given by the observation J,, s 5 t, and y, = E[(O, - r,(0))2] is the mean square error of the estimate with yo = E[(Oo - T ~ ( O ) ) ~Again ]. by Theorem 215 it is already known that (7.56) has a unique solution (rt(O),y,),Vt 2 0. Here, we are interested in how to get the explicit formulas for the solution. First, from the practical point of view, let us replace { Z t }by {l,)from the formula: dl, = A(t)rt(O)dt B(t)fit, because our observation is {f,} . Thus we get the following filtering SDE system:

+

~ ~ (=0ro(e) )

+ J,"(a(s) - B - 2 ( ~ ) A 2 ( ~ ) T s ) r 8 ( Q+) dJot~ B-2(s)A(s)TsdEs,

yt = yo + J,"2a(s)y,ds

-

Jot ~ - ~ ( s ) ~ ( s ) ~ y t d s .

(7.57) Obviously, if we can find a formula for 7, solving the second ordinary differential equation - the so-called Riccati equation in (7.57) then the estimate, or say, the filter rt(0), can also be obtained from the following formula:

204

7. Linear and Non-linear Filtering

(In fact, one can use the Ito formula to check that rt(0) defined above satisfies the first SDE in (7.57)). Fortunately, the solution of the Riccati equation does have an explicit formula, if we make some further assumptions that a(s) = ao, A(s) = Ao, B(s) = Bo > 0, To = E[(00 - ~ o ( @ )> ) ~0 ] are all constants. In this case one easily checks that the following yt satisfies the second Riccati - ODE in (7.57):

These tell us that if we can know "the mean square error of the initial estimate" yo, which is larger than zero, then the estimate ~ ~ (=0E[o~~$] ) by observation, and 7,-the mean square error of the estimate, can be calculated by formulas (7.58) and (7.59), respectively. One naturally asks = 80 and one finds that yt = 0,Vt 2 0 how about yo = 0. In this case TO(@) is the unique solution of the Riccati equation, that is the second equation of (7.57). So xt(0) satisfies the following equation ~ t ( @=)00 f a(s)n~(0)ds, that is, the same equation as the signal equation. So one immediately gets the solution formulfted by ,,(@) = 6, = 0,e& a(s)ds. This means that the estimate is exactly equal to the signal process. This is quite reasonable. Because the initial value can be explicitly observed, so one can directly use the known signal dynamics to get the explicit signal process. However, one should notice that if the signal process satisfies an SDE (Not an ODE!) dBt = a(t)Otdt b1(t)dw;, 00 = Oo, and the obsevable process J t still satisfies the second SDE in (7.55), then by Theorem 215 ;Jt will satisfy a more complicated Riccati equation: yt = To ~,"[2a(s)y, b l ( ~ ) ~ ]-d s B - ~ ( s ) A ( s ) ~ ~ : ~ s . In this case even if 7, = 0, we still cannot get ;l;, = 0. So for continuous linear filtering problems one asks in what cases we can get explicit formulas for the filterings, then by the above discussion one sees that this completely depends on how many explicit formulas we have for solutions of the Riccati equations.

+

+

+

Jot

Option Pricing in a Financial Market and BSDE

In this chapter we will discuss option pricing in the financial market and how this problem will draw us to study the backward stochastic differential equations (BSDEs) and how the problem can be solved by a BSDE. Furthermore, we will also use the partial differentail equation (PDE) technique to solve the option pricing problem and to establish the famous Black-Scholes formula.

8.1 Introduction 1. Hedging contingent claim, option pricing and BSDE In a financial market there are two kinds of securities: one kind is without risk. We call it a bond and if, for example, you deposit your money in a bank, you will get a bond that will pay some interest at an agreed rate. It is natural to assume that the bond price equation is:

where r ( t ) is the rate function. Another kind of security in the financial market is with risk. We call it a stock. Since in the market there can be many stocks, say, for example, N different kinds of stocks, and they will usually be disturbed by some stochastic perturbations, for simplicity we assume that the stochastic perturbations are continuous, so it is also natural to assume that the stochastic differential equations for the prices

206

8. Option Pricing in a Financial Market and BSDE

of stocks are:

where wt = (w:, w:, . . - ,w:)* is a standard d-dimensional Brownian Motion process, and A* means the transpose of A. Now suppose that a small investor who wants his money (or say his wealth) from the market at the future time T when it reaches X. (Notice that X is not necessary a constant, for example, X = co clP;, where q,and cl are non-negative constants, and P; is the price for the first stock at the furture time T, because the investor has confidence that the first stock can help him to earn money). How much money xt should he invest in the market, and how could he choose the right investment portfolio at time t? Suppose the right portfolio (n!, nf . , . ,n?) exists, where n! is the money invested in the bond, and IT: is the money invested in the ith stock. Then he should have

+

where 77: is the number of bond units bought by the investor, and is the amount of units for the ith stock. We call xt the wealth process for this investor in the market. Now let us derive intuitively the stochastic differential equation (SDE) for the wealth process as follows: Suppose the portfolio is self-financed, i.e. in a short time dt the investor doesn't put in or withdraw any money from the market. He only lets the money xt change in the market due to the market own performance, i.e. self-finance produces dxt = @dPf 77idP;. Now substituting (8.1) and (8.2) into the above equation, after a simple calculation we arrive at the following backward SDE (BSDE), where the wealth process xt and the portfolio .rrt = (T: ... ,T?) (actually it is the risk part of the portfolio) should satisfy:

+ EL,

dxt = rtxtdt

+ nt(bt - rtl)dt + ntatdwt,

XT = X,

t E [O,TI,

(8.4)

where I=(1,. . . , I ) * is an N-dimensional constant vector. In a financial market if we let X be a contingent claim, then the solution (xt, nt) of the BSDE (8.4) actually tells us the following fact: At any time t, let us invest a total amount of money xt, and dividing it into two parts: One part of the mone is for the non-risky investment; that is we invest the money n: = xt ~f into the bond. The other part of the money is for the risky investment nt = (n: . . . ,n?); that is we invest the money T: into the the i-th stock, i = 1,2,. . . ,N. Then, eventually, at the terminal time T our total money xt, t = T, will arrive at the contingent

8.1 Introduction

< < < <

207

claim X ; that is, such an xt with xt, 0 t T, can produce the contingent claim X, and we will say that (xt,x t ) ,0 t T, hedges the contingent claim X. In a financial market we can see that to price an option will also lead to the solution of a BSDE like (8.4). The so-called "option" actually is a contract, which provides the right for the contract owner to do the following thing. "At the given future time T, by the contract, one can use a fixed price K to buy a unit of some fixed goods or some fixed stock in the market." Suppose that the price of the fixed goods or the fixed stock for one unit at the future time T becomes PT.Then at the future time T the option will help its owner to earn the money XT = (PT - K)+. Therefore, if somebody wants to own this option now he has to pay for it. That is to say, at the time t, the option should have a price xt. This raises an important and interesting problem: how can we price the option at the time t as xt, in a way that is reasonable and fair to both the seller and the buyer? To solve this problem, and to show the idea involved, for simplicity we assume that in the market there is only one stock which we mentioned and one bond. We can imagine that if we put the money xt into the market as follows: one part x: of it is deposited in the bank (with no risk), and another part n l of it is used to buy the fixed stock, or say, to buy the stock of the fixed goods (whose price changes by some stochastic perturbation, so it is with risk). Then as time t evolves to T, if the option price xt is a fair price, it should arrive at XT = (PT - K ) + at the time T. So, actually, we should require that xt =x:+.rrtl,t E [O,T), XT = (PT- K)+. Thus we have the same condition as (8.3) for X = (PT - K)+ and N = d = 1. (N = 1 means there is only one stock in the market, and d = 1 means there is only one continuous real-valued stochastic perturbation or, say, the dimension of the BM is 1). So a samilar discussion will lead to (xt,nt), 0 t T, satisfying BSDE (8.4) for N = d = 1. 2. Difference between forward SDE and BSDE. We see that (8.4) is actually a backward SDE, that is, the terminal condition XT = X is given, and we want to find a solution to satisfy (8.4). So actually we need to solve the SDE backwards. However, there is a big difference between solving the forward SDE (given the initial condition xo = x and then solving the SDE to find the solution xt,O 5 t T ) and solving the backward SDE. In fact, even in the continuous case, if we want to solve the following forward SDE (FSDE): dxt = b(t, xt)dt a(t, xt)dwt, xo = xo, t E [0,TI, then by the standard SDE theory in Chapter 3 under a less than linear growth condition and a Lipschitzian condition on b(t, x) and a ( t , x) with respect to x, this SDE will have a pathwise unique e - a d a p t e d solution xt, t E [0,TI. So there is no difficulty at all in solving an FSDE under the

{

< <

<

+

208

8. Option Pricing in a Financial Market and BSDE

usual standard condition. However, if we want to solve an SDE similar to the one above, but bakwards:

+

(8.5a) dxt = b(t, xt)dt a(t, xt)dwt, XT = X, t E [0,TI, a big difficulties arise. In fact, since the terminal condition XT = X is given first, and usually X is ST-measurable, one can imagine that the solution x t , t E [O,Tl, if it exists, will depend on the given X, so xt will also be ST-measurable for all t E [0,TI. However, this will make the stochastic a(s, x,)dw, have no meaning in general, because the integrand integral J~~ of a stochatic integral should be at least 8,-adapted. (But now a(s,x,) is ST-mea~urable!) So, usually we cannot solve the BSDE in (8.5a) even when b and a are bounded and Lipschitz continuous! Fortunately, for the BSDE (8.4) in the financial market, we know that actually the solution is a one-pair radom processes ( s t , at),and from practical financial experience it seems that it can exist and satisfy the BSDE (8.4). This motivates us to discuss the general BSDE theory and its applications to Finanacial markets. 3. The developement of BSDE The adapted solution for a linear backward stochastic differential equation (BSDE), which appeared as the adjoint process for a stochastic control problem, was first investigated by Bismut (1973), and then by Bensoussan (1982), and others, while the first result for the existence of an adapted solution to a continuous non-linear BSDE with Lipschitzian-coefficient was obtained by Pardoux and Peng (1990). Today the BSDE has become a powerful tool in the study of financial markets and stochastic control. (See, for example, El Karoui et al. 1997, and References in However, one also finds that almost all interesting problems studied in BSDE can be initiated by mathematical finance.

8.2 A More Detailed Derivation of the BSDE for Option Pricing To help the reader understand clearly what is involved we give the detail of the derivation of the BSDE satisfied by the option pricing problem. As in the introduction, we have an equation for the bond price PO(t)and an SDE for the stock price P ( t , w) as follows: dPo(t) = Po(t)r(t)dt,PO(0)= 1, dP(t, w) = P(t, w)[(b(t,w)dt a(t, w)dwt],P(0, w) = P(0, w). The fair price xt for the option at any time t should satisfy xt = a: +n-: = $P,o +v;pt, X T = (PT - K)+, where n-: is the investment for the bond and n-: for the stock, and r$ is the amount of bond units bought by the investor, and 77i is the amount of units of the stock. In a self-financed market we have

+

{

8.3 Existence of Solutions with Bounded Stopping Times

209

+

dxt = $dP? 77:dPi. Now replacing dP? and d P i by their equations we find that dxt = $PO(t)r(t)dt 11: P(t, w)[(b(t,w)dt a(t, w)dwt] = r:r(t)dt x:[(b(t, w)dt a(t, w)dwt] = (x: r:)r(t)dt r:(b(t, W ) - r(t))dt r:a(t, w)dwt = rtxtdt rt(bt - rt)dt ntatdwt. So st-the price for the option, with rtl-the portfolio, should satisfy the following BSDE

+

+

+

+

+

+

+ +

+

+

+

dxt = rtxtdt rt(bt - rt)dt ntatdwt, = ( P T - K ) + , 0 5 t ST.

XT

(84

More generally, we may consider the BSDE dxt = g(t, xt, nt)dt Z(t, xt)dwt, XT=X, O < t < T , and find its Et-adapted solution ( x t , r t ) . Or, to simplify the notation, we set qt = Z(t, r t ) , and if this equation can be solved to give nt = f (t,qt), then we may consider the BSDE

+

{

-

where b(t, xt, qt) = b(t, s t , f (t, qt)), and then find the zt-adapted solution (xt, qt). After that, setting rt = f (t, qt), we know that (xt, r t ) will satisfy the original BSDE. In the next section we will consider the BSDE in (8.7) and also a more general case.

8.3 Existence of Solutions with Bounded Stopping Times 8.3.1

The General Model and its Explanation

For more general usage let us consider the following backward stochastic differential equation (BSDE) with jumps in R~ :

WT

<

where = (wtl,. . - ,w$), 0 t, is a dl-dimensional standard Brownian motion (BM), wT is the transpose of wt; kT = (Icl,- . .,kd,) is a d2-dimensional stationary Poisson point process with independent components, N ~(ds, , dz) is the Poisson martingale measure generated by Ici satisfying

210

8. Option Pricing in a Financial Market and BSDE

fiki (ds,d z ) = Nki (ds,dx) - n(dx)ds, i = 1, - . . ,d z , where n(.)is a a-finite measure on a measurable space (2,% ( 2 ) ) ,Nki (ds,dx) is the Poisson counting measure generated by ki, and T is a bounded St stopping time, where St is the a-algebra generated (and completed) by ws,ks,s < t.

c,

Remark 225 1) If X E and T is a SY-stopping time, then BSDE (8.8) reduces to a continuous BSDE

I n fact, i f (xt,qt) solves (8.9), then (xt,qt,O) solves (8.8). 2) Set r = T. If X =a constant ko, and the coeficient b = b(t,x,q, p) does not depend on w, then BSDE (8.8) reduces to a usual continuous ordinary differential equation (ODE)

In fact, i f ( x t ) solves (8.10), then ( x t ,0,0) solves (8.8). So (8.8) is a general form which includes the ODE, the continuous BSDE and the terminal time which is non-random. The reason we consider a jump term, as i n (8.8), is that the perturbations usually occur at finite discontinuities. If the perturbation is a random process, then the number of the points where jumps occur is finite but random. Moreover, the jump values taken at the jump points are also random. For such a kind of random point process the simplest and most frequently encountered one is the Poisson point process. (See the explanation below). So it is natural to consider that the jump term is an integral over a martingale measure (it will have a nice property) generated by a Poisson point process k(.). Now let us explain why a Poisson point process is the one most frequently met among all point processes. In fact, i f p(.) is a finite Poisson point process, let N ( t ) = N U ( t )= Np((0,tl, 0 where Np((O,t],U ) is the counting measure generated by p(.), that is, it is the numbers of p(s) dropped in U during s E (O,t],then N ( t ) , t 2 0 , is a Poisson process, which has the following nice properties: 1) It has an independent increments, that is, if 0 < ttl < tz < ... < t,, then all increnents N ( t i ) - N(ti-l), i = 1 , . . . ,n, are independent. In other words, the increments of N ( t ) in disjoint intervals occur independently. 2) The probability of { N ( t h ) - N ( t ) 2 2 ) is a higher degree infinitesmal of h. Or, i n other words, in a suficiently small time interval the likelihood that more than two jumps happen is almost impossible (the probability is very small compared to the event that only one jump happens).

+

8.3 Existence of Solutions with Bounded Stopping Times

211

3) N ( t ) only takes values 0,1,2, . . . (Because N ( t ) is the numbers of jumps counted). So i n any finite interval there are only finite jumps. (Otherwise, there exists an interval ( t l ,t2] such that N ( t 2 ) - N ( t l ) = co. This is impossible. In fact, i n this case N ( t 2 ) = N ( t l ) ( N ( t z )- N ( t l ) ) = co. This is a contradiction). 4) The probability of { N ( a t ) - N ( a ) = k ) is the same as that of { N ( t )= k ) , for all k = 0,1,2,. . . , and all t 2 0 , and, moreover, it cannot be that P { N ( t )= 0 ) 1,Vt 2 0. I n other words, i n any time interval (a,a t] the probability of that k-jumps happen only depends on the length of the interval t , and not o n the starting point a. Moreover, it cannot always be that no jumps occur, V t 2 0. (Otherwise, N ( t ) = 0 , 'dt 2 0 , and we cannot call it a Poisson (jump) process. At most it can only be called a degenerate Poisson process or a constant process). One can prove that the above four properties characterize a (stationary) Poisson process. (See [&@). That is, it is a necessary and suficient conditions for a (stationary) Poisson process. Furthermore, i f we are also concerned with the jump value falling i n a set U , then we will meet the counting measure Np((O,t],U)generated by a random point process p(.). Suppose that N U ( t ) = Np((O,t ] ,U ) is a (stationary) Poisson process for any U E B z (that is, U is a measurable set). The by the above property 4 ) there exists a function X(U) > 0 such that ENp((O,t ] ,U ) = X(U)t. Suppose that, Np((O,t ] ,U ) also has the following property: 5) Np((O,t ] ,U ) is a non-negative finite measure on U E B z , and as U l , . . - , Un E B z , the random variables Np((O,t ] U , l ) ,. . . , Np((O,t ] ,Un) are independent. In other words, the events that the jump values fall in disjoint sets occur independently. Then Np((O,t],U)is a finite Poisson random measure, so p(.) is a finite Poisson random point process. (See Definition 51 i n Chapter 1). However, since in some cases we will also meet the case that the function X(U) > 0 is only u-finite, (that is, there exists a sequence of measurable sets Z1 c . . . c Zn c . - . such that Zn E B z , UF=lZn = Z , and A(&) < co);and for VU E Bz such that X(U) < oo, then properties 1 ) - 5) hold for Np((O,t ] U , ) . For such a general case we will call the random point process a a-finite Poisson point process, or simply just a Poisson Point process. (Actually, if X(U) = oo, then one understands that Np((O,t],U ) = oo,Vt 2 0. So U is only a very extreme special set. We do not need to check its other property).

+

+

+

Now let us assume that 0 < T < To,where To is a fixed number, and b in (8.8) is a Rd-valued function, and X E 5,. It is known that the study of (8.8) is useful for the option pricing in the financial market (See the Introduction above and @ I ) . For the precise definition of solution (8.8) we need the following notation.

8. Option Pricing in a Financial Market and BSDE

212

S;(Rd) =

{

f ( t ,w) : f (t,w ) is St - adapted, Rd - valued such that ~ s u ~ ( Af 7,w)12 ( t < co t>O f ( t ,w) : f ( t ,w) is Xt - adapted, Rdmdl - valued such that E

JT1f ( t ,w)12dt < co 0

and

f (t,z , W ) : f ( t ,z , W ) is RdBd2- valued, St-predictable such that E

J: / z l f ( t , z , w)12rr(dz)dt < co

Definition 226 . (xt,qt,pt) is said to be a solution of (8.8), i f f(xt,qt,pt) E S;(Rd) x L;(RdBd1)x F ; ( R ~ @ ~and ~ ) it, satisfies (8.8). Remark 227 As i n Remark 225, if X E 5:, and T is a %-stopping time, then (8.8) will reduce to a continuous BSDE (8.9) with the coeficient b(t,x , q, 0 ,w). So the solution for a continuous BSDE also reduces to ' ) , it satisfies (8.9) with the coeficient (xt,qt) E S;(Rd) x L $ ( R ~ @ ~and b(t, x , 970, w).. From Definition 226 it is seen that when discussing the solution o f (8.8) we always need t o assume that b satisfies the following assumption ( A ) 1 b : [0,To]x Rd x RdBdlx L;(.)(Rdwd2)x !2 -+ Rd is jointly measurable, St-adapted, where f ( z ) : f ( z ) is Rdmd2- valued, and L2 (Rd@d2)= 4.) I I ~ I =I ~ z +I < co Moreover, t o simplify the discussion o f (8.8) we also suppose that all Nk,(ds, Z ) , 1 < i < d2, have no common jump time; i.e. we always make the following assumption (A)2 Nki ( { t ) ,U)Nkj( { t ) ,U ) = 0, as i # j, for all U E % ( Z ) , such that T ( U ) < 00. By Definition 226 it is also seen that the following remark holds:

{

J 1ml2

Remark 228 . If ( x t ,qt, pt) satisfies (8.8) , and

-

[(I;

E I ~ ( ~ , X ~ , ~ ~dS)2] , P ~< ,W m) I, ~ l<~0 ,l ~ then ( x t lqt, pt) E s ; ( R ~x) L ; ( R ~ @x~F~; )( R ~ @ ~ ~ ) ( x t ,qt, pt) E L ; ( R ~ x) L $ ( R ~ @ xF ~ ;~()R ~ @ ~ ~ ) In fact, since r < To, S;(Rd) C L;(Rd). Hence the implication is true. Conversely, by (8.8) and by the martingale inequality, one also easily sees that under the assumption o f this remark 2 Esuptao l x t ~ , l < ~ 4 ( 1x~1' E Ib(s, x s , qslps, w)l d ~ ) +4EJ; (qs12ds 4~ Jz ~ p ~ ( zn(dz)ds) )l~ < co. Hence

+

+ (Jl

8.3 Existence of Solutions with Bounded Stopping Times

8.3.2

213

A Priori Estimate and Uniqueness of a Solution

As in the SDE case we have the a priori estimate and the uniqueness results on solutions. Lemma 229 Assume that ( x t ,qt,pt) solves (8.8), T < TO,E 1x1~ <m and assume that b satisfies that 1" x . b(t,x,q,p,w)< ~ l ( t ) ( l1+ x1 + 1 x 1 ~+)c2(t)1x1 (191 + llpll)), ci (t),i = 1,2, are where a . b means the inner product of a, b E R d ,0 non-random, such that co = J , q ( t ) d t J , ( ~ 2 ( t dt ) )< ~ m. Then E ( S U P lxtAr12 + S l ( 1 q t 1+ ~I I P ~ I I < ~ k0 ) ~ <~ m, ) t>O where ko > 0 is a constant depending on co and E 1x1~ only.

<

+

Proof. By Ito's formula

Then for each N = 1,2, . . . , we have that

tArAur~

{ So

square integrable martingale, i.e. VA E gT,0 < r < t, TATAUN xs.qsdwsI~. (*) SotATAUNxs.qsdwsI~= E J o So by the Fatou lemma tArAo~ ' qsdwsl = E!bN+m ISo x~ . qsdwsl ~ATAUN 1x8 . qs12~ l ~ 1 ~ / ~ < colimN,,EIJo 2EsupOGt1w,12+ 2EJ; lqSl2ds< m. Moreover, by the Lebesgue dominated convergence theorem as N E xs . qS1~(IS
is a

I./iAr

<

1~:"'

--+ m

214

8. Option Pricing in a Financial Market and BSDE

Hence letting N -+ co in (*) one finds that J'"' xs . qsdw, is a martingale. A similar conclusion holds for Sz x, p, ( z )~k (ds,dz). Hence

sot/\,

;Jk,

;SkT

~ [ l x t+l ~ 19s12ds + SZ Ips(z)I24dz)dsI < E 1x1' E J & , ( ~ c I ( s+) ~ c z ( s ) 'lxs12 ) ds cl(s)ds, + E J k T C I ( S ) lxS12ds + 2E where we have used the result that 2 J;, c ~ ( slxsl ) d~ < J;, c1(s)1xs12ds + c~(s)~s.

+

JkT

Sk,

x t as t x,, as t

[O,T],

> 7.

i JkT

i

E[lutli + lqs12ds + SI,, Sz IpS(4l24dz)dsl < k& e ( 3 c l ( s ) 4 c ~ ( s ) ~ ) ~ l u ~ l ~ d s , where c1(s)ds. k&= E 1x1'+ 2 By Gronwall's inequality one easily finds that

+

+

Lemma 230 Under the assumption in Lemma 229, i f ("1 - 2 2 ) ' (b(t,Xl,Qlr P ~ W) b(t,x2)92,P2,~)) c i ( t ) p ( l x~ x212) ~ 2 ( t 1x1 ) - 2 2 1 (191 - 921 ((pi- ~ 2 1 1 ) 1 , where ci(t),i = 1,2, satisfy the same condition as that in Lemma 229, and p(u) 2 0, as u 2 0 is non-random, increasing, continuous and concave such that So+ dulp(u>= w then the solution of (8.8) is unique.

<

Proof. Let

+

+

8.3 Existence of Solutions with Bounded Stopping Times

215

x t = x ; - x , , 2 Qt=qtl-q;, pt=p;-p;. Then as in (3.2) by Ito's formula one finds that zt = E[lxt12 J,',, 1 ~ ds ~ SAT 1 JZ~ I P ~ ( Zr(dz)ds] )I~ E kc1 (s)p (IXSI') 4c2(s)' IXS12] ds

+i + < h,: + < ~J?(CI(S) + C ~ ( S ) ~(lzs12) ) P ~ ds

+

where we define Zs = 0, for s > T, and pl(u) = p(u) u.Hence by Lemma 235 below P - a s . Zt =O,WE [O,r]. Unlike the SDE theory, an interesting thing is that for BSDE we have the following result: If the terminal variable X is bounded, then the solution is always bounded.

Lemma 231 Assume that loin Lemma 229 holds, T a constant ko 2 0 such that 1x1< ko, P - a.s. If ( s t , qt,pt) is a solution of (8.8) , then P - a.s. lxtl < No,vt 't [ o , ~, W ] E 0, where No 2 0 is a constant depending on cl(t)dt, only.

< To,and there exists

fl( ~ ~ ( dtt ) and ) ~ ko

Proof. Similar to (8.12) one obtains lxt12 E ~ ~ [ I X ~ J,' I ~ ,, lqSl2 d~ J;, J~I P ~ ( ZT)(I~~Z ) ~ S I [ ~ 51x1~ t + C1(S)dS]eS,TO(3~ds)+4~z(s)z)ds, where we write ESt [.] = E [. IZt]. Hence the conclusion now follows.

<

<

+;

+;

Remark 232 By the proof one sees that the condition in Lemma 231 can be weakened to E5t 1x1~ 5 ko,Vt E [O,T].

8.3.3

Existence of Solutions for the Lipschitzian Case

In this section we are going to discuss the existence of a solution to BSDE (8.8) . As usual, to show the existence of a solution for some equation, if the coefficient in this equation is Lipschitzian, one can use the contraction principle. However, to let the contraction principle work, we first need to discuss the following simple BSDE:

We have

8. Option Pricing in a Financial Market and BSDE

216

zT-

Lemma 233 . If T is a bounded &- stopping time, X is measurable and Rd- valued, f ( t ,w ) is St- adapted and Rd- valued such that 2 E 1x1~ < 00, E 11; I f ( s ,w>Idsl < m, then (8.13) has a unique solution. Proof. Let xt = Egt ( X f (s)~s). This makes sense, since by assumption E5' f (s)ds12)< m , V t Z 0. Hence xo = E g O ( X+ f (s)ds). Notice that Mt = E g t ( X + f (s)ds) is a square integrable martingale. Hence by the martingale representation theorem (see Theorem 103) there exists a Z ) that unique (qt,pt) E L;(Rdmd1)x F ; ( R ~ @ ~such

+ Jk,

( I x + Jk,

However M t ~ 7= EZth7(X J: f ( s ) ~ s=) xt and by (8.14) X f f (s)ds = XO qsdws Hence

+

+

1;

X

Jr

+ f (s)~s, + SZ ps(z)Nk(ds,d ~ ) . - ./k7JZ PS ( r ) ~(ds, k dz)

+ Jk7f (3, W ) ~ -S Jh7 + lotA7 qsdws + JZ pS (z)fik(ds,d z ) - ~7 f (s)ds tA7

sotA7

Q S ~ W S

= x0

= MtAT - JO f ( s ) ~=s xt. We have now that ( x t ,qt,pt) satisfies (8.13). Now by (8.13)

Esupt,[o.,

< 00.

lxt12

< ~ [11x31 + 1s: l f (s)l dsl12 + 6J:(lqSl2 + I I ~ ~ 1 1 ~ ) d s l

Hence xt E S$( R d ) .Therefore ( x t,qt ,pt ) is a solution of (8.13) . The uniqueness of the solution to (8.13) follows from Ito's formula (Theorem 93).Lab* m Now we can show an existence and uniqueness result for the solution to (8.8) by using the contraction principle.

Theorem 234 Assume that T Q To lo b(t, x , q, p, w ) : [0,To]x Rd x RdBd1 x L 2 ( ~ d @ dx2M) e--t Rd adapted and measurable process such that P - a s . is a Ib(t,x,q,p,w)l < ~ l ( t ) ( l + 1x1) + c2(t)(l + I91 + llpll), where c l ( t ) and ~ ( tare ) non-negative and non-random such that cl (t)dt ~ 2 ( t ) ~
st-

+ ~2

8.3 Existence of Solutions with Bounded Stopping Times

217

Proof. Let us use the contractive mapping principle t o show this result. Introduce a new norm as follows: xF ~ $~()R ~ @ ' let ~~) For (x., q., p.) E B = L;(Rd) x L $ ( R ~ @ 2 Il (x.,q., P.)II& = sup,,^, c b A @ ) EI x ~ A , ~ e-bA(t)EJ&, lqt12dt + SUP,,T~ e - b A ( t ) ~ J kllpt112 T dt, where b 2 0 is a constant, which will be determined later, and A ( t ) = J: (cl(s) c2(s)')ds. Write

+

+

H = { ( x . , qv1p.)E : l l ( ~ . , q., p.)llM < m, . Then H is a Banach space. Now by Lemma 233 for any ($.,$.,pi.) E H, i = 1,2, there exist unique solutions (xi., qi., pi.) ,i = 1,2, of the following BSDE xi = X b ( s , % , $ , g , w)ds q:dwS - J ~ , J Z p ~ ( z ) f i ~ ( d s , dOz( ) ,t , i = 1,2; moreover, by Lemma 229 (xi., qi.,pi.) E H, i = 1,2. Let Xt=xZ-x:,-QtZqt1-Qt2, Pt=pZ-pt-1 Similarly, define X t , Q,, and Pt. Then by Ito's formula as in (3.2) one gets that ~ [ l x +t l ~ 1&sI2 ds + l l ~ s l l ~ d ~ l (y l ~ [ J cl(s) ;, ds + J;, I V , Ids ~+ / 1 ~ 1 / ~ d s 1 +~EJ;,(cI(s) + C Z ( ~ ) I~X) , I ds ~ = y-l1,1+ c 2 ( ~ ) ' lxs12 ) d~ G y-l1,1+ ~ J ? ( c I ( s ) C Z ( S ) ~I )XE, A ~ I ~ ~ S exp(y(A(t)- A(s)))(cl(s) ~ z ( s ) ~ ) I j d s , < y-'I,1 where we have applied Lemma 235 below. Notice that 0 ( A(t) is decreasing, so e-bA(t)E J k T cl ( s ) ds ( e-bA(t) cl ((sE I X , ~ , ds $, cl(s)ds = up,^^^ ecbA(')E G S~P,,T, e - b A ( s )1fisA,l2 ~

+

&,

Jk,

Jk, 1 ~ 1 ~ YEJL,(C~(S) +

+

~k~

+

+

lxsl2

JT

L~~

+

I

l ~ ~ ~ , l ~

Ix,-[~

e-bA(t) exp(y(A(t)- A(s)))(cl(s) c ~ ( s ) ~Js:T) E cl(r) drds ( e-(b-l)(A(t)-A(s))(~I(Q c )~ ( s ) ~ ) ~ s e - b A ( s )lFSA, ~

R

+

'1

e - b A ( s )~~~ i , ~ , l ~ Z oy)-', ( b for0 < y < b. 2 Now write ut = E J;, 1Qsl2ds or E Il'?jgll ds. Then eFbA(,) exp(y(A(t)- A(s)))(cl(s) c ~ ( s ) ~ ) u ( s ) d s e-"A(")U(s) e-(b-l)(A(t) (cl -A ( s()+ s )c)~ ( s ) ~ ) ~ s ( SUPSGT, ( SUP,
'7

JLT +

~7

+

Lemma 235 . (Gronwall's inequality). If0 6 yt ( y v t + ~ 2c(s)y,ds,Vt 2 0, where. y > 0 is a constant, and c(s) 2 0, then

8. Option Pricing in a Financial Market and BSDE

218 ?/t

< yvt + y

Proof. Let 0

exp (J: c(r)dr)c(s)vSds.

< zt = J?

c(s)y,ds, gt = yt - yvt - zt

+ +

< 0. T h e n

-zt = c(t)yt = c(t) (gt yvt zt) . From this one gets To zt = St exp (J: c(r)dr)c ( s ) (g, yv,) ds < exp (S,8 c(r)dr)c(s)yv,ds. Hence ~t < yvt zt Q yvt y exp (J: c(r)dr)c(s)v,ds. Now let us give some examples t o show that the conditions on c l ( t ) and c2(t) in Theorem 234 cannot be weakened.

+

~7

+ ~7

+

.

cl(t)dt < co cannot be weakened). ConExample 236 . (Condition sider the following BSDE 0 t T , T T T Xt =1 Is#~s-"xsds - St qsdws SZ ~ s ( z ) ~ k ( dz). ds, Obviously, i f a < 1, then by Theorem 234 it has a unique solution ( x t ,0 , O ) , where xt is non-random. However, if a 2 1, it has no solution. Otherwise, for the solution (xt,qt,pt) one has EX: = 1 + J,T I , # ~ S - ~ E X=~eJtTIa#os-"ds,~i ~S = I , 2,. . . ,d. Hence E x ; = m , V i = 1,2,... ,d, as a 2 1, which is a contradiction.

< <

+ St

~2

Example 237 (Condition ~ ~ ( t ) <~ CQ d t cannot be weakened). Now suppose that all processes n BSDE (8.8) are real-valued. Let X = ISZo(l ~ ) - ~ / ~ ( l o g s))-"'dw, (l+

+

, + & IS#0(1 + ~ ) - l / ~ ( l o g ( ls+) ) - I~U (~Z ) $ ~ ( ~ S dz), 6 = ISjo_(l+ + s ) ) - ~ ' ( &Id+ k Sz lp(z)I Iu(z)*(dz)), where El, kg 2 0 are constants, and we assume that 0 < a3 < 4, and 0 < K ( U )< Obviously, if 0 < a1 < a, then Theorem 234 applies, and (8.8) has a unique solution. However, if a1 > 4, and kl > 0 , a l + 2 1, a2,

CQ.

a2

z2

or > 0 , a1 + a3 2 1, then (8.8) has no solution. Otherwise, the solution ( x t ,qt ,pt ) should satisfy xt.t = SOtIs# O ( l s)-1/2(log(l s))-~"w, JotJz I,#o(l+ s)-lI2(log(l+ s ) ) - ~~u~( z ) G k ( d ds z, ) I ~ # ~ ( ~ + S )( -l ~ og(1+~))-(a~~a2)+~2(1~g(1+~))~~a~~a~~~(~)]d qs = IS#0(l ~ ) - l / ~ ( l o g ( l3))+ , p,(z) = IS#0(l ~ ) - l / ~ ( l o g ( l + I&). Hence one has that Exo = I , # o ( ~ +~)-~@1(10g(1+ S))-(~I+"~) +z2(log(l S ) ) - ( " ~ + " ~ ) T ( U ) ] ~ S= co,

+

+

+

+

+

+

[&

+

8.4 Explanation of the Solution of BSDE to Option Pricing

-

in the case kl contradiction.

> 0,al

+a 2

-

2 1, or k2 > 0 , a l

+ as

219

2 1, which is a

8.4 Explanation of the Solution of BSDE to Option Pricing 8.4.1

Continuous Case

1. Option pricing in the continuous case. Now suppose that in a financial market there is a bond and only one stock, and this stock is only disturbed by a continuou stochastic perturbation; that is, it is a 1-dimensional BM. Now we are going to price an option related to this stock. As in the previous sections we know that xt (the price for the option) with n: (the portfolio) should satisfy the following BSDE in (8.6):

Now let qt = r t a t . Then (8.15) becomes dxt = rtxtdt qtu,l(bt - rt)dt qtdwt, XT = (PT - K)+, 0 5 t T. By Theorem 234 this BSDE has a unique solution (xt,qt),O 5 t 5 T, provided the following conditions hold: (A) rt, bt, and at are all non-random such that a,' exists and g[lrtl+ la;'(bt - rt) ]dt < m ; moreover, E[(PT - K)+] < co. Hence (8.15) also has a unique solution (xt,r:),O 5 t 5 T, where nt = qta;', provided assumption (A) holds. A particular case for assumption (A) to be true is that rt, bt, and a,' are all non-random and bounded. This means precisely that under assumption (A) the option can be priced as xt. The same idea can be applied to use the BSDE solution to hedge the contingent claim. 2. Hedging contingent claims in the continuous case. For simplicity, as in the Introduction of this chapter, suppose that in a financial market there is a bond, its price equation is (8.1), and there are two kinds of stocks disturbed by two independent contiuous stochastic perturbations, with their price SDEs described by (8.2) (that is, N = 2, and d = 2 in (8.2)): dPi(t, w) = Pi(t, w) [(bi(t,w)dt oilc(t, w)dwt], Pi(O,w)=Pi(O,w), i = 1 , 2 ; where wt = (w,', w : ) ~ is a %-dimensional BM. Then by means of a d e r i m tion similar to that in (8.6), hedging a contingent claim X E will lead

{

+

<

+

P

+ c:,~

220

8. Option Pricing in a Financial blarlet and BSDE

to finding a 5y-adapted solution ( x t ,n t ) E R1 x R2 (that is: r1= ( n : ,n ; ) ) solving the following BSDE

where I= (1,1)* is a 2-dimensional constant vector, and

4=

(g)

E ~ ~ g ~ , aI' ~ d u22 =2

( ~ j ,

)

E~2&2.

<

In other words. if the solution ( z t ,nt),O 5 t T of (8.16)exists. then it can reproduce X. Now let qt = rtut. Then (8.16)becomes dxt = rtxtdt qtutl(bt- r&dt qtdwt, xT=X, OLt
+

{

+

+

2

s:[lrtl lnt1(bt- rtl)l ldt < 03; moreover. E 1x1~ < 00. Hence (8.16) also has a unique solution ( z t , n t ) E R1 x R2,0 5 t 5 T , where nt = qtu;', provided that assumption ( A ) holds. Notice that t are not real numbers. A particular caw. when here bt E ~ ~ @ l E, aR262 assumption ( A ) is true. is when r t ,bt, and a;' are all non-random and bounded. This means precisely that under assumption ( A ) the claim can be priced as xt, and can be hedged by ( x t ,x t ) ,0 5 t 5 T . The conclusion is easily generalized to the case N = d > 2. Ia a financial market. if every contingent claim can be hrdgetl. (that is. can be reproduced). tlieli this lnarket is called a conipletr (fi1tancia1) market. By the discussion ~ b o v eone sees that a necessary condition for a col~tinuousn~arketto be complete is that N 2 d, that is. thv ~ir~~ttl)c~r of kinds of stocks should at least bc equal to the number of conti11uous independent stochastic perturbations. (That is. the dimension of thc B l l ) . Otherwise. a;' does not exist. and we cannot transform (8.16)to a solvabl~ BSDE as here.

8.4.2 Discontinuous Case 1. Option pricing in markets with jumps. It is easily imagined that in a financial market on many occasions the stochastic perturbation will also have discontinuities. So we nerd to discuss a market with jumps. Suppose that in a market there is a bond, and that its price equation is still (8.1).and there are two kinds of stocks. each disturlwtl by a contiuous stocliastic perturbation (i.e, a BM) and a jump stochastic perturbation (i.e. a centralized Poisson process). where their price SDEs are described as follows: dPZ(t, w) = P y t , w)[(b"t,w)dt a 2 ( tw)dwt , p2(t,w)dGt],

+

+

8.4 Explanation of the Solution of BSDE to Option Pricing

221

Pi(O,w) = Pi(O,w), i = 1,2; is a centralized Poisson process, where wt is a 1-dimensional BM, and which cag be seen as Nt = Nk((0, t], 2 ) = Nk((0, t], 2 ) - t r ( 2 ) = Nk((O,t], 2 ) - t, where k(.) is a 1-dimensional Poisson point process, and for simplicity we assume that r ( Z ) = 1. Recall that if k(-) is a 1-dimensional Poisson point process, then Nk((O,t], 2 ) is a Poisson process with the intensity function ENk((O, t], 2 ) = tn(Z) = t. Here the term "centralized" will mean to make it become a martingale process. Based on these ideas we see that the above stock price SDE can actually be rewritten as dPi(t, w) = Pi(t, w)[(bi(t, w)dt ai(t, w)dwt pi(t, w) JZ zk(dt, dz)], Pi(O,w)=Pi(O,w), i = 1 , 2 . So it is a special case of our SDE with jumps. (See (3.1)). Again by means of a similar derivation, in a self-financed market, pricing an option of the first stock will lead to the solution of the following BSDE

zt

+

dxt = rtxtdt

+

+ nt(bt - rtl)dt + ~ t a t d t u +t rtptdfit,

XT

= (P; - K)',

t E [0,TI,

(8.17)

where I=(1,1)* is a 2-dimensional constant vector, bt =

(2

),at=

(2

),pt=

(2)

ER'"',

and rt = ( r i ,R:). NOWlet qt = Ttat, pt = rtpt. Then (8.17) becomes dxt = rtztdt (qt pt) 5,l(bt - rt1)dt qtdwt ptdfit, XT = (P&- K)+, 0 I: t S T . where

+

+

( ;I ) 1

.=(ot,pt)

=

+

1

-

Since by Theorem 234 this BSDE has a unique s l N - a d a p t e d solution (xt,qt,pt) E R1 x R1 x R1,O t T, provided the following conditions hold: (A) rt, bt, at and pt are all non-random and such that 5,' exists, and

< <

2

15r1(bt - rtl)l ldt < m; and, moreover, E[(P& - K)+I2 < w. Hence (8.17) also has a unique solution (xt,nt) E R1 x R2,0 t 5 T, where r t = (qt, pt) Z,', and (qt,pt) E R1@ 2,' provided assumption (A) holds. Notice that here bt, at,pt E R2@l,SO they are not real numbers. A particular case in which assumption (A) is true is that r t , bt, at1 and pt are all non-random and bounded. This means that under assumption (A) the option of the first stock can be priced as xt. However, in a financial market with jumps, unlike the continuous case, the option price xt actually depends on the prices of two stocks, because in the BSDE (8.17) for the option price b:, a:, and p: are involved. Moreover, if there is only one stock and one bond in the financial market with jumps, then in general J?[l~tl+

222

8. Option Pricing in a Financial Market and BSDE

the option for this stock cannot be priced by this way because, in general, qt = IT,1et,pt = xtPt has no solution for x i . 2. Hedging contingent claims in a market with jumps. Now suppose that in a market there is a bond, its price equation is still (8.1), and there are three kinds of stocks disturbed by a contiuous and two independent jump stochastic perturbations, and that their price SDEs are described as follows: dPi(t, w) = Pi(t, w)[(bi(t, w)dt+ci(t, w)dwt+pf (t, w)d&'+&(t, w)dfi:], Pi(O, w) = Pi(O, w ) , i = 1,2,3; where wt is a 1-dimensional BM, and fit = ( f i i , 2;) is a 2-dimensional centralized Poisson process with the same density function E@ = t; that is, for each i, (i = 1,2), fil is a Poisson process with the same density function t, and f i i , t 2 0 is independent of fi;, t 2 0; moreover, they have no common jump times, that is, Afi; . Afi: = 0,Vt 2 0. For a given contingent claim X E $$' in this market, if the market is self-financed and we want to hedge it, then this will lead to the solution of the following BSDE

where I= (1,1,1)* is a 3-dimensional constant vector,

and xt = (xi, x:, x:), and (xt,xt) E 3 r , N . NOWlet qt = xtct,pt = ntpt. Then (8.17) becomes dxt = rtxtdt (qt pt) 5t1(bt - rt1)dt qtdwt ptdfit, xT=X, O < t < T .

+

+

+

Since by Theorem 234 this BSDE has a unique $""-adapted solution T, provided the following conditions (xt,qt,pt) E R1 x R1 x R2,0 t hold: ( A ) rt, bt, at and pt are all non-random such that 5,' exists, and

< <

< m. mooreover, E 1x1~ Hence (8.18) also has a unique ~ r ' ~ - a d a ~ tsolution ed (xt,xt) E R1 x R3, 0 t T, where xt = (qt pt) and (qt ~ t =) (qt,plt,p2t) E RlB3,

< <

z,',

8.5 Black-Scholes Formula for Option Pricing. Two Approaches

223

provided assumption ( A ) holds. Notice here that bt, ct E R3@',pt E R3@2 are not real numbers. A particular case in which assumption ( A ) is true is that rt, bt, CTand ,' pt are all non-random and bounded. This means that under assumption ( A ) the contingent claim can be hedged by ( x t ,qt,pt). By the same token one easily obtains the following theorem. Theorem 238 In a self-financed market, suppose that there is one bond, whose price satisfies (8.1), and that there exist N-stocks, whose prices satisfy the following SDEs, respectively:

where wt = (w,', . - - ,w$)is a d-dimensional BM, and fit = (fii,. . . ,fitm) is a m-dimensional centralized Poisson process with the same density func-. tion EN," = t , i = I , . . . ,m. Assume that (A1) N = d m, and rt, bt,at and pt are all non-random such that Zt' exists, and

+

2

~ ? ~ l r t+l l ~ ; ' ( b t - r t ~ Ildt < m, where gt E R ~ @ ER - ~~ ,@ ~= ~ (ut ~, zpt)~ E R~ @ ( d + ~ ) ; moreover, X E 3YtNand E 1x1~ < co. Then the following BSDE has a unique 3r'N-adapted solution ( x t ,r t ) E RI x R ~ , OI I T ,~ dxt = rtxtdt + nt(bt - rtl)dt + rtatdwt ntpdfit, xT=X, OItIT. Moreover, ( x t ,n t ) ,0 I t I T , hedges the contingent claim X E &.

+

{

Obviously the above theorem is very general, and it includes all previous results in the financial market obtained previously in this book. For example, set d = l , m = O , X = (PT- K)+, then we get the option pricing result for the continuous case at the beginning of this section.

8.5 Black-Scholes Formula for Option Pricing. Two Approaches In this section we are going to price an option using a probaility approach, and also by the partial differential equation (PDE) technique to establish

224

8. Option Pricing in a Financial Market and BSDE

the famous Black-Scholes formula. First, we see that the stock price SDE (8.2) for i = 1, N = d = 1 equals d P l = P$-[rtdt at(acl(bt - rt)dt dwt)],P,1 = P,1; P - a.s. and the SDE (8.16) for hedging the contingent claim X equals the dxt = rtxtdt xtat(a,l(bt - rt)dt + dwt),zT = X , P - a.s. If we take X = (PT - K)+, then it becomes the option price SDE. Hence, applying Girsanov's theorem (Theorem 124) we immediately obtain the following lemma.

+

+

+

Lemma 239 Assume that (B) a;' exists, and that there exists a constant ko > 0 such that (a,l(bs - rs)l 5 ko Then 1) the stock price SDE can be rewritten as: FT - a s .

and the - SDE for hedging the contingent claim X also can be rewritten to be a PT - a.s. dxt = rtxtdt ntatdGt, (8.21) X T = X , O
+

t

P:

= P,' exp[J r,ds 0

+J

0

t

&

t

o,dG -

lo. l2 ds],

3) Since the bond price equation is non-random dPf = Pfrtdt, P: = 1, where for simplicity we take the initial value equal to one unit, so that he solution oft the bond price equation is pf = Jo r e d s . 4) If the contingent claim is X E ZT, then by (8.22) the price of this contingent claim xt is xt = E& [ X exp[rsds]lZt]. This pricing formula, even though is theoretical, has a nice physical meaning: it simply says that i f we want the contingent claim to be X at the fwture time T , at time t we can expect (that is, we take the conditional rexT pectation) that deposite money X exp[- St rsds] will produce the claim X at time T.

8.5 Black-Scholes Formula for Option Pricing. Two Approaches

225

Proof. 3) is obvious. 4) is true by taking the conditional expectation under the probability measure FTon both sides of (8.22). So we only need to prove 1) and 2). However, 1) follows immediately from Girsanov's theorem (Theorem 124 or Theorem 191). 2): Applying Ito's formula to (8.22) one sees that the @-adapted processes P; and xt satisfy (8.20) and (8.21), respectively. w Now we are going to establish the famous Black-Scholes formula for the price of option related to the one stock price Pt ,t E [0,TI. First, we will use the probability approach. One will see that this approach has more proba bility meaning (hence also more physical meaning). From this approach we can guess the result; that is, the formula, can actually be constructed. Theorem 240 (Black-Scholes formula). Assume that r, b, a are all constants, then the option price related to the first stock is +m a ) -K~-'(~-~)N(~~(P/)), xt = P:N(do(P:)

J2

1

-hd

N(x) = d m e YThis can also be written in the more symetric form: - ~( ~) - ( T - t, P;)), xt = P:N(p+(T - t, P t ) ) - ~ e - " ( ~ N where -n-I-[ ( r -t )-)I. p * ( ~- t, P:) =&

$+

Proof. By (8.22) Pi = exp[Jot rsds P$ = PJ exp[Cr,ds

Pt

f

+ JotTa,d& - $ laSl2 ds], + Jo ~ s d G -s $SoT I U ~ I ~ ~ S ] .

and noticing that r, b, a are all constants, one finds that -PT- a.s.Vt q, E [0,TI,

Evaluating

pt

Hence one sees that the price of the option is (Write h(P$) = (P$ - K)+) xt = ~ ~ ( e - " ( ~ - ~ ) h ( ~ + ) 1 ~ t ) = E"(~-'(T-~) h ( ~ + ) l & )

- e-'(T-t) -

J

R

,h(ptleIoV-*(IoIa-")(T-t)I)

Ae hdY

- e-~(T--t)JR1 h ( ~ : ~ [ o ~ ~ - + ( 1 ~ 1 ~ - ~ ) ( ~ - t&.I ) l ) - l - ~ -

./z;; "y, where we have applied the facts that St = and Gt, t E [0,TI, is a stationary Markov process such that GT - Gt N(0, ( T - t)). That is, GT - Gt is a Normally distributed random variable with the mean 0 and variance T - t, because Gt, t E [0,TI, is also a BM under the probability PT (See Corollary 102 in the First Part). Hence we have found a formula

z,

-

226

8. Option Pricing in a Financial Market and BSDE

for the evaluation of the conditional expectation:

Actually this formula is also true for any Bore1 measurable function h(.). Notice that P$ > K is equivalent to -(GT-Gt)<-ln$--$la] 2 (T-t)+r(T-t)

AT

= (-a& ln - 40dFi)c= d0(p:)~-. So E = {P; > K ) = a m < do(P:)), where &(pi)= ln - $0dT7. Thus the -option price is xt = - (P$ - K)+ = e - r ( T - t ) ~ P [ ~ $I & ~ ] E- Ke-r(Tpt)~'[I Notice that by (8.24)

{--

Ist)

la

lEJ2 03 E ' [ I E I ~= ~ JRl I ~y>-ciO(P~)&e 2 dy = J-&(p;) &e-$dy - do(^,') 1-J.d - Lm &$ dY = N(do(P:)), where as usual we write the standard Normal distribution function as

-*

J 2

dy. N(x)= & q e 2 On the other hand, again by (8.24) e-r(T-t)E'[P$IE

1st]

- e-r(T-t)

J R ~Pt

1 [amy-i(la12-r)(T-t)] I

e

1

-ld

y>-do(P,')xe By the technique of completing the square one finds that o m Y - +(lot2- r)(T - t ) = - i ( y - -o)~ + ( ( T - t ) 1012- $(lot2- r ) ( T - t ) = - i ( y - m o ) ? f H r ( T- t ) . Hence let y = y - -0,

qI

d ~ .

e-r(T-t)~'[~$= ~ ~JRll ~Ptl1 x 1e -GC4(~;)+madG = P: ~ ( d ~ ( + ~ , 1 ) Therefore, xt = P,'N(do(P$) -0) -K~-'(~-~)N(~~(P$)), and the Black-Scholes formula is proved. This can also be written in the more symetric form. do(^:) = ln - $0-

mO). +

- l0-p n $ + ( r -

do(^/) + -o=

$)-)I -&[ln$

=p(~-t,~i), + (r - $)&Ti)]

8.5 Black-Scholes Formula for Option Pricing. Two Approaches

227

$+ + f

+a =& -nl[ (r )-I = p+(T - t, P)). Thus we also have that the option price equals ( ~-- t, P,")). w xt = Pi N(p+(T - t, P l ) ) - K ~ - " ( ~ - ~ ) N(T Now we are going to establish the Black-Scholes formula using the partial differential equation (PDE) approach. For this we will first use 60's formula and the theory of BSDE to establish the relation between the solution of some PDE and the price of a more general contingent claim in a selffinanced market. Suppose that at is non-random, and X = g(P$). If we introduce a function u(t, y) E C112(R+x R1), and we require that u(t, P)) = xt, then by applying Ito's formula to u(t, P)) on t E [O,T] with (8.20) one easily derives the following lemma.

Lemma 241 Under assumption (B) in Lemma 239 and the condition that Sotlrtl dt < co, if u(t, y) E C1*2([0,T)x R1) satisfies the following partial differential equation

2

and g(y) is continuous such that E Ig(p$)I < co, then u(t, P)) = xt and XT = u(T, P;) = g(P$) holds true, where g(P4) E 8; is a contingent claim and xt is its price at time t. Moreover, if we let .rrt = %(t,P:)Pl, then , ), (t, P))Ptl), t E [0,t], hedges the contingent claim (xt ,.rrt) = ( ~ ( tPl g(P$).

%

Proof. Applying Ito's formula to u(t, P)) on t E [0,TI, where Pi satisfies (8.20), one finds that FT - a.s.W E [O,T], T au u(t, P i ) = u(T, P;) - [&, P,l)f &s, P:)P:G

St

+

,

2

w ]~d s - ~ ~ ~ ( s , ~ P : ) ~ : ~ s d u ? , - :J %(s, PP:)P:osdu?.. On the other hand, the wealth process xt (or say, the price for the contingent claim X = g(P;)) satisfies the BSDE: FT- a s . xt = g ( P ~ ) - ~ r , z , d s - ~ ~ . r r , a , d G ~E, V[O,T]. t However, the above BSDE has a unique @-adapted solution (xt, qt) = (xt, .rrtat). Hence by the uniqueness of the solution of a BSDE, comparing the two BSDEs one finds that xt = u(t,P,1),7rt = %(~,P))P). In particular, XT = u(T, P;) = g(P;) holds. rn This lemma tells us that the option pricing problem can be solved by the PDE technique. Actually we can obtain the Black-Scholes formula by this approach. However, from such a PDE approach it is not easy to guess the formula, though we can always verify whether a given formula is true. = g(P$) -

I

aZu

StT rsu(s, P:)ds

228

8. Option Pricing in a Financial Market and BSDE

Theorem 242 Assume that r > 0, b, a are all constants, and g(y) = ( y K)+,y 2 0. Let u(t,y) = BS(t,T,y, K , a , r ) , as 0 t < T ,y 2 0; and u(t,y ) = g(y), as t = T ,y 0; where BS(t,T,y,K,a,r) = yN(p+(T - t , ~ )-)K ~ - ' ( ~ - ~ ) N ( ~ t-,(yT )), and p*(t, y ) = -&[log j$ t(r f )I, N ( y ) = J_Y_e - ~ ' / ~ d z . Then u(t,y) E C1r2([0,T)x (0, co)) and satisfies (8.21). Furthermore, u(t,P i ) is the price of the option, and $ ( t , P,')P,' is the portfolio such that (u(t,P,'), e ( t ,Pi)P,') can duplicate the money (P: - K)+ which the option owner can earn at the future time T . Such an explicit formula ~ ( P,') t , = BS(t,T ,P i , K , a , r ) for the option price is called the BlackScholes formula.

<

>

+

f

-&

Proof. It is a rutine matter to show that u(t,y ) E C1,2([0, T )x ( 0 , ~ ) ) and that it satisfies (8.21).The reader can verify those results for u(t,y ) by himself.Now let us check the terminal condition u(T,y ) = ( y - K)+,Vy > 0 also holds true. In fact,

Obviously, as t 1 T,O 5 If 5 K ( l - e-T(T-t))-, 0. Observe that in the expression for p+(T-t, y), if y > K then as t T,p+(T-t, y ) -,co; on the other hand, if y 5 K, then as t 1T ,p+(T-t, y) -, -co. Therefore,as t 1T , I: -+ ( y -K)+. The terminal condition holds for u(t,y ) . Finally, let us show that the proof of Lemma 241 goes through. In fact, notice that by definition u(t,O) = 0. So &(t,O) = 0. Now applying Ito's formula to u(t,P,') on t E [O,T],where P,' satisfies (8.20),one finds that FT - a.s.W E [O,T], Bu u(t,P,') = u(T,P$) - J,T Iz(s, P i ) + $(s, p,1)p,1rS las121p:l2 @ ] ~ p : & - :J e(s,p:)p:aSd"', = g(P$) r,u(s, ~:)lp:+ods- :J $(s, P:)P:~,~G,

++

JT

=g ( ~ $ ) :J rsU(s, p:)dS - :J %(s, P:)P:c,~~?,. So the proof of Lemma 241 still goes through. The Black-Scholes formula is easily evaluated pratically, because in Statistics books there are tables for the values of N ( x ) . Furthermore, the Black-Scholes formula can also be converted into a computer program, so the option price can be found immediately from the computer, if it is provided with the necessary data.

8.6 Black-Scholes Formula for Markets with Jumps

229

8.6 Black-Scholes Formula for Markets with Jumps Since financial markets with jumps are more realistic, in this section we will briefly discuss how to obtain a Black-Scholes formula for option pricing in such markets. Recall that from Theorem 238 we have considered a self-financed market. There is a bond, whose price satisfies the bond price equation:

where r(t) is the rate function, and there are N different kinds of stocks, with their prices satisfying the following general stochastic differential equations:

where wt = (wf,w:, . . . ,w:)~ is a standard d-dimensional Brownian Motion process, Nt = (N), - . . ,i@)T is a m-dimensional centralized Poisson process, i.e. all components are indzpendent and any two components have non-common jumps, such that dN," = dN," - dt, and 0 = (aik)F;il E

-

-

-

RNmd,p = (Pilk)t& E RNmm.In such a market if X E TT = g'N is a contingent claim, and (xt, xt), t E [O,T] can hedge this contingent claim, then they should satisfy:

where i= (1,. - . , l)T is a N-dimensional constant vector, and xt is called the price of the contingent claim X, (it is also called a wealth process in many cases), and xt = (xi . . . ,x y ) is called a portfolio, (actually it is the risk part of the portfolio). According to Theorem 238, under the condition (Al), (8.28) has a unique CfN-adapted solution (xt,nt). Now let [qt,pt] = xt[at, pt] = mat, d where qt = (q,',-,qt),pt = (pZ,.-,pZn). If 5,' = [ ~ ~ , ~ , ] - ~ e x i(for s t ssimplicity, , to ensure this, let us assume that N = d + m), then (8.28) can be rewritten as

230

8. Option Pricing in a Financial Market and BSDE

where we set

with 0: E R ~ @0"8, E RmQ1.Then the existence of the solution ( x t ,qt,pt) for BSDE (8.29),or, more precisely, the existence of a solution ( x t ,n t ) for BSDE (8.28) will help the investor to make his wealth XT arrive at X at the future time T (or say XT can duplicate the contingent claim X ) , if at time t by using the portfolio nt he invests the money xt. Moreover, if one calls X the value of some contingent claim, then the existence of xt also shows that one can give the contingent claim a price xt at time t. W e can give another existence theorem (the sufficient conditions are a little different from Theorem 238) for BSDE (8.29) and BSDE (8.28) as follows (its proof is similar to that of Theorem 238):

-

Theorem 243 .[l64 Suppose that b, r, a , p are all bounded ~ ' ~ - ~ r e d i c t a b l e processes and -5,' exists and is also bounded, and assume that N = d + m , X E g;'* and E 1x1~ < oo. Then (8.29) has a unique solution (xt,qt ,pt) E Lg ( R 1 )x Lg (RIBd)x Fg (RIBrn),and there also exist a unique wealth process xt E Lg(R1) and a unique portfolio process nt E L g ( R N ) satisfying (8.28), where . --1 xt = xt, nf = q:(~:l)ji ~ y = ~ )(j+d)& g ( ~i =~1,.. . ,N .

c;=,

+

Recall that i f the contingent claim X can be duplicated by ( x t ,~ t (i.e. ) putting such ( z t ,nt) into the market, when t evolves to time T , then XT = X ) , then we will say that ( x t , n t ) hedges the contingent claim X . So we see that under the conditions of Theorem 243 any contingent claim can be hedged. Such a market can be called a complete market. Now we will use the Girsanov type thoerm to simplify these equations: (8.27) and (8.28). First we know that (8.27) also can be rewritten simply as dPt Po

= Pt-[rtdt =

Po,

+ (bt - rtl)dt + atdwt + ptdfit],

(8.31)

t€[O,Tl,

where Pt E RN,rt E R1,at E E RNBrn.So we can establish the following theorem by using the Girsanov type transformation.

Theorem 244 Under the assumptions of the above theorem 2J in addition, = -(Bt iG. )' > -1,Vi = I , . . . , m ; then 1) the stock price SDE can be rewritten as: FT - a.s. dPt Po

= Pt[rtdt =

+ atdGt + p t d r t ] ,

Po, O < t < T ;

8.6 Black-Scholes Formula for Markets with Jumps

231

and - the SDE for hedging the contingent claim X also can be rewritten as PT - a s . dxt = rtxtdt rtatdGt r t p t d T t , XT=X, OstST. (8.33)

+

+

whzre we write p t , ~= exp[(e;, dw,) le:12 ds - ( ~ , , 1ds]. ) Ht<.s<Ti1 + ( P S ,A N S ) ) d P ? and PT = PO,T is a new probability measure. Moreover, where Gt and are a new BM and a new centralized pure jump process, respectively, such that e:= W: s,'(e:)%s, i = 1,. - . ,d; -. t fii -. N t = N , 2 + S O ( e s ) d s = ~ , 2 - S i p $ d ~ , i = 1 , . ,m; .. and is with the new compensator $ ( I + p$)ds. 2) PT - a.s.Vt 't [O, TI,

ST

LT

s',

rt

+

-2

where the compensator of AN: is f i . -i

ri ~ : d s= s i ( 1 + p:)ds,

- ass. and

p%= -(es ) = , x=~ I + & . 3) Since the bond price equation is still dpt0 = P:rtdt, P t = 1, where for simplicity we take the initial value equal to one unit, the solution price equation is of the pf = eSo r e d s .

bent

4)

If the contingent claim is X E contingent claim xt is

eN,

then by (8.34) the price of this

Eeven though this pricing formula is theoretical, it has a nice physical meaning: It says that if we want the contingent claim to be X at the future time

8. Option Pricing in a Financial Market and BSDE

232

T, then now at time t we can expect (that is, take the conditional expectation) that depositing the money X exp[- :J r,ds] will produce the claim X at time T.

Proof. Theorem 244 can be proved by the Girsanov theorem for SDE with jumps. In fact, formally, (8.29) can be rewritten as T T xt = X r,x,ds q,(Oyds dw,) - :J p,(O"s d~(s)) = X - ~ r , x , d s- cq,diZi, - ~ : ~ , d ~ ( s ) , 0t T, where we write wt, = w: + r(Oy)'ds, i = 1, . . . ,d; -2 -. -. t . t iVi N t = N,Z Jo(8, ) ds = N," - Sop",s,i = I , . . . ,m. However, by the Girsanov type theorem (Theorem 124) Gt, t t [O, TI, and N t , t E [0,TI, are a BM and a centralized pure jump process under the new probability measure FT defined in the conclusion 1) of Theorem 244, respectively. Notice that [qt,pt] = 7rt [at,p,]. Hence (8.33) now follows. (8.32) is similarly established. So 1) is proved. Notice that by Theorem 122 the solution of the linear SDE (8.32) is F - a s . pi - exp[Jot r,ds a p d ~ -t ~ ; f = la:k12 ~ ds]. a N:),xyZl1; X:pikds I . (1 + CL P, Hence (8.34) holds. On the other hand, applying Ito's formula (Theorem 94) one sees that (8.35) satisfies (8.33). So 2) now follows. 3) is obvious. Finally, the first equality sign is obtained by taking the conditional expectation EP~[.l&]on both sides of (8.33). The second equality sign can be seen as follows: VA E Zt, JA EPT[x exp[rSds]l3t]d& = JA X exp[- 1 : r,ds]d& T = JAXexp[- ~ ~ r , d s ] % d P= JA E[Xexp[r,d~]po,~18t]d~

+

< <

+

-'

+

no,,, r

= JA E[Xexp[-

sA

+ x;fzlSot

:J

r&]

(&)

R

4

pt,~)

st

I8tIdP

= (F0,t) E[X =PIr,ds] (a,T) 18tldp. On the other hand, T E ~ T [ exp[X r,ds]18t]d&- = EPT[x exp[-

fr,ds]18t]Fo,~d~

(Fo,~) ER [Xexp[- :J r s d s ] ~ 8 t ] ~ [ F t , ~ l 8 t ] d p . Notice that E[P~,TI~~] = 1. Hence = JA

EFT[xexp[rSds]I8t]= E[Xexp[- f r , d s ] ~ t , ~ l 8rnt l . Now suppose that u,p, X are all non-random, and X = ~ ( P T ) If . we introduce a function u(t, y) E C 1 7 2 ( ~x+R ~ )and , we require that u(t, Pt) = zt, then by using Ito's formula to e-1; T s d S ~Pt) ( t , on t E [0, T] with (8.32) one easily derives the following fact: if u(t, y) satisfies the following partial

8.6 Black-Scholes Formula for Markets with Jumps

233

differential equation au +tu =

-= +

t d = l alkdkyiyj& + rt ELl yi@ + E;cnZ1 ~:l[u(t,Y + (0, .. . ,0, pfk,O,.. . -u(~,Y) i i k OU(~;Y) ] i t ,

-Y Pt

ay

u(T, Y) = d y ) , t E [O,TI,

(8.37) then u(t, Pt) = xt and XT = u(T, PT) = ~ ( P Tholds ) true. This tells us that the option pricing problem in the market with jumps still can be solved by the PDE technique. To get a more explicit formula for the price of the option we should assume more. Suppose, further, that ~ ( P T = ) (P; - K)+, that is, the contingent claim only depends on the 1st stock, or say we want to price a call option on stock 1, and suppose that N = d m = 1 1= 2, and T > 0, a, p are all non-random constants. Then by (8.30) oN also is a = Nt - (1-oN)t becomes a centralized constant. Under these conditions Poisson process with the intensity function (1 - oN)t with respect to the probability measure &. Since by (8.34) FT - a s .

+

+

xt

Notice that now (6t) and {N,) are independent under the probability measure FT,and that they have independent increments. Hence by using Ito's formula it is not difficult to verify the fact that if we write

..

.

(8.39) where BS(t, T, y, K, a,T) = y@(p+(T- t, 9)) - K ~ - ' ( ~ - ~ ) @ ( ~ - (t,Ty)), (8.40) asOO,and pl(t, y) = h [ l o g 5 t ( r 3~ )I, @(y)= 1 6 -m e-'V2dz, then u(t, y) E C1>'([O,T) x (0,oo)) and ~ ( ty), satisfies (8.37) with g(y) = (y - K)+ and Ic = 1; i,j = 1. So after a discussion similar to the one in the previous section one sees that

+

f

ry

8. Option Pricing in a Financial Market and BSDE

234

is the price o f the option for the first stock. Noticing that in (8.41)the price of the option for stock 1 actually - depends on both stocks, because 1does so. (Actually, depends on ( I N . By (8.30) (IA is solved by both coefficientsof the two stocks). Hence, if there exists only one stock with a continuous and a jump perturbations, then usually we cannot have a formula, because in this case X usually does not exist. Formula (8.41) is a generalization o f the Black-Scholes formula for option pricing. In the case that p = 0 and N = 1, i.e. the stock is without any jump perturbation, and we only consider one stock in the market, then formula (8.41) is reduced to

-

xt = E ~ ( ( P; K ) +I&)

= B S ( ~ , TP:, ; K , a ,r ) .

(8.42)

That is the standard Black-Scholes formula which we found in the previous section. So the option pricing formula also can be evaluated by solving a PDE, even in a self-financed market with jumps, but under the assumption that 2 = N = d+ m = 1 1 and all coefficientsare constants. That is, we arrive at the following theorem.

+

Theorem 245 Assume that r > 0, b, a are all constants, d = m = 1, N = 2 and g(y) = ( y - K ) + , y L 0. Let u ( t , y ) define by ( 8 . 3 9 ) ~0~< t < T ,y 2 0. Then u(t,y) E C1,2([0, T ) x ( 0 , ~ )and ) satisfies (8.37), with u ( T ,y) = ( y-K)+ ,Vy 2 0. Furthermore, u ( t ,P)) is the price of the option, ( t ,Pl)P: is the portfolio, such that (u(t,P)), ( t ,P))P)) can duplicate the money (P; - K ) + , which the option owner can earn at the future time T . Such an explicit formula u(t,P l ) (defined by (8.41)) for the option price is called a generalized Black-Scholes formula.

8.7 More General Wealth Processes and BSDEs In a financial market sometimes we will meet more general wealth processes. Suppose that during the time interval [0,T ] the investor also consumes the money dCt 2 0 in a small time dt with Co = 0, and CT < W , and borrows the money from the bank with the interest rate Rt at time t. Then his wealth process xt developed in the financial market should satisfy the following SDE: dxt = rtxtdt - dCt

-

N

rf)-dt

-ko(Rt - rt)(xt i=l

+ 7~t[bt+ bt - r J ] d t

+ atctdwt + x t p , d ~ t0, < t < T ,

(8.43)

8.7 More General Wealth Processes and BSDEs

235

+

where Zt = [at,pt] is the N x (d m) volatility matrix process, 6t = (bit, ...,6N)T is a dividend rate process, and KO 2 0 is a constant, and we write a- = m u ( - a , 0). (8.43) is a general model in the financial market, which includes many interesting cases. For example, let ko = 0, then we get the model in [8];and if we let ko = 1, and pf O,1 i G m, (i.e. no jumps), we get a model discussed in [221J331. Suppose now the investor expects that his wealth will arrive at X at the future time T, then his present wealth (or say, his present investment) xt should satisfy the following BSDE from (8.43):

-

<

As in the previous section we write [qt,ptl = rt[at, ~ t =l r t z t , whereqt = ( q f , . . . , qdt ) , p t = (pt,...,pF), andassume that N = d + m . If --I at = [at, pt]-lexists, then the above BSDE can be rewritten as

where as before we set

Bt =

[$]

= z;'[bt

+ 6t - rJ].

Now suppose that the investor wants his consumption to satisfy the rule:

Then the existence of the solution (xt,qt,pt) for BSDE (8.45) or, more precisely, the existence of the solution (xt, r t ) for BSDE (8.44), will help the investor to make his wealth XT reach X a t the future time T, if he at time t by using the portfolio rt invests the money xt. Moreover, if one takes X as the value of some contingent claim, then the existence of xt also

236

8. Option Pricing in a Financial Market and BSDE

shows that one can give the contingent claim a price xt at time t. However, later one will see that the consumption density function c(s,x , q, p) may, in some cases, be non-Lipschitzian: in particular i f we consider the optimal consumption problem. So we need to consider the BSDEs with more general coefficients.

8.8 Existence of Solutions for Non-Lipschitzian Case Consider a general BSDE in R~ as (8.8). W e have the following theorem for the existence of solution of BSDE with a non-Lipschitzian coefficient.The method o f deriving such a kind of theorem involves using the smoothness and limit technique.

+

Theorem 246 Assume that T < TO,b = bl b2, lo bi = bi(t,X , q,p, W ) : [O,To]x Rd x RdBdlx L : ( . ) ( R ~ @x ~Q~+ ) R ~ , i = 1,2, are St - adapted and measurable processes such that P - a.s. Ibl(t,x,q, p,w>l < ~ l ( t > ( lIxI), + ( l1x1) + ~ 2 ( t ) ( l +191 + IIPII), Ib2(t,x, 9, p,w)l < ~ l ( t ) + where cl(t) and c2(t) are non-negative and non-random such that J? cl(t)dt J? ~ 2 ( t ) ~
+

<

+

<

First, let us give an example for the existence of a solution to BSDE in the case that cl ( t ) d t + e ~ 2 ( t ) ~
8.8 Existence of Solutions for Non-Lipschitzian Case

237

Example 247 Let b = - I . + ~ S - ~ ~ X (xl-@ IsZos-LY2q I ~ # ~ Js Z- p~( ~~ ) ~ ~ ( ~ ) n ( d z ) , where a1 < 1; a 2 < 1/2;0 < P < 1; n(U) < co; and assume that X E ~ T , E I< ~co.I ~ Obviously, by Theorem 246 (8.8) has a unique solution. However, cl(s) = Is+0~-"1,c ~ ( s )= Is+O~-"2are unbounded in s, and b1 = -Is#o~-fflx is also unbounded in s and x, and is non-Lipschitzian continuous in x. Notice that here we have not assumed that X is bounded. (cf. [153]).

+

+

XI-'

Now let us show Theorem 246. Proof. We can show this Theorem by using the smoothness technique. However, for simplicity we can assume that b2 = 0. (Otherwise we can smooth out bl and then proceed the same way, as follows.) Case 1. Under condition (i) in 3'

In this case we can make a stronger assumption on bl :

where q ( t ) is non-negative and non-random such that c ~ ( t ) d< t co; and make a stronger global condition than (i) of 3' for bl and b2, i.e. assume that in (i) of 3'

~2

cfJ(t) = cl (t), c g (t) = c2(t) and pN(u) = p(u) do not depend on N. (8.49) Indeed, if the conclusion is proved under these stronger conditions, then one can derive the same conclusion under the weaker condition Ibl(t, x,q, P, w)l s ( t ) ( l + lxl), where cl(t) is non-negative and non-random such that :J cl(t)dt < co, and under the weaker local condition as (i) of 3'. For this for each N let WN(x) E C ~ ( R be ~ such ) that 1, for 1x1 N 2, WN(x) = 0, for 1x1 2 N 3,

<

< + +

<

<

and 0 WN(x) 1, where & is a constant. Define ~ N ( ~ , x , Q , P=bi(t,x,q,p,w)WN(x),i= ,~) 172. Then it is easily seen that if one lets ?h(t) = ~ ( t ) ( l +N 3)ko ~ r + ~ ( it = ) , 1,2, PN(u) = P ~ + ~ (+u2)1, then for each N by satisfies the stronger assumption lbrl c(t)(4 N).

+

-

<

+

+

8. Option Pricing in a Financial Market and BSDE

238

Moreover, for each N the condition 3O holds globally for b?, i = 1,2, with these new Zb(t), i = 1,2, and FN(u). Hence there exists a unique solution ( x r , q?, p r ) of (8.8)with such bN for each N. Applying Lemma 231, one sees that there exists a natural number No, which depends on /co in (8.47),

1

To

1

To

cl(t)dt and

0

cz(t)'dt in condition loonly, such that

0

I x ~ I < No + 2,VN = 1 , 2 , - . -. .

.

)

Therefore ( x p , q,",p? is a solution of (8.8). The uniqueness is easily derived by Lemma 231 and 115. Now let us show the result under additional conditions (8.47) , (8.48) , and (8.49) . Recall that we have already assumed that b2 = 0. In this case for simplicity denote- bl by b. Let us smooth out b to get bn, i.e. let for all u E Rd cd exp(-(1 - lu12)-I), for luI < 1, J d ( 4 = 0, otherwise, such that the constant cd satisfies JRd Jd(u)du = 1; and for u E ~ ~ (it is @ viewed ~ asl a d - dl-dimensional vector), J d . T ( ~is) similarly defined. Set for (x,q) E Rd x ~ ~ @ ~ l J(x, q) = Jd(x) ' Jd.dl (Q). Define bn(t, x, 9, P , W ) = JRdxRdmil b(t, x - n-l- x, q - n-'g,p, w) J(T, ?j)&d?j. Since for each n = 1,2, . - . it is easily seen that as (x, q,p) ,(x', q1,p') E R~ x ~ ~ x L;(,)(R~) @ ~ l Ibn(t,x,q, P,W)- bn(t, 2 ' 7 ql, PI, w)l Sncl(t) - x'l +cz(t>(kI - + lip - ~'11). Hence by Theorem 234 there exists a unique solution (x;,qr,p?) of the following BSDE (denote t' = t A T below)

{

<

8.9 Convergence of Solutions

< kb ~7{cl (s) Ji

239

(E lxm - x: - (m-' - n-') 11) ' J(1, ij)&@

+

+

+cz(s)'E Ixa - x;12)ds k6 (m-I n-l) . Since by Lemma 114 for all n ~ S < 00. E(suptG71x;12 ~;(l9;1~ I I P ? I I ~ ) k~ Hence by Fatou's lemma it is easily seen that lim E[IxF - x?l2 lim JLT(lq? - q;12 IIpY - p?l12)ds] m,n+ca m,n--00

+

+

<

+

+

+

where pl(u) = p(u) U. heref fore, 2 lim EI1.T - xI: +m ! i ~ , J ~ 7 ( l -~ k ~12 llpa - p;l12)ds = 0. m,n+m From this it is not difficult to show that there exists a (xt, qt, pt) E Szt (Rd) X Lit (RdBd') X Fgt (RdBdz) such that as n + m , (XI",q?, P?) (xt, qt, pt) , in sgt(Rd) X Lit (Rd@d')X Fit (RdBd2). Notice that Ibn(%x;,9;, P?,w) - b(s,xs, qs,ps,w)l b(s, x r - n-'3, qr - n-l- x>p37 w )- b(s7xs,qs,~s,w) I J(Z,Q)&4. Hence, letting n + m (take a subsequence, if necessary) in (8.50) one finds that (xt, qt,pt) is a solution of (8.8). The uniqueness can be derived by Lemma 231 and 115. Case 2. Suppose that we have X E E 1x1~ < m . Let

+

+

< Jgt I

z7,

Then by step 1there exists a unique solution (xy, q?,p?) satisfying as 0

< t,

with coefficient b and terminal random variable Xn. By Ito's formula it is easily seen (cf. the proof of step 1 above) that (x;,qI",pF) is a Cauchy sequence in Sit (Rd) x L i t (RdBdl)x FZt (RdBd2).Hence there exists a limit x ' ) ( R ~ @ ~ Taking Z). limit in (8.51) by (xt,qt, pt) E Sgt (Rd) x L $ (~R ~ @ ~ letting n -+ CCI (if necessary taking a subsequence), one finds that (xt, qt,pt) is a solution of (8.8). The uniqueness can be derived by Lemma 230. rn

8.9 Convergence of Solutions We have the following convergence theorems for solutions of BSDE (8.8).

8. Option Pricing in a Financial Market and BSDE

240

Theorem 248 Assume that T is a St-stopping time for n = 0,1,2, - . . lo bn = bn(t,x , q,p, w ) : [O,T]x R d x Rdmdl x Li(.)(Rdmda)+ R~ are St-adapted such that P - a.s. ( x , b n ( t , x , q , p , w ) ) c l ( t ) ( l + 1x1) cz(t) 1x1 ( 1 + 141 + IIPII), where c l ( t ) and c 2 ( t ) have the same properties as that i n lo of Theorem 24 6; ,f? ( ( X I - x 2 ) , ( b 0 ( t , x l , q l , ~ l , w )-b0(t,x2,q2,p2,w))) G ci(t)p(lxi - x212) + 2 ( t ) 1x1 - 2 2 1 (191 - 421 1 1 ~ 1 - ~ 2 1 1 ) ~ where p(u) has the same property as that i n 2' of Theorem 246; S X n is ~TAT-measurablesuch that E 1xnl2 < oo,for all n = 0,1,2,. .- , and 2 lim E (xn - xOI = 0 ;

<

+

+

+

n-w

dOlimS,T n-m

sup Ibn(t,x,q,p,w)-bo(t,x,q,p,w)12dt=0. (Z,~)E x ~R ~ ~ @ ~ l PEL:(,) ( R d m d 2 )

WER

< <

If (x;, qr,p;) is a solution of the following BSDE: 0 t T XI" = X n ~ , ' t ~ b " ( s , x : , q r , ~ y , w ) d-s~ & ~ ~ q r d ' w ,

+

s,',:~

P R L ( Z ) Q ~ d.4, n = 0,1,2, . . . , respectively, then lim E [ S U P ~ ~ [ x;12 ~ , + ~ .&lAT(lqr ~ n-ca

1%;

&I2

+1

1 -p:l12)ds~ ~ ~

W e also have some other useful convergence theorem as follows:

8.10 Explanation of Solutions of BSDEs to Financial Markets

241

Proof. 1 ) In the condition 2"' letting n -+ m one sees that 2"' is also true for n = 0. Furthermore, by bol = b0 - bo2 one sees 2"' is also true for bol at most with some other q ( t ) and cz(t). Thus by Theorem 246 1 ) is obtained. 2): B y Ito's formula

8.10 Explanation of Solutions of BSDEs to Financial Markets In the previous sections we have already introduced the following BSDE for a general wealth process xt :

242

8. Option Pricing in a Financial Market and

BSDE

where xt is the wealth of a small investor at time t , Zt = [at,pt]is the N x (d m ) volatility matrix process, dt = (bit, ..., ~ 5 is a~dividend ) ~rate 2 0 is a constant; and during the time interval [O,T]the process, and investor spends the money dCt 2 0 in a small time dt with Co = 0, and CT < co, and borrows the money from the bank with the interest rate Rt at time t. Now by using the existence theorem of BSDE (Theorem 246) we can show the following fact, that under appropriate conditions (8.52) can be solved. That is, there exists a solution (xt,rt),where xt is the total invest money of the investor at time t and rt is the portfolio of the investment (actually it is the part o f investment for the stocks) at time t , such that as the time evolves they can produce the money X in the market at the future time T . Or say, i f the small investor invests his money xt with the portfolio rt at each time t in this market, even if he spends the money dCt at a small time dt, and he borrows the money from the bank with the rate Rt, his wealth can still eventually arrive at the amount X in the market at the future time T . Actually we have the following theorem.

+

zo

Theorem 250 In a financial market suppose that there is one bond, which price satisfies (8.26), and there exist N-stocks, whose prices satisfying the SDEs (8.27). Assume that ( A I ) N = d m , and rt, bt, &, at, Rt and pt are all non-random such that 5;' exists and is bounded, that is, 15;' 5 ko, and

+

2

+

+

1.

.

~ T [ l r t+l I ~ ~ l ( b C5r t - r t ~ 1 1Rt - rt12]dt< m, where at E R ~ @ E RNBrn,Zt ~ , ~= (~ at pt ) E R ~ @ ( ~ + ~ ) ; X E 3r1' and E 1x1~ < co; and, moreover, one of the following conditions is satisfied: (i) 0 5 Ct 1 is Tt-adapted such that EC$ < co; (ii) 0 Ct = E(s, x,)ds, where 0 5 E(t,x ) satisfies the properties: E(t, x ) is continuous in x , and IE(t,x)l 5 c o ( W + 14, (E(t,x)-E(t ) ) ( ~ - ~ ) < c o ( t ) l ~ - y l , W [O,T],Vx,!/ER1, E , and 0 5 ~ ( t So ) , ~ ( t ) d
<

Ji

YT

{

+

zE1 <

+

+

8.11 Comparison Theorem for BSDE with Jumps.

Proof. Write [qt,pt] = nt[ut,pt] = r t z t , where qt (Pi,. . .,P?). Then the above BSDE can be rewritten as

=

(q:,.

243

. .,qf),pt

=

oreo over, in Ease (ii) one sees that CT - Ct = C E ( S ,x,)ds. Hence, for either of cases (i) and (ii), applying Theorem 246, one finds that (8.53) has a unique gt-adapted solution (xt, qt, pt). Let nt = [qt, p t ] ~ t 1 . Then (xt, nt) is the unique &-adapted solution of (8.52). rn

8.11 Comparison Theorem for BSDE with Jumps. Comparison theorems for solutions of BSDEs are very important in both theory and applications. For example, in a financial market we naturally require the following facts: If an investor wants his money to increase to a higher level at the future time T, or during the time interval s E [O,T]he wants to consume more, then now, at the present time t , should he invest more in the market. That is, X1 2 x2, C: 2 C:, s E [0,T] +x: 2 xf, t E [O,T]. This is exactly the content of the comparison theorem of solutions of BSDEs. In this section we will discuss such kind of problem. We will see that the answers to thess problems can not only be applied to the financial market, but can also be used to discuss some deeper theorectical problems in BSDEs. Suppose that (xi, qf,pf ), i = 1,2, are solutions of the following BSDEs in 1-dimensional space for t 2 0; i = 1,2

where bi in (8.54) is a real function, i = 1,2, q: E ~ ' @ ' ~ ~-, p ;E( zR) ~ @ ' ~ ~ ; and for simplicity we assume that T is a bounded Zt = ~ : ' ~ ~ - s t o p p i n g time. Then we have the following

244

8. Option Pricing in a Financial Market and

BSDE

Theorem 251 can be applied to explain the option pricing problem in a financial market. If we comsider Xi to be the contingent claim with respect to a given option at the future time T, and let xf be the present option price, then Theorem 251 tells us that for a bigger contingent claim it is reasonable to make the option price bigger at the present time. One notices that in assumption 2" of Theorem 251 the condition for b with respect to p is stronger than the usual Lipschitz condition, and later we will give an example to show that such a stronger condition cannot be weakened to the usual Lipschitzian condition for p. Now let us give a simpler comparison theorem. However, from this simpler comparison theorem we can derive the above Theorem 251. Theorem 252 Under assumption lo and 3" in Theorem 251 and the assumption that 2" one of the following conditions is satisfied: J z IP(t, x, 9, W)- P(t, x', Q', w)I k o b - $'I + I9 - 9'11, where ko 2 0 is a constant, and C i (z, w) satisfies the condition (B)1 IC,l(z,w)l b,Cft(z) 2 -1,i = 1,. . .,d2; c,l(z) = (c;, (z),. . .,cia(z)) E F; ( ~ ~ 8; ~ 2 )

<

<

J z IP(t,x,9,w) - P(t,xl,9',w>l ko[Ix - 2'1 + I9 - ~ ' l l , where ko 2 0 is a constant, and C? (z, W) satisfies the condition (B)2 IC?(Z,W)I kolCzt(~) 1,i = 1 , . ' .,d2; ct2(z) = (eft (z) ,- . .,C& (2)) E F; ( ~ ~ 8; ~ 2 ) the conclusion of Theorem 251 still holds.

<

<

<

Before we establish Theorem 251 by using Theorem 252, let us explain the following fact: The condition 2"' in Theorem 252 actually cannot be weakened. Intuitively, the condition C t! (z, w) 1 or C; (z, w) 2 - 1means that the coefficient cannot be influenced too much by the jumps. Otherwise,

<

8.11 Comparison Theorem for BSDE with Jumps.

245

the conclusion of the above theorems would fail. The following is a counter example when the condition is violated.

Example 253 Assume that 0 < r ( U ) < oo,EO > 0. Notice that P(Nk([O,t ] ,U ) = 0 ) = e-t"(u) > 0 , W 2 0. Hence we can construct an example as follows: Consider BSDE (with all random processes i n R 1 )

with b(p) = ( 1 + ~ 0JZ) I u ( ~ P ( z ) ~ ( ~ z ) . Obviously, (0,0 , 0 ) solves the above BSDE with X = 0 , and - t ) r ( U ) ,0 , - I Z E U ) ( x t , qt, ~ t=)(-Nk([O, t ] ,U ) solves the above BSDE with X = -Nk ([0,To],U ) < 0. In fact, by Nk([t,To],U ) = N/c([t,To],U ) - (TO- t ) r ( U ) one easily checks this result. However, > 1, as z E U ; C? ( z , ~=) ( 1 EO)IU(Z) and P ( x t > 0 ) 2 P(Nk([O,t],U)= 0 ) > 0,W E [O,To).

+

-

+

A counter example for c,' ( z ,w ) = -(1 E,,) < -1

+

can also be similarly constructed. Now we will give an example to show that the condition in 2" of Theorem 251 for b with respect to p cannot be weakened to the usual Lipschitz condition.

Example 254 . Assume that n ( U ) = 112, r ( V ) = 1, U C V . Consider the BSDE (8.55) with b(p) = Jz I v ( z ) p ( z ) r ( d z ) . Obviously, (O,0,0 ) solves the above BSDE with X = 0, and ( x t , ~ ~t , t =)(-Nk([O,t],U),O,- I Z E U ) solves the above BSDE with X = -Nk([O,To],U) < 0. Moreover, here C: ( z ,W ) = IV ( z ) and P ( x t 6 0 ) = 1,Vt E [O,To]. Hence the comparison theorem holds. Now if we consider the BSDE (8.55) with ) l ~ b(p) = (Sz l ~ ( ~ +))1'2. then the coeficzent b satisfies that

\lSz

with C t ( z ) = I v ( z ) ,IlpII = l p ( ~ ) l ~ n ( d ~ )= , n i(, ~n )( V ) = l , U c V. (Notice that the condition (8.56) is weaker than that Ib(pl) - b(p2)1 6

246

8. Option Pricing in a Financial Market and BSDE

JzlCt(z)I I(pl(z) - p2(z))l ~ ( d z ) )Obviously, . (O,0, 0 ) still solves the above BSDE with X = 0 , and (xt,'?t,pt) = ( - ~ k ( [ o , t ] y u+) [ m - n ( U ) ] ( T o - t ) , O , - ~ z ~ " ) solves the above BSDE with X = -Nk([O,To],U ) 6 0. However, P ( x t > 0) 2 P(Nk([O,t],U)= 0 ) > O,vt € [&To), and so the comparison theorem fails. Now, let us establish Theorem 251 by using Theorem 252. Proof. Notice that ( S t ,q2,p2) = (x: - xi,q: - q,',p: - p i ) satisfies the BSDE as follows:

C t ( z , w ) = G t ( w ) Icit(z,w)l ~ n @ i t ( z ) )IQt(w)l ,

< 1 , i = 1, ...,4;

<

fo(t,w) = b2(t,y:,9,l,P:,w) -bl(t,y,l,a,l,P:,w) 0, and 3 1 1 2 2 qi,t - (qlt,..., qit, q(i+l)t,...,qdlt)> 1 a: = ( ~ l t...,, afi,t), 6 = (?lt,.. . ,Gdlt), Ct ( z ,W ) = (Clt( 2 ,w ) , . . . ,Cdzt( z ,w ) ) , etc. Obviously, condition (ii) of 2'' in Theorem 252 is satisfied. Hence Theorem 252 applies. One has P - a.s. x t = X: - x i 0,Vt E [O,T]. The proof using Theorem 252 to derive Theorem 251 is complete. w Before we establish Theorem 252, some preparation is necessary. Introduce another condition for C t ( z ) : (C)1 Cit(z) -1, i = 1,. . .,d2; ~ ( 2=)( c l t ( z ) ,. . .,c d z t ( z ) )E F ; ( R ~ @ ~ ~ ) , A

<

<

Jz 11 - ~ m ) l ~ n ( d ko z )< m. Suppose that (st,qt,pt) solves the following BSDE for 0


8.11 Comparison Theorem for BSDE with Jumps.

247

where at E Lg(R1), bt E L;(R'@~'), fo E Lk(R1), moreover, at and bt are bounded, i.e.latl lbtl ko. We have the following simple comparison lemma. However, this is a basic comparison result for the solutions of BSDEs, and actually, all other comparison theorems can be derived from it.

+

<

Lemma 255 If fo 2 0, X 2 0, and Ct(z, w) satisfies the condition (B)1, then P - a.s. xt 2 0, for all t E [O, T]. The comparison result for solutions of BSDEs cannot be derived by Tanaka's formula as in the forward SDE case. Since now we cannot know whether the coefficients qt satisfies the Holder-continuity with the index required by the Tanaka formula. However, there are two methods that can be used to solve such kinds of problem. One is by using Girsanov's theorem to simplify the BSDE to make it become the summation of a non-negative process and a new martingale under a new probability measure, and then to take the conditional expectation under this new probability measure to obtain the result. This method will also be seen later in the dicussion of arbitrage-free markets. Here we will first use "a duality method" to derive the comparison result. The idea is that we introduce a relative forward SDE. Then, by using Ito' formula to the multiplication of the solutions of BSDE and the forward SDE, we can show that this multiplication process is actually the summation of a non-negative process and a martingale under the original probability measure. Thus we can then take the conditional expectation under the original probability measure to obtain the result. We will also use the Girsanov's theorem to again establish the result. Let us now discuss the details of such approaches. We now give the proof of Lemma 255. Proof. Method 1. (A Duality Method). First of all, suppose that the condition (C)1 holds. For any given (t, x) E [0, oo) x R1 assume that ys,t s T, solves the following forward SDE aryrdr ys = x : :J bryrdw, : :J Jz D,Y,-~k(dr, dz) 1 SAT - 8 JtAT lbr12yrdr - : :J JZI D C~rd.dz)dr, I~ where D, E RlNd2will be determined later, b, E Rmdl and

i

+i

< <

+ Ly\

+3

etc.-Then lySl2 satisfies the following forward SDE: 2 Iy.1 = 1zl2+J:ar 1~r1~dr+~,":~~ br ~ Y , ~ ~ ~ w , + J : : ~ ~ J ~l D ~~ r-[~fik(dr~d~)

8. Option Pricing in a Financial Market and BSDE

248

wecan let Di,(z) = -2 +2d=,i = 1,s. .,d2. Thus one has that ~ lyr12 fo(~,w)dr ISt~rl2 0. xt 1x12 = E[xl ~ r+l J&, The conclusion is now derived under condition (C)l. Now, for the general condition (B)l, one can take 2, t Z such that ~ ( 2 , ) < m . By assumption there exist a unique solution (xy,qT,py) that solves BSDE (8.58) with Z, replacing 2. Obviously, the result applies to x r . Letting n + m , one arrives at the required conclusion under condition (B)l. Method 2. (Girsanov's Transformation). By the Girsanov type theorem (8.58) can be rewritten as P - a.s. xt = x + J & ~ ( ~ s ( w ) x sfo(s,w))ds-JkT + Q S ~ ~JZ SpS(z)Nk(d~, - J ~ ~ dz), where t Gt = wt - SOb,(w)ds, and Rki(dt, dz) = f i k i (dt, dz) - Cit(z)n(dz), i = 1,2,. .. ,d2, are a new_ BM and new martingale measures under the new probability measure P : dP p 1st = (exp[Jot (bs,dws) - g lbSl2d ~ l .

nn d2

J,t

+

(1 Jz c ~ ~ ( z ) ({s), N ~ ~dz))e- So' Jz C,c(z)a(dz)ds)d~ ; i=lO<s -1 and n(Z) < m. (In this case Jz 1 ~ i t ( z )n(dz) 1 ~ I kon(Z), and for any 0 T < m I ~ ~ A ( dt z )
+

<

Gr

- JkTe l l aa(w)dsqsd~s - JS7 AT Jz eJ: a~(w)dsp,(z)F(ds, dz), F - a.s.

8.11 Comparison Theorem for BSDE with Jumps.

249

Hence, taking the conditi-1 expectation under the new probability measure P , one obtains that P - a s . , t E [O, TI,

>

F-

Notice that P. One also has that P-a.s.,t E [O,r],xt 0. Now for the case that Cit(z) 2 -1 and n ( Z ) < m we can let > -1. CG(z) = C&) Then by Theorem 234 there exists a unique solution (xT,qr,pF) solving the following BSDE XF = JLr(as(w)x: bS(w). (CIT+ JZ C : ( z , w ) p : ( ~ ) ~ ~ ( d z ) s J;, JZ p:Nk(d~,dz). + f O ( ~ , w ) ) ds J&, ~ r d w -

+

x+

+

Then by the result just proved P - a s . , t E [O, TI, 22" 2 0. Letting n -+ m, by Theorem 249 one obtains that E sup,, O,TAT] 1x7 - xt12 -+ 0 , for any T < m. Hence there exists a subsequence { n k ) of { n ) , that for simplicity we again denote by { n ) again, such that P - a s . 2 limn-+, S U P t E [ O I,x ? ~ ~-~~] t l= This shows that P - a.s., t E [0,TI, xt 2 0. Finally, for the case that Cit(z) 2 -1, letting Zn f Z such that A(&) < m, then by Theorem 234 there exists a unique solution ( x r ,qr,py) solving the following BSDE xt" = J&,(as(w)x: + b&) . (9,nIT JZn C s ( z , w ) P : ( ~ ) ~ ~ ( d z ) + f o ( ~ , w ) ) ds S&, Q F ~ W S Jzn p ? f i k ( d ~d, z ) ,

x+

S&,

+

and by the result just proved P - a.s., t E [O, T ] , x7 2 0. Notice that (27,qr,py(.)IZn(.))solves the following BSDE x7 = J&,(as(w)x: bs(w) . (qFIT JZ Cs(~,~)~:(z)TI~n(z).lr(dz) + f O ( ~ , w ) ) ds J,', qrdws - J,', JZn P ~ I Z (, z ) N k ( d ~d ,z ) , and latl+ lbtl 5 ko, E J ~ I CL~ ( Z , W ) I ~ A ( ~ Z ) k~ ~ S < m, E JZ lp:(z, w)12a(dz)ds 5 kT < m , V n . Hence applying Ito's formula to lxr - xt12, and proceeding as in the proof of Theorem 234, one easily sees that as n + m, E SUPtEIO,TAT~ lx; - xt12 -+ 0. From this one easily finds that P - a.s., t E [O,T I , xt 2 0. Now let us use Lemma 255 to establish Theorem 252. Proof. One can establish Theorem 252 by constructing a BSDE similar to the one in (8.57), and then using Lemma 255 under the condition ( B ) l .

x+

pr

+

+

<

8. Option Pricing in a Financial Market and BSDE

250

The proof for condition (B)2 is similar. Thus the proof of Theorem 252 is complete.

Corollary 256 Under the assumption of Theorem 251

Proof. Recall that

$

=

8+

(&, q2, p2)

llT

= (x: - xt ,q; - q:, p: - pt ) satisfies

(as(w)$

+4 ( w ) ~ +T

(i, ~)

F~(i)~r(dz)

where

G(.,

W ) = G ( w ) l C t ( ~ , ~sgn(p^t(z)), )l 1 1 1 f o ( ~ ,= ~ b2(~,x;,9sl,p;,w) ) -~ ~ ( ~ , X , , Q ~ , P ~ , W ) . See the notation in (8.57). Comparing (8.60) with 8.58, from (8.59) one finds that

where dF

pIa,

T-

= [exp(Jo bsdws -

4J,T lKsl

2

ds).

+

n o < s < ~( 1 Jz e s ( i ,w ) N ( { s ) , d z ) ) e - loT SZ Es(z~w)a(dz)dsdp provided that e s ( z , w ) < 1, and n ( Z ) < m. Notice that 8 j7 0 and fO(s,W ) 5 0. Hence - a, s , 0 x=o,f0(s,w)=O, provided e s ( z ,w ) < 1, and n ( Z ) < oo. Since P P, the conclusion is true provided that & ( z , w ) < 1, and ~ ( 2<)m. Now for the general case the conclusion can be drived just as in the proof of Lemma 255. A similar corollary can also be obtained for Theorem 252.

z0=o

A

8.12 Explanation of Comparison Theorem. Arbitrage-Free Market Now let us apply Theorem 251 to the financial market. Suppose we have two wealth processes xi, i = 1,2, satisfying the BSDEs as (8.52) in a financial market, as in the previous sections, respectively: i = 1,2,

8.12 Explanation of Comparison Theorem. Arbitrage-Free Market

+

251

+

dxf = rtxfdt - d ~ : r f[bt St - rtlldt

-

N

-ko(Rt - rt)(xf - )ry)-dt + rfetdwt + n f p , d ~ ~ , j=1 x k = xi, 0 Q t Q T ; (8.62) where ( x i ,rf) is the solution of (8.62) corresponding to the given terminal wealth X i at the time T and the given consumption process C:, t E [0,TI. Write [qt,ptl = nt[ct,ptl = rtzt, d whereqt = (q;,,..,q,),pt = ( p : , . - . , p y ) , andassume that N = d + m . If --I = [ct, pt]-'exists, then the above BSDE can be rewritten as i = 1,2, ct

where as before we write Bt =

[ B) ] 0"

--I

= ct [bt

+ 6t - rt11.

Thus, by using Theorem 251, we immediately obtain the following theorem.

Theorem 257 Under assumptions of Theorem 250 if X 1 X 2 , C l 2 Ct2, Vt lt [O,T],and 1 , = 2 , - ,m; 0 ( 8 -6 ) , l j + d ) i 9 then P - a.s. xt' 2 x;, w E [O, q. In particular, i f X 1 2 0 , and C i 2 0, Vt E [0,TI, then P - a s . xt' 1 0,vt E [0,q. Furthermore, i f io = 0, all coefficients are bounded, and is also bounded, and P ( X 1 > 0 ) > 0, X 1 2 0, then P - a.s. x: > 0,Vt E [O,T],VwE { w : x l ( w ) > 0 ) .

>

z,'

Proof. We only need to show the last conclusion. Suppose that P ( X 1 > 0 ) > 0. Now applying Girsanov's theorem we find that (8.63) can be rewritten as PT - a.s. 0 Q t < T , xi =X 1 r,x:ds + C; - C j q:dG, - c p ; d ~ , , where e

fl

8. Option Pricing in a Financial Market and BSDE

252

+ + +;

Gt = wt J: Oyds, N t = Et Ofds, are a new BM and a new centralized (not necessary stationary) Poisson process - under theT new probability measure &, respectivvely, such that PT =exp[-& ( ~ ~ , d w s ) - ~ ~ l ~ ~ 1 2 d s - f ( ~ s , ~ ) d s ] . . I l o < s < ~ j l +( ~ snNs))dP, , p =-ON s

-

-

1

F?

-

N t = Nt - So(1 - 6, )ds, PT- a.s. Hence applying Ito's formula to xieone sees that PT- a s . 1 ; = xle-gr.ds + e- :$ rddsdc: -

t

JT e- ./:

-

~Js&ij,

-

J,T e- .ToY radsp@r,.

Now taking the conditional expectation E'T(-I&), one finds that FT- a s . vt E [O,Tl, 1 - Sot 'ads = E&(Xle- :$ Fads + e- ./: r&dc;[&) xte - joTr.ds - $ , r,ds 2 EPT(X1e, 1%) 2 E ~ ( X ~ I X ~ > O Wt). ~ ~ So FT- a s . Vt E [O,T], E ~ ( X i e ; . f ? ~ s d sIxl>olZt>. Now let A = {w : X1(w) > 0) , and pA = P(.IA). Then pA is still a probability measure. Replacing P by p A , and discussing as in the above case, one finds that PAT- a.s. > EP"T(XI~; ./?'ads t -I X ~ > O ~> &0. ) Since PAT pA.SO pA- a.s. Vt 't [O,T], xi > 0. That is, P - a.s. W E [O,T], Vu E A, x: > 0. This theorem has a nice physical meaning. Actually, it tells us that in a financial market the following facts are true. 1) If a small investor wants his wealth in the market at the future time T to reach a higher level, and his consumption rule does not change, then he must invest much money now; that is, X1 X 2 ; C i = C:,Vt E [O,T]d x i 2 x:,Vt E [O,T]. 2) If a small investor wants to consume more during the time interval [0,TI, and his terminal target does not change, then he also must invest more money now; that is, C l 2 C:,Vt E [O,T];X1 = X 2 ===+xi >x:,Vt E [O,T]. 3) If someone wants his money in the future in the market to be nonnegative, and he also wants to consume money in the market, then he must invest now. Naturally, if a zero investment in a market can produce a positive gain with a positive chance (that is, with a positive probability), then such a market has an arbitrage chance, and we will call it an arbitrage market. Otherwise, we will call the market an arbitrage-free market.

fl

-

N

>

-

8.12 Explanation of Comparison Theorem. Arbitrage-Free Market

253

To explain the concept of "arbitrage" more clearly, let us simplify the market. Suppose that in the market there is one bond, and its price Pf satisfies the equation as before: dPf = Pfrtdt, P$ = 1. There are N-stocks, and as before they satisfy-the following SDE: dP," = ~ , i [ ( b f d t af%w: CF="=,fkd~,k], Pi-Pi 00 -> O 7 i = 1 , . . . ,N. Applying Ito's formula, one can verify that the solution of the bond price is pf = eiotradS> 0, as t 2 0, and the solutions of stock prices are P,"= Pi exp[sOtb,ds ~ t = Ca"fdw," , -$ ds]. t ikds . n ~ = , n o , , , t ( l + p ~ k A N sIc) e- Sopa >O,i=1,2,...,N, provided that pF > -1, and P; > 0. Obviously, the wealth process of a small investor in the market under the condition that p;Ic > -1 and Pi 2 0, is xt = CEO T,f P; = CEO 7rf 2 0, where rl!is the number of bond units bought by the investor, and -the amount of units for the i-th stock. Now suppose that the market is selffinanced, then the wealth process xt should satisfy N dzt = C i = o vfdP,". This deduces that

+ c%=, +

+

dxt = rtxtdt

Sot

+ 7rt(bt - rtL)dt + ntatdwt + rtPtd~tTt.

(8.64)

Now let us give the explicit definition of an arbitrage market and an arbitrage-free market.

Definition 258 Suppose in a market the wealth process is described by (8.64), and the conditions stated above are satisfied. Thus, if for xo = 0, P - a s . , there exists a portfolio r t , t E [0,T] such that the wealth process xt developed by SDE (8.64) reaches X at the future time T, such that one of the following results holds: (i) X 2 0 and P ( X > 0) > 0; (ii) X 0 and P ( X < 0) > 0; then 7rt, t E [O,T], is called an arbitrage portfolio, and the market is called an arbitrage market. Otherwise, the market is called an arbitrage-free market.

<

An arbitrage-free market actually means that in such a market, without any investment; that is, xo = 0, we cannot have a gain with a positive chance; that is, X 2 0 and P ( X > 0) > 0; and we also cannot have a loss with a positive chance; that is, X 5 0 and P ( X < 0) > 0. Otherwise, somebody can use such a chance to make money. Hence applying Theorem 250 and the above theorem we immediately obtain the following result.

254

8. Option Pricing in a Financial Market and BSDE

Theorem 25%If for a financial market all of the assumptions i n Theorem 257 hold and ko = 0, plk > -l,Vi,k,t, then this market is a complete market, (that is, any contingent claim i n this market can be duplicated), and moreover, it is an arbitrage-free market.

Notice that xie- 5; = x:/Pf. SOwe may call xie- 5; 'ads a discount (w.r.t the bond) wealth process. By the proof of the arbitrage-free market, or say, the proof of the last conclusion of Theorem 257, one immediately sees that the following theorem is true.

-

Theorem 260 For a self-financed market, i f there exists an equivalent probaility measure FT (that is, FT is a probability measure and FT P) such that under this measure & the discount wealth process becomes a martingale, then this self-financed market is an arbitrage-free market.

In fact in this case x:e-l; = E ~ T ( X ~ ~ - ~ ~ ~SO ' *the ~ proof ' I ~ ~of) . the last conclusion of Theorem 257 still goes through.

8.13 Solutions for Unbounded (Terminal) Stopping Times It is quite possible that in a financial market some investor will stop his all activities a t some random time according to his rule for investment. However, it is not known beforehand when he will stop. Moreover, sometimes it is also unreasonable to assume that he will stop before some fixed time; that is, the terminal stopping time is not neccessary bounded. So we need to consider the BSDE with an unbounded stopping time as a terminal time. Now let us discuss BSDE (8.8) with an unbounded stopping time T E [0, w]. In this case we still use Definition 226 as the definition of the solution to (8.8). Moreover, one easily sees that Lemma 229, Lemma 231, and Lemma 230 still hold, if we substitute To by w in their proofs. However, the proof of Lemma 233 needs a little more discussion. Let us rewrite it as follows:

zT-

Lemma 261 If T is a zt- stopping time, X is measurable and Rdvalued, f ( t ,w) is adapted and Rd- valued such that

zt-

E 1x1~ < w, E IS,' If(3,w)l dsI2 < w, then (8.8) has a unique solution. Proof. Let xt = E5t(X f (s)ds),V t 2 0. Then, in the same way as before, one sees that it makes sense. Similarly, Mt = E5t(X f ( s ) ~ st) 2 , 0, is a square integrable martingale. Hence by the martingale representation theorem (see Theorem 103 in Chapter 2) there exists a unique

+ SLT +

8.13 Solutions for Unbounded (Terminal)Stopping Times

where

255

{ f (t,w)suchf that ( t ,w) is 7jt adapted, Rd@dl- valued : ( t ,w)12dt < m,QT < m E1

~ ; l o c ( ~ d @ d= ~)

:

-

If

etc. Let us show that E

J: lqt12

dt < m, and E

J: Jz

1

'

lpt(z)12r(dz)dt < m.

In fact, by assumption one finds that E 1Mrl2 < m and E 1Mol2 < m. So by (8.65) one sees that E

ITAT

+E

Jz bt(z)12n(dr)dt = E IMrAT - Mo12 A+TE I1Mol2] ~ 6 2[E1Mrl2+ E 1 ~ 0 1 ~<1 m.

~ q ( w)12 t , dt

0

JrAT 0

I 2 [~ ~M ~ Letting T f m. By Fatou's lemma one finds that E J r [q(t,w)12dt+E J r 0

JZipt(z)l2r(dz)dt 5 2[EIM,~'+E I M O ~< ~m.]

0

Now the rest of the proof is just the same as that of Lemma 233. Theorem 234 is also true when T is unbounded. For this we only need to put To = m in the prozf of Theorem 234, and introduce a new norm as follows: For (x.,q.,p.) E B let S U P ~ e ~ -Ob A ( t ) ~ lqt12dt II (x.,q., P . ) I I =~ supt>oe-bA(t)E + SUPt>0 C b A @ ) E llptl12dt, where B = L t (0,T ;R d ) x L; (0, T ;~ ~ " x~ F i1 (0,) T ;R ~ @ ~ z ) , b 2 0 is a constant, and A(t) = Jtm(cl(s) c ~ ( s ) ~ and )~s. f (t,w) : f (t,w) is 7jt - adapted, RdBdl - valued p ( 0 , T ;~ d @ d l = such that E 1f(t,w)l2dt< m,QT < m

L;,

+

Gr

+

{

IT 0

then the proof can be completed as that of Theorem 234. Similarly, if one takes To = m, and changes the interval [O,To]to be [0,m),then Theorem 246 is also true. cl(t)dt+ Furthermore, we also have examples to show that the condition J r ~ ~ ( t )<~md can t not be weakened.

Jr

. (Condition J r ~ 2 ( t ) ~
Example 262

+

<

+

+

256

8. Option Pricing in a Financial Market and BSDE

X t = I1(tloo) + Iz(O,t) + k ( O , t ) , gt = f ( t ) , p t ( z ) = f ( t ) I u ( z ) , T where I l ( t , T ) = St f (s)ds. However, this is impossible, since i n this case we have f (s)ds = J r [ ( 2 t )ln(2 + t)]-ldt = oo, 191. ds = which is a contradiction.

Jr

+

Example 263 (Condition J r cl (t)dt < oo can not be weakened). Use the above notation. Then BSDE xt = 1 S,OO f (s)xsds - J," qsdws - J ' Jz ~ s ( z ) f i k ( d s , d z 2 ) , 0~, where c l ( t ) = If (t)l = [(2+ t ) ln(2 + t ) ] - l ,has no solution. Indeed, i f it has a solution, it should satisfy E x t = 1 + J,OO f (s)Ex,ds = e-ftmf(s)ds = oo, which is a contradiction.

+

Now we have that all o f the existence theorems o f solutions including Lipschitzian and non-Lipschitzian conditions, the a priori estimates o f solutions, and the uniqueness o f solutions for BSDEs in this case still hold true. However, the proofs o f the convergence theorems require more discussion. Actually, we will have the following convergence theorems: Theorem 264 . Assume that conditions l o-3' i n Theorem 248 hold, and assume that 4"' lim Sooo sup Ibn(tl x,q, P, w ) - bO(t,2 , q, P, w ) I dt = 0. n+w (. x. -. . u ~ XR ~ R ~ @ ~ ~ PEL:(,) ( R d B d 2 )

wEa

Then the following conclusion holds: lim E[suplxyAT - xRTI2 J;(lq," - q!12 n-+w t>O

+

Remark 265 If lim SUP n+w

+ lip: - p:1l2)d~1= 0.

Ibn(t, x,q, P, w ) - bO(tlx,q, P, w ) I = 0 ,

( x , ~ ) EXR ~ R ~ @ ~ ~ PEL:(.)

(

~

~

~

~

2

)

wen then under the condition l o ,40' is satisfied. In fact, in this case Lebesgue's dominated convergence theorem applies. Now let us prove Theorem 264. Proof. W e use the previous notation. Note that , q:, p,",w) - bO(slx:, 92, P%)) 2(x: - x : ) ( b n ( ~x:, = 2(x: - x!)(bn(s,x:, Q,",P:, W ) - bO(slx:, q,", P:,w)) + 2 ( C - x:>(bO(s,x:, q:, p:,w) - b o b ,x:, q:, P:,w)) = I,'>" I,">" < I,"#" 4(c1( s ) C ~ ( S )([(x: ~ ) ~ 2:) ~ 12) +2-l(19," - dl2+ - p:1I2), where pl(u) = p(u) u. However, one can show that lim E I;+ds = 0.

+

+

+

n-oo

+

8.13 Solutions for Unbounded (Terminal) Stopping Times

257

Indeed, by the Schwarz inequality and Lemma 229 for unbounded stopping times E S,' I;rnds 2(E sup Ixk, - xyAT 4

<

12)

t>O

Theorem 266 All assumptions are the same as in Theorem 249 for solutions to BSDE with bounded stopping times, but now T E [O,CQ] and T = To = co. Then the conclusion of Theorem 264 still holds. Proof. Notice that now we have

However, now one finds that by Lebesgue's dominated convergence theorem

EL'

I,"lnds

< 2(E sup Ix&, t>

-X:,,~I~)~

0

The proof is now completed as above. w The results of the BSDEs for the terminal time being an unbounded stopping time can also be explained in the financial market. One only needs to replace the terminal time T by the stopping time T and to replace X E ZT by X E ZT, then a similar explanation can still be made. We leave this as an exercise for the reader.

258

8. Option Pricing in a Financial Market and BSDE

8.14 Minimal Solution for BSDE with Discontinuous Drift Applying Theorem 251 we can derive a new existence theorem for solutions to BSDE (8.8) with discontinuous coefficient in R1 (i.e. we assume that d = 1 in (8.8)), where for simplicity we also assume that T is a bounded stopping time: T 5 To < co.

Theorem 267 Consider (8.8). Assume that lo b(t,x , q, p, w ) is jointly continuous i n ( x ,q, p) E R 1 x Rlad1 x L:(.) (R1mda) \G, where G C R1 x Rladl x L : ( . ) ( R ' @ ~is~ a) Borel measurable set such that ( x , q , p ) E G + Iql > 0 , and m l G l = 0, where G1 = { x : ( x , q , p ) E G ) , and ml is the Lebesgue measure i n R1; moreover, b(t,x,q,p,w) is a separable process with respect to ( x , q ) , (2.e. there exists a countable set { ( x i , q i ) ) z l such that for any Borel set A c R1 and any open rectangle B c R 1 x Rladl the w-sets {W : b(t, x , 9, P, w ) E A , ( x , d E B), and {w : b(t,X , q,P, w ) E A, ( x i ,qi) E B,Qi) only differ a zero-probability w-set; 9 Ib(t,x,q,~,w)l< c l ( t ) ( l + 1x1) + cz(t)(lqI + IIPII)), Ib(t,x,q, Pi,w) - b(t,x,q,P2,w)l < JZ ICt(z,w)l IPi(z) - P2(z)l ~ ( d z ) , dt where cl (t),c2(t)2 0 are non-random such that J?(cl(t) + ~ ~ ( t ) ~< )co, and C t ( z ,w ) satisfies the condition (B) i n P of Theorem ,251. Then (8.8) has a minimal solution. ( A minimal solution xt is one such that for any solution yt it holds true that P - a s . yt 2 x t , V t E [O, 71). Furthermore, if b satisfies 30 (el - ~ 2 ) ( b ( t , ~ l , q l , ~ l-b(t,x2,q2, ,w) ~2,w)) < ci(t)p(Ixi - 2212 ) + c2(t)1x1 - 221 (191 - 421 + llpi - ~ 2 1 1 ) 1 , where cl(t),c2(t) 0 have the same properties as those i n P ; and p(u) 2 0,as u 2 0, is non-random, increasing, continuous and concave such that I L00; ) So+ ~ ~ I P ( ' = then the solution of (8.8) is unique. Remark 268 I n the case that T ( . ) = 0 (a zero measure) and X E $! (8.8) is reduced to the following continuous BSDE: b(s,x s , qS, - h>TqSdwg,0 < t, xt = x where T is a St-stopping time such that 0 < T < To, and To is a &ed number.

+ JkT

The reader easily obtains the existence of solutions to the continuous BSDE with discontinuous coefficients b from Theorem 267. Before proving Theorem 267 let us give some examples.

8.14 Minimal Solution for BSDE with Discontinuous Drift

259

+ h q T + SZ C t ( ~ ) ~ ( z ) r ( d z ) , where kl = ( k i , ...,k f ' ) ; ko,k(,,kg 3 0 , k f , 1 < i < dl,O < S < 1 are all constants, and C t ( z ) satisfies the condition (B). Then conditions l o - 2O in Theorem 267 are satisfied. Hence (8.8) has a minimal solution. However, b& is discontinuous at ( x ,q,p) , where x = 0, q # 0 , and p is arbitrary given. I n particular, for 0 < a < 1 b& is also non-Lipschitzian continuous i n q. However, i f we let a = 1, and b(t,x , 9, P, w ) = -b; ( t ,x , 9, P, w ) , then 3" i n Theorem 267 is also satisfied. Hence (8.8) has a unique solution. Example 270 Assume that C;=l arc < 00, where arc > 0. Let { r k ) E l = the totality of rational numbers in R1, f ($1 = Cn:r,
Proof. B y condition l oone easily sees that bn(t,x , q,p, w ) is jointly measurable and &- adapted. Obviously, b, b,+land b, < b < c i ( t ) ( l + 1x1) + ~ 2 ( t ) ( l +191 + llpll) Moreover, since ( nV cl ( t ) ) 3 c1 ( t ), (nV cz( t ) )3 c2 ( t ), bn 3 in&,,-(-cl(t)(l+ IYI> - c2(t>(l+Id + IIPII) +(nv cr(t>>IY - X I + ( n V c : ! ( t ) )I9 - @I) >/ i n f y , a ( - c ~ ( t ) (+ l Ivl) - ~ 2 ( t ) ( l +Id + llpll) + ~ l ( t I Y) )- 21 + c2(t>>I9 - 4) 3 -c1(t)(l + 1x1) - ~ 2 ( t ) ( + l Iql + IIPII).

<

8. Option Pricing in a Financial Market and BSDE

260

Notice that from the elemantary inequality infx f ( x ) - infx g ( x ) SUP, I f ($1 - g(x)l one has that linfx f (4- infx g ( x )1 SUPX I f ( 5 )- d x )1 . From this 2 ) is easily proved. Moreover, one also has that Ibn(t, x , ~ , P , w) bn(t,x,q,pl,w)l SUP^,^ Ib(t,x, ~ , P , w ) b ( t , x , q,p1,w)I . Hence 3 ) is also obtained. Finally, notice that Ibn(t,xn,qn,Pn,w) -b(t,x,q,P,w)I 6 Ibn(t,xn,qn,Pn,w) - bn(t,xn,qn,P,w)I Ibn(t,xn,qn, P , W ) - b(t,x,q,p,w)I Ct(z,w)(Pn(z)- P ( z ) ) ~ ( d z )+l Ibn(t,xn, qn, P,W) - b ( t , x ,9, P,w)I . ( x ,q ) , by that b(t,xn, q,, p, w ) has a less than linear growth If ( x n ,qn) in ( x n ,q,, p), so b(t, x,, q,, p, w ) is bounded as w is fixed. By the definition of an infimum there exists a sequence (y,, Cn) such that b(t, ~ nqn,, P, w ) 2 bn(t,xn, qn, P,w) 2 b(t,yn, Cn,P , W ) + ( n v s ( t )l)~ n - ~ n l (+n v c ~ ( t ) ) I C-qnln 2 b(t,yn,Fn,P,w) - 1 Notice that we have already shown that b(t,xn,qn,p,w) is bounded, as t and w are fixed, so by the above inequaity one finds that lyn - x,I 4 0 , IFn - qnl --+ 0 , as n 4 m. From this one has that limn,, b,(t, xn, q,,p, w ) = limn-+, b(t,yn, Cn,P, w ) = limn-+, b(t,xn, qn, P, w ) = b ( t , x , 9, P,w). Now applying condition 2' in Theorem 267 4 ) then follows. The proof is complete. w By using Lemma 271 and the local time technique of treatment for disone can derive Theorem continuous points of b with respect to x in 267. Proof. Take bn(t, x , q,p, w ) from Lemma 271. Obviously, b,(t, x , q,p, w ) is jointly Lipschitzian continuous in ( x , q,p) E R1 x R'@~'x L ; ( . ) ( R ' @ ~ ~ ) . Hence by Theorem 234 there exists a unique solution (x;, q?, p;) satisfying (8.8) with coefficients bn for each n = 1,2,. . .. Set h ( t ,x , 9, P, w ) = k o ( l + 1x1 + lal + llpll). Again by Theorem 234 there exists a unique solution (Et, u t , v t ) satisfying the BSDE ~ s d ws JZ v s ( z ) f i k ( d ~d > z). [t = X h ( ~ , U s , 2197 W ) ~ -S Moreover, by property 1)-3) of Lemma 271, applying the comparison Theorem 251 one finds that x? %:+' t t , V n . Hence there exists a limit P - a.s. xt = lim x7,W 2 0. n-+m Moreover, by Lebesgue's dominated convergence theorem one also has that 2 lim EJ: 1x7 - xtl ds = 0. n+m Applying Ito's formula, one easily shows that there exists a unique (qs,p,) E ~25t (~ l @ d) l x Fit (RIBdz) such that

< <

<

+ < IJz

A

JkT <

+ JkT

<

lirn E(J; Iq; - qs12ds

n-+m

+ J:

JZ

JkT

Ip; -ps12 T ( ~ z ) ~=s0.)

8.14 Minimal Solution for BSDE with Discontinuous Drift

261

From these one can take a subsequence {nk) of {n) , denoted again by {n) again, such that dt x d P - a.e. q: -+ qs, in R1md1; p: -+ ps, in L;(-)(R'@~~); and P:I -PSI 2 r ( d z ) d ~ ) 1/2",n = 1.2, ... E(S; 19;: qs12ds Hence E(So sup(lqrl + llP;11)2dt < 00.A similar conclusion holds for x?.

+

sZ

<

n

Now notice that Sz p:(z)q(ds, dz), xZ" = + AT Sot CdwS where A; = bn(s, x:, q,: p:, w)ds. Hence there exists a limit (if necessary, take a subsequence) At = lim A;, n--+a and we have s t = 20 At qsdws Sz pS(z)q(ds, dz). Notice that Ibn(~,x:,s$n,p:,w)l k0[1+supn(lx2"I IqTI + IIPZ"II)I. Hence AT is a finite variational process, and so also is At. Let us show that

+

+

+ sot

+ +$

<

+

Indeed, from ml(a : (a,q,p) E G) = 0, applying the occupation density formula for the local time of a semi-martingale (Lemma 159), one finds that 2 t 2 O = SG, L?(x)da = Sot I { Z ~1981 E Gd~~2) So I{(X.,~,,~~)EG) ~QS] d ~ . Now by &sumption lqSl2 > 0, aa (xs,qs,ps) E G, so m l ( s E [O,t]:(xs,qs,Ps) E G ) = O . Therefore, by conclusion 4) of Lemma 271, Ib(s,xs,qs,~s,w)-bn(s,x!,q,",p:,w)lds J ~ ~ ~ c x s , q , , p I, b ) ~( s~,)x s , q s , ~ s ,-~b)n ( ~ , x ? , q ? , ~ : , w ) l d ~ and (8.66) is derived. Now suppose that (x:, qi,pi) is another solution of (8.8). By Lemma 271 bn(4 x,q, P, w) b(t, X, 9, P, w). Hence applying Theorem 251 one has that x? x:,Vn. Letting n -+ oo, we obtain xt xi. Therefore xt is a minimal solution of (8.8). By Lemma 230, under condition 3' the solution is also unique. H Applying Theorem 267 we can get a better comparison theorem for solutions to (8.8) with a discontinuous coefficient b2 (t, x, q, w). Actually, we have

Jot

<

<

<

<

Theorem 272 Assume that conditions lo and 3' of Theorem 251 hold and, moreover, assume that b2 satisfies all conditions in Theorem 267. Then the conclusion in Theorem 251 still holds. Proof. As the proof of Theorem 267 we construct bn from b2 by Lemma 271. Denote by (xin,qin,pin) the unique solution for (8.8) with the coef-

262

8. Option Pricing in a Financial Market and BSDE

<

<

<

ficients b,. Since b, b2 bl, by comparison Theorem 251 xin xi. Hence lim x'" < xl. n-+ca 't However, by the uniqueness of solutions to (8.8) with coefficient b2, one has xf = lim xkn. Hence xf x i , and the proof is complete. n-+m

<

8.15 Existence of Non-Lipschitzian Optimal Control. BSDE Case In some practical cases one needs to consider the optimal consumption problem. Suppose the wealth process of a small investor satisfies the standard BSDE:

which is a special case of (8.52) with jEb = 0. Then under appropriate conditions (for example under the conditions in Theorem 244 with 6, = 0), we can rewrite it as (8.32). Or we can rewrite it as FT - a.s

if dCt = csds, and we denote Zt = ee--fotrsdSxt, etc. Therefore, for simplicity, we can consider the simpler BSDE (8.68) directly. To simplify the notation let us still write P for FT, (xt, r t ) for (Zt, Ft), and (wt, Nt) for ( G t , r t ) . Then we can consider the following wealth process for a small investor: For 0 t 5 T, P - a.s.

<

Furthermore, if we denote qt = rtot, pt as

where we write u(t) = c(t).

= ntp,,

then (8.69) can be rewritten

8.15 Existence of Non-Lipschitzian Optimal Control. BSDE Case

263

In this section we are going to apply the results found above to solve some special kinds of optimal stochastic control problems with respect to (8.70).We obtain results on the existence of some non-Lipschitzian optimal controls for some special stochastic control problems with respect to such BSDE systems with jumps. Some of the results can also be explained in the financial market. Let us consider the following more general d-dimensional BSDE system as (8.8):for 0 t T ,

< <

where xy,u(s,x~,q,",py) E R~,Q," E~~"~l,pE y (Rz ~ ) " and ~ ~u E ,

U,

u = ~ ( x, t ,q,p) : ~ ( x, t ,q,p) is jointly measurable such that

'=

{ (8.71) has a unique solution (xt,qt,pr),and lu(t,x,q,p)l< 1x(8.72) 1' <

where 0 < 0 1 is a given fixed constant. We have the following

Lemma 273 . Let V ( t ,x ) = $ lx12 T - t , for x E R ~t ,E [0,TI, then V ( t ,x) satisfies inf (u.&v)+l(a/at)V-

+

=0, 0 < t < T ,

UER~,~UI
V ( T ,x ) = $ 1x12. Moreover, if uO(x)= -IXfox/ , then IuO(x) 1x1' u o ( x ) . & ~ ( t , x ) = inf (U.&V)=-IXI~+~.

I<

XI'-^

, and

UER~,~UI
Proof. The conclusion can be checked directly. Now let

Then uO(x)is continuous and such that ( x - Y ) (uO(x)- uO(y)) --(I4- I Y I ) ( I ~ ' lvlP>G 0. Hence, by Theorem 246, (8.71) has a unique solution for such coefficient uO.However, in the case that p is given and satisfies 0 < p < 1, then such u0 is non-Lipschitzian in x. Denote this solution by ( x f ,q:,p:). Then by Ito's formula v uO(x:). &V 1 - 1x911+P)ds V(T,x$) - V(t.x:) = J : ( ~ -

<

+

8. Option Pricing in a Financial Market and BSDE

264

~ : ( 1 ~ ; 1 ~+

+martingale+; Jz lp:12 r ( d z ) ) d s- ~ : ( 1 - lx:ll+')ds =martingale+: J:(1q,012 + Jz r ( d z ) ) d s- J T ( 1 - ~x;l~+')ds. Similarly, for any u E U V(T,x$)- V(t.xy) <martingale+% c ( 1 q ; 1 ~ JZ lp;12 r ( d z ) ) d s- JT(1 - (x;ll+')ds. Therefore, for all u E U martingale+; 1xy12 J:(lq;12 Jz 1p;l2 r(dz))ds + J:(-l+ lxyll+@)ds2 - T t =martingale

lp:l2

+

+

+

+% +

+

+

+

1 ~ 1 1 ~ JtT(lq;12 Jz Ip;12 r(dz))ds !:(-I+ 1~;1~+')ds. Hence EZr(; 1x?12 J:(1q;12 & lp;12 r ( d z ) ) d s lx:ll+'ds) EX.(% lx:12 J:(1q;12 + J z I p ; ~ ~ r ( d z ) ) d sJ: Ix!l1+'ds). Thus we have proved the following Theorem 274, which shows that a nonLipschitzian feedback optimal stochastic control exists.

+1

>

+

+

+3

+

Theorem 274 . Define u O ( x )by (8.73), and for any u E U, where U is defined by (8.72), let J ( u ) = E(; lx$12 !j ~oT(1q;l~ JZ lp;12 r ( d z ) ) d s J: 1x;l1+' ds), where (xy, q?, p?) is the unique solution of (8.71) for u E U. Then 1) u0 E U, 2) J ( u ) 2 J ( U O ) ; for all u E U.

+

+

+

The above target functional J ( u ) can be explained as an energy functional. We can also discuss another kind of optimal control problem which can be explained in the financial market. Still consider the BSDE system (8.71), but this time with xy,u(s,x;,q:,p;) E R1,q; E R1@dl,p;(z)E RIBda,and consider the admissible control set

u = u ( t ,x , q, P ) : u ( t ,x , q, p) is jointly measurable such that = (8.71) has at least a solution ( x t ,qt, pt), and lu(t,x , q, p)l 1x1' (8.74) where 0 < ,O < 1 is a given fixed constant. Denote the target functional by J ( u ) = S U P { E ( 1 ~~ ~~ 1 ~d~" - l$l2 - g ( l @ 1 2 + Jz lp;l2 ~ ( d z ) ) d s ) ) = sup{E[~?(xyll+'ds - 1x$12

{

+(J?

<

g(lg.:12JZ

lx;ll+'ds + I~;l~r(dz))d~)I}, for each u E U, where ( x y ,q?, py) is any solution corresponding to the same u. We have the following Theorem 275 Let , x,/ lxll-' u O ( x )= I+ Then 1) u0 E U,

.

8.15 Existence of Non-Lipschitzian Optimal Control. BSDE Case

265

To prove Theorem 275 one only needs to use the following

i

+

Lemma 276 V(t, x) = 1x12 T - t satisfies (d/dt)V 1 1x1~'' = 0,0 t < T, (u . &v) sup u€R1,lul<1x18 V(T,x) = 1 x 1 ~ . Moreover, if uo(x) = Ix+ox/ 1x11-8 , then luo(x)1 5 1x1' , and 118 uO(x).&v(t,x)= sup (u.&v)=IxI u€R1,lul< l ~ l o

+ + 4

{

<

'

The proof for 2) of Theorem 275 is similar to the above. However, since now the BSDE is in R1, and uo(x) is continuous and depends on x only, by Theorem 267 it is seen that (8.71) has a solution for u = uO. Hence 1) of Theorem 275 is proved. Now let us explain Theorem 275 in terms of the financial market. If we see (8.71) as the equation for the wealth process xt of a small investor, the control u(t, at, qt,pt) can be explained as his generalized feedback consumption process, and (qt,pt(.)) as some generalized portfolio process for stocks, then in the case p = 1 we can interprete the target functional J(u) as a subtraction of the total summation of the squares of wealth and the squares of money of bonds in the total time interval from the square of initial wealth (or say initial investment) of the investor. So J ( u ) can be seen as some generalized utility functional for the investor. Theorem 275 tells the investor that he can get a maximum utility if he choose the consumption law to be u(xt) = xt/ lxtll-' , i.e. when the wealth process xt 2 0, he should consume the money xt/ ~xtll-' ; and when xt 0, he should borrow the money -xt/ ~xtll-'. Let us explain this in more detail. Suppose that in the market there is a bond with a price that satisfies an equation: d ~ : = Pfrtdt, P$ = 1; and there are N = d + m stocks, with prices that-satisfy the following SDEs d P j = Pj- [bidt crikdwt CrZlpikd@], pi - p o, a i = 1 , 2 , . . . ,N. If the market is self-financed, then the wealth process of a small investor will satisfy dxt = rtxtdt nt[bt - rJ]dt ntatdwt n t p t d ~ 0t , t T, where nt = (n:, . . . ,T?) is the profolio of the investment of the investor. (See (8.4)). Now suppose that the investor also consumes the money dCt = ctdt during the time dt, then his wealth process will satisfy

<

+ c:=,

+

+

+

+

< <

266

8. Option Pricing in a Financial Market and BSDE

According to Theorem - 243 we have the following fact that assumes b, r, a,p are all bounded ~ : ' ~ - ~ r e d i c t a b lprocesses, e and 5,' exists and is also bounded and, furthermore, that ct is predictable such-that E [el2 dt < oo, and we will assume that N = d m, X E GtNand E 1x1~ < oo. Then there exists a unique wealth process xt E L$(R1) and a unique port-

+

~ ) (8.75). Furthermore, assume (B:)~ folio process xt € L ~ ( Rsatisfying 1 , V i = I , . . . , m , where Bt =

[$]

= 5,' [bt

<

- rJ], then by Theorem 244

(8.75) can be simplified to & - a.s. dxt = rtxtdt - ctdt ntatdGt x t p t d x t , xT=X, O I t < T , where FT P and, Gt and are respectively, a d-dimensional BM and an m-dimensional pure jump martingale with a new compensator $(A - 8,)ds. (See (8.33)). In the case that Bt is non-random, then Tt is an m-dimensional centralized Poisson process. Furthermore, if 8 = Bt is an m-dimensional constant vector, which does not depend on t, then Nt is a m-dimensional centralized stationary Poisson process with the conpensator (1-8)t. Now let Et = ee--fat Tadsxt.Then the above BSDE can be rewritten as PT- a.s Et = X f z ~ ( s ) d s J~~ Z,a,dGs Z S P , d ~ ( s )t, E [O,T]. For simplicity of notation we still write P for &, (xt, xt) for (Et, Zt), and (wt, Kt) for ( G t , r t ) . Then we can write the above BSDE as P - a s .

+

{

+

rt

N

-

+

where we set U(S)= c(s). Again by Theorem 244 let [qt,ptl = xt[at, ~ t =l r z t . Then (8.76) is equivalent to

and ~ ,I , , . . , N . xt = s t , n: = ~ 3 d , ~ q ! ( ~ ~ '~) j~mi_ ~ & ( 5 : ' ) ( ~ + ~i )= Thus the admissible control set in (8.74) is

+

u = ~ ( tx,, xu, xp) : u(t, x, ra, xp) is jointly measurable such that (8.76) has at least a solution (xt, xt), and lu(t, 2, xu, xp)l lzlP (8.78) where 0 < p 1is a given fixed constant. The target functional in Theorem 275 is

<

<

8.16 Existence of Discontinuous Optimal Control. BSDEs in

J(u) = sup(E(2 :J 1x:11+8 d ~ - 1 x $ 1 ~ - f l ( 1 ~ : a , 1 ~ ln:p, +~ = s u p { E [ r IX:I"~ ds - lx$12 lx:ll+' ds

+

(c

R1

267

l2 ~ ( d z ) ) d s ) )

- g(1.:asl2 + Sz lr:p,l2 4 d ~ ) ) d ~ ) l l for each u E U, where (xr, T?) is any solution of (8.76) corresponding to the same u, and Theorem 275 tells us that the following theorem holds true.

Theorem 277 Let uO(x)= Izz0x/ ~xll-fi. Then 1) u0 E U, 2) J ( u ) < J(uO),Vu E U. For the special case that P = 1,p = 0, a, = I d x d , where I d x d is the d x d unit matrix, and N = d, X E the target functional becomes J(u) = ~UP{E[(J: 1x:12 ds - 1x,"I2)+ (f'(lx:12 - I T : I ~ ) ~ S ) I ) We can explain this as an ultility functional for the investor. Obviously, this ultility has contributions from two terms: the first term is the subtraction of the summation of the wealth square 1x:12 in the total time interval from the initial wealth square 1x,"12 of the investor, while the second term is the summation of the subtraction of the wealth square 1x:12 from the in the total time interval. Or, roughly speaking, the porfolio square ultility is positively proportional to the money square deposited in the bank and the total money square subtracted from the initial investment square. The surprising thing is that for such a ultility functional the zero consumption is not the best. To make this ultility attain its supremum the optimal consumption law should be: u(xt) = xt/ ~ x ~ l l i.e. - ~ when , the wealth process xt 2 0, the investor should consume the money zt/ lxtll-P (a positive consumption st/~ x ~ l l - and ~ ) ; when zt 0, he should borrow or, say, he should sell some stocks to get the money the money -xt/ -xt/ ]xtll-fl (a negative consumption xt/ \xtll-') (Notice that here -the amount of units for the i-th stock bought by the investor can be negative, that is, he may sell it to someone else, so the investment at time t may be negative: xt 5 0).

zT,

<

8.16 Existence of Discontinuous Optimal Control. BSDEs in R1 Consider the following BSDE system: for 0

< t < T,

8. Option Pricing in a Financial Market and BSDE

268

where ko is aconstant, x ~ , u ( s , x ~ , q ~E,R1,qy p ~ ) E R'@'~', p:(z) E R'@'~z, X E 5;'k,and u E U, u = u(t,x , q,p) : u(t,x , q, p) is jointly measurable such that (8.79) has a unique solution (xt,q t pt ), and I ~ ( t 1 ~ 1p)l9 ,G 1. U= (8.80) W e have the following

{

Lemma 278 . Let V ( t ,x ) = 1x12 T - t , for x E R , t E [0,TI, then V ( t ,x ) satisfies

4 +

+ - Iqllxl - ko 1x12 = 0,O < t < T ; V ( T ,X ) = 4 1 x 1 ~ . (a/at)V -

inf

[(lqlu - kox) . &v] 1

u€R,IuI
Moreover, if uO(x)= -Ix20x/ 1x1, then luo(x)l 5 1, and u O ( x ) . & ~ ( t , x ) = inf ( U -B ~ V ) = - ) X ) . u€Rd,luI
Proof. The conclusion can be checked directly. Now let

Then uO(x)is discontinuous in x and does not depend on q,p such that ( x - x') . (191uO( x ) - 1q'l u0(x')) lql ( x - x') . (uO( x ) - u0(2')) Ix - ~'114- 4'1 Ix - x'l la - 9'1. Hence, by Theorem 267, (8.79) has a unique solution for such a coefficient uO.Denote this solution by (x!,~:,~!). Then by Ito's formula 0 V ( T ,x!) - v(t.2:)= - 0 u0 (x,) - koxt)&V 1 2 1x81 - 4 ~x~l~)ds+martingale+~ Jz ~(dz))ds 2 -c ( 1 - kO1x81 )ds =martingale

+

<

<

C(&V (Iq, 1

I$/

J:(IcI~

IqtI I X ~ I

+ /P!I

J:(IBI~

+

+

+& lZlptI2 ~ ( d -w1 3 1 - lqtl Ix:I - ko lx:I2)ds. Similarly, for any u E U V ( T ,x?) - V ( t . x r )<martingale++f(lq212 JZ lp$ a(dz))ds +;

J:(lel2+ Jz

+

Ip:12 +,)ds

- !:(I

- In:lIx81 - ko Ix:lZ)ds.

8.16 Existence of Discontinuous Optimal Control. BSDEs in R1

269

Hence E5"+ 1x;12 + S,T(1q:12 + JZ lp:12 a ( d 4 ) d s ~:(1q:11x:1+ ko lx:12)ds) 2 E V + 1x:I2 f(ld12 + l ~ : l ~ + z ) ) d s + S:(l&l lxOl+ ko Thus we have proved the following Theorem 279, which shows that an optimal feedback Bang-Bang stochastic control exists.

;

+

+i

h

Theorem 279 . Define u O ( x )by (8.81), and for any u E U, where U is defined by (8.80), let J ( 4 = E ( ; 1x;12 + s:(19:12 + SZ lp:12 ~ ( d 4 ) d ~ + s:(19:11~:1+ ko Ix:I2)d4 where (xy, qy,py) is the unique solution of (8.79) for u E U. Then 1) u0 E U, 2) J ( u ) 2 J(uO),for all u E U.

;

Similarly, Consider the following BSDE system: for 0

where ko is a constant, xy, u(s,x:, q,",p:) E R1,q; E X E 5;lk, and u E U,

~


~

p:(z) ~ 1 E RIBdz, ,

8

u = u(t,x, q,p) : u(t,x,q,p) is jointly measurable such that (8.79) has a unique solution (it, qt,pt), and Iu(t,x, 9,p)l < 1. U= (8.83) We have the following

{

Lemma 280 . Let V ( t , z ) = 1x12 T - t, for x E R, t E [0,T ] , then V ( t ,x ) satisfies

+

270

8. Option Pricing in a Financial Market and BSDE

Moreover, if uO(x)= -Iz#ox/ 1x1 , then IuO(x)I5 1, and U~(X),&V(~,X)= inf (u.&v)=-1x1. u€Rd,(ul<1

Proof. The conclusion can be checked directly. Now let

Then u O ( x )is discontinuous at x = 0 and does not depend on q,p. Hence, by Theorem 267, (8.82) has a unique solution for such a coefficient uO.Denote this solution by (x!, q:,p:). Then by Ito's formula 0 0 0 V ( T ,x$) - V ( t . x f )= (lqs u (x,) - kox!)&v 1 1x91 ko 1~:1~)ds+martingale+i f(Iq:l2 lp9l2 ~ ( d z ) ) d s 2 -f(l+ 1x91 +kg 1x91 )ds =martingale

+

C(&V+ I

+ Iq:1

+

+

+

ff(l&l2

+$ + JZ ~p:l~w(dz))ds - c ( 1 + lq:1 1x91 ko lx912)ds. Similarly, for any u E U V ( T ,x:) - V(t.x;) >martingale+? r(lq,"l2 JZ lP:l2 r(dz))ds - J T ( l + 1q:I Ix:I + ko lx:12)ds. Therefore, for all u E U martingale+: 1x211~ J:(lq,"12 JZ lp:12 .rr(dz))ds

+

+

++

+

IxUl+ ko ~ x : ~ ~ )5d s -~ : ( 1 =martingale+; 1x:I2 f hT(Iq:l2

- T +t

+ JZ Ip:l

+

2

~(dz))ds

optimal feedback Bang-Bang stochastic control exists. Theorem 281 . Define u O ( x )by (8.84), and for any u E U, where U is defined by (8.83), let 1xo"12 g(lq,"12 Jz lp:12 ~ ( d z ) ) d s J(u)= - ~,(ln:lIx:l+ ko Ix:l2)ds), where (xy ,qr, py) is the unique solution of (8.82) for u E U. Then 1) u0 E U, 2) J ( u ) 5 J(uO),for all u E U.

~ ( 4 +4

+

The target functionals in Theorem 279 and Theorem 281 can be explained as some energy functionals.

8.17 Application to PDE. Feynman-Kac Formula

271

8.17 Application to PDE. Feynman-Kac Formula In this section we will discuss the following problem. If we have a classical solution for some partial differential equation and with integral term, how could we express it as the solution of some BSDE or FSDE under some appropriate conditions. Such an expression is usually called a FeymanKac formula. Suppose that D c Rd is a bounded open region, dD is its boundary thatt is supposed to be sufficiently smooth, and set D C= Rd- D. Consider the following integral-differential equation (IDE)

=

f (t, x, u(t, x), uk(t, x) . a(t, x), u(t, x + c(t, x, .)) - u(t, x)),

(8.85)

where for the coefficients b, a , c we make the following assumptions (C): C.l b : [0,T] x Rd --+ Rd is bounded and measurable; C.2 a : [0,T] x Rd Rd@' is bounded, measurable and, moreover, there (A, aX) ko lX12 , for all exist constants SO,ko > 0 such that So [X12 X E Rd,where a = aa* = ( ~ i j ) & ~ , a n d(., .) is the inner product in R ~ ; C.3 c(t, x, z ) : [0,T] x Rd x Z Rd is bounded, measurable, and -+

<

<

-+

Jz

114s~ x, 4121~

( d 54 h

C.4 2 (Yl - Y2, b(s, Yl) - b(s, ~ 2 )+ ) 14% Yl) - 4% Y2)12 + Jz IC(S,YI,Z) - C(S,YZ,2)12 ~ ( d z ) CT,N(S)PT,N(~Y~ - y212), as l ~ l l , l v 2 l< N a n d t ~[(),TI, where CT,N(~) 2 0 is non-random; and pN(u) 2 0 defined on u 2 0 is non-random, increasing, continuous and concave, and such that u) c ~ . ~ ( t ) 2< d tm, So+d u / ~ ~ , ~=( m, for each N = 1,2, ... and each T < co. For functions f , 4, $ we make the following assumption (D): D.1. f : [O,T] x D x R1 x R1@' x L;(.)(R1) -+ R1 is measurable such that

<

8. Option Pricing in a Financial Market and BSDE

272

for all u E C112([0, T ] x Rd)satisfying (8.86); D.2. I f ( t , x , r l , q l , p d - f ( t , x , n , q 2 , ~ 2 ) 1 k o ( h - r21 + 141 - 921 llpi - pzll)), for all r1,r2 E R1;q1,q2 E R1@",p~,p2E L:(.)(R1); D.3. 4 E C(Rd); (4 : Rd -+ R1); $ E C([O,T ]x D C ) ;($ : [0,TI x DC -+ R'); D.4. the boundary d D is sufficiently smooth; D.5. I f ( 4 2 , r, q, p)l k0(1+ Irl 191 IIPII). Consider now the following forward SDE with Poisson jumps in R~ for any given (t,x ) E [0,TI x D:

<

+

+ +

<

< <

< <

as t s T , where w,, 0 s T , is an r-dimensional BM, and for simplicity fik(dr, d z ) is 1-dimensional Poisson martingale measure defined in (8.8). Under condition (C) by Theorem 185, (8.88) has a pathwise unique strong solution. Here "strong" means that the solution y?" is St,,-measurable, where St,' = u { w ( r ) - ~ ( t N) k,( ( t , r ] ,U ) ;for a11 t r S , U E !R(Z)). Now for any given (t,x ) E [0,T]x D let T = rt,= = inf { s > t : &" 4 D ) ,and T = ~ t = ,T, for ~ inf{0). Write f ( s ,w ) : f ( s ,w ) is &,, - adapted, R~ - valued such that E SUP If ( s A T, W )l2 < m

< <

s> t

f ( s ,W ) : f ( s ,W ) is St,s- adapted, R ~ -@ valued ~ L $ ~( , ~~ ~

8

=~

1

)

suchthat

etc. Then we have the following

lf(t,w)12dt<m

E J I

tA7

Theorem 282 Under assumption (C) and (D) for ( f ,I$,$) there exists a unique (x,, q,,p,) E S&, (R1)x Lkt,. (R'@")x FZt,+(R1),which solves the

8.17 Application to PDE. Feynman-Kac Formula

273

following BSDE

where y,

= y;",t

< s < T,is the solution of

(8.88).

Theorem 282 is true by virtue o f Theorem 246 in this Chapter. Now b y Ito's formula one finds that

Hence we can arrive at a new Feynman-Kac formula as follows:

Theorem 283 (Feynman-Kac formula). Under assumption (C) and (D) xs = u (s,Ys) ,9s = 4%y s ) ~ x (us ,u s ) , Ps (.) = u ( 8 , Ys- + 4% YE-,-1) - u ( s ,ys-) , where y, is the unique strong solution of the FSDE (8.88), and ( 2 5 , qs, P S ) E S$,,. ( R 1 ) X L&t,+(R1@T) X Fzt,, ( R 1 ) is the unique solution of the BSDE (8.89). Hence x , has a "Markov property". Furthermore, we have a Feynman-Kac formula ~ ( 2t ),= xs Is=t = Et,xxt, where u ( t , x ) E C 1 ' 2 ( [T] ~ , x R ~Rm) , is the unique classical solution of (8.85) and (8.86), if it exists, we denote by w ) = EF(y>", w). Et,,F(y,, w ) = WY,, Proof. Applying Ito's formula to u ( r , y T ) on r E [S A T , T ] , and using Theorem 282 the conclusion is derived.

.

Remark 284 We will say that x,,t 5 s, has a "Markov property", i f it can be expressed by some function of y,, where y, is a Markov process, i. e. there exist Bore1 functions fi such that x , = fi ( s ,y,) , W 5 s. Naturally, we can also have another Feynman-Kac formula for the classical solution o f (8.85) and (8.86), which only needs t o know the existence o f the solution o f the FSDE (8.88). However, in this case the function f i n the free term (in some cases we also call it a "force")cannot be so general. In fact, let us suppose that in all o f the above discussion f only depends o n t and x. That is, f = f (t,x).In this case (8.85) becomes

274

8. Option Pricing in a Financial Market and BSDE

Assumptions ( D ) becomes (D)I: D.l'. f : [0,T ] x D -,R1 is bounded and measurable; D.2'. the same as D.3; D.3'. the same as D.4. Then applying Ito's formula to u ( s ,y,) we immediately obtain the following theorem.

Theorem 285 (Feynman-Kac formula). Under assumptions (C) and (D)I if u ( t , x ) E C1y2([0,T]x R ~Rm) , is a classical solution of (8.91) and (8.86), then 4 4 x) = E t , o [ L < ~ $ ( 7y7) , + G = T ~ ( Y T-) f ( r ,yT)dr1, where y,, s 2 t is the unique strong solution of the FSDE (8.88).

.JL7

Notice that the conditions in Theorem 285 are weaker than those in Theorem 283, so the conclusion of Theorem 283 still holds in this case. However, obtaining a Sobolev solution for (8.85) and (8.86) is much easier than finding a classical solution. So one natually asks can we have similar Feyman-Kac formulas for the Sobolev solutions of (8.85) and (8.86)? The answer is positive. For this we first change the condition on 4 and $ to satisfy the following Assumption (D)" D.1". f : [O,T]x D x R1 x RIWPx L;(.)(R') -+ R1 is measurable such that g" E Lp([O,Tl x D l , where

for all u E WJ12([0,T ] x Rd)satisfying (8.86); D.2". The same as D.2. 20-$) d D.3". 4 E Wp ( R ); (4 : Rd -+ R1); $ E W ; * 2 ( [ ~ x, TDC); ] ($ : [O,T]x DC + R1); D.4. d D E 02;(for the definition of O2 see [68]). Then we have the following theorem.

Theorem 286 (Feynman-Kac formula). Under assumption (C) and (D)" i f p > 2d+4 then , ( s ,Y,) , x s = u ( 3 , Y , ) , QS = ~ ( syS)dxu Ps (.I = u ( s ,ys- + c(s1 ys-, - u ( 8 , ys-) , where y, is the unique strong solution of the FSDE (8.88), and ( X ~ , Q ~ , PE~S) $ ( [ ~ , TRm) ] ; x L $ ( [ ~ , TRmBdl) ]; x F g ( [ t , ~RmWda) ]; is the unique solution of the BSDE (8.89), which exists. Hence (x,) has a "Markov property". Furthermore, we have a Feynman-Kac formula u(t12 ) = 2 s Is=t = Et,xxt, ~ ,] x R ~Rm) , is the unique Sobolev solution of where u(t,x ) E W i . 2 ( [ T (8.85) and (8.86), which exists.

8.17 Application to PDE. Feynman-Kac Formula

275

However, to prove this theorem we first need to generalize the Ito formula to functions with Sobolev derivatives, and then to prove it.[15313[1661 Moreover, we also need to explain the meaning of a Sobolev These matters need more space than we have here, so they will not be discussed here.

Optimal Consumption by H-J-B Equation and Lagrange Method

In this chapter we will show how to use the forward Hamilton-JacobiBellman (H-J-B) equation technique to solve the the optimal consumption problem. We also establish the necessary conditions for the optimal consumption by a stochastic Lagrange method, which also helps us to find the optimal consumption. Some examples for such applications are also given.

9.1 Optimal Consumption In the last sections of the previous chapter we have already seen that in a financial market the wealth process of a small investor under appropriate conditions can be written as: P - a.s.

Suppose that the investor has a target functional J(c) = E[g(xo) f (s,xs, ns, cs)dsI, and he has an admissible consumption set: U = c = c(t) : c ( t ) is - adapted such that E ~ c ( tl2) dt < 00. He wants to find out an optimal consumption cO(.)E U such that it can minimize the target functional:

+ S,T

{

9. Optimal Consumption by H-J-B Equation and Lagrange Method

278

Let us consider the following forward Hamilton-Jacobi-Bellman equation:

+f (t,x , n , c)]= 0, V ( 0 ,x) = g(x); 0 < t < T ;

(9.2)

where we assume that f (t,x, n , c) is jointly measurable, and 2

I7T~tl

d

=

Ic~=~ N

2 TiOikI

.

(9.2) will give us an approach by which to find the optimal consumption if (9.2) has a solution V ( t ,x ) E C1~'.In fact, we have the following Theorem 287 Suppose that (9.2) has a solution V ( t , x ) E C112,by the measurable version theorem[l8Q there exists a jointly measurable function c0 = c0(t,x, X ) such that

+ c & v ( ~ , x )- &v(t,x)(17ratlz EL, n"pfk) - V ( t , - qg xL1nipik)

- infc[-zV(t,x) a

- CF="=1V(t,x +

2)

+ f(tl 2 , n1c)l. Moreover, if there exists a unique solution (x:, 7r:) satisfying T T T xt = X J cO(s,xs,ns)ds- Jt K ~ o -~ St ~ T w ~ / J~, ~ @ (ts€) ,[O, TI, 5 then x: and nt are the optimal trajectory and optimal portfolio, respectively, such that T E [ 9 ( 4 )+ So f (s,x!, n:, c o b ,x:, 4 ) ) d s l = infc(.),v E[g(xo)+ f ( s ,xs, x S , cs)dsI.

+

JOT

Proof. Applying Ito's formula to V ( t ,x!), t E [0,TI, where (x:, 7r:) satisfies T x: = X cO(s,x!, n:)ds - St n!asdur, ntpsd@(s),t E [O, TI, one finds that E[V(T,X ) - V(O,x8)l= f ( 3 , x!, n:, ~ ' ( 3x:, , n!))d~, where we have applied the forward H-J-B equation (9.2). So E[V(T,X ) = g(xg) E f ( s ,x!, nz, cO(s,x!, n!))ds. Applying Ito's formula to V ( t , x i ) t, E [O,T],where ( x i ,x i ) satisfies T X; = X StT csds - S,T - Jt ~:p,d@(s), t E [O, TI, one also has that Vc(.)E U E[V(T,X ) - V(O,231 E f ( s , ~ ;T:,, c.)ds. where we have also applied the forward H-J-B equation (9.2). So E [ W ,X ) g&) f (s,x:, a;, cs)ds. The conclusion follows.

+ stT

EST

+

fl

+

<

<

+E S ~ T

9.2 Optimization for a Financial Market with Jumps by the Lagrange Method

279

In general the solution of (9.2) is not easy to obtain. However, for some special cases we can solve the problem. Let us always assume that i~,' = [at,pt]-l exists and is bounded. Write c = c(t, x , s) : ~ ( xt ,,s) is jointly measurable such that (I = (9.1) has a unique solution ( x t ,st), and lc(t,x , n)l where 0 < /3 5 1 is a given fixed constant, and U is an admissible consumption set. The following Theorem shows that a non-Lipschitzian feedback optimal stochastic consumption exists.

{

< XI'

Theorem 288

. Define co(x) = cO(t,x , s) = -Ix#ox/

+;

= E(; 1 x 6 1 ~

+ x;l

1 ~ : ~

.and let

e-ff~(1s:os12 stptkl2)ds 1x;12 [x:~l+')ds), where a is a constant, (xz,s:) is the unique solution of (9.1) for c E U. Then 1) c0 E U , 2) J ( c ) 2 J ( c O ) for , all c E U. J(C)

+ S,T e-as(-z

+

The above target functional J ( c ) can be explained as an energy functional, provided a 0. To show Theorem 288 we need the following lemma.

<

Lemma 289 V ( t ,x ) = 4 e p f f t1x12 + t satisfies inf (u . &v) ax - 1 2 ae 1x12 + 1 + e-"t -(d/dt)V + UER~,IUI
IX~"'

= 0,

-

Moreover, if u O ( x )= -Ix#ox/ 1x1~-' , then luo(x)l 5 lxlP, and 8 v ) -- -e-at 1x11+8. u0(x).&v(t,x)= inf UER~,~U~G~X~~ Lemma 289 can be checked directly. Now to show Theorem 288 one only needs to apply Ito's formula to V ( t , x z ) and V ( t , x f ) , where x: satisfies (9.1), and x! satisfies (9.1) with co(x),respectively, and then using Lemma 289, in exactly the same way as in the proof of Theorem 274, one arrives at the conclusion of Theorem 288. We can also consider some other target functional similar to Theorem 275, but to save space we omit it.

9.2

Optimization for a Financial Market with Jumps by the Lagrange Method

In this section we discuss the optimal consumption problems for a financial market with jumps. We provide a proof of the necessary conditions for optimal consumption by using a Lagrange method for models driven by Brownian motions as well as Poisson jump processes. Such a method was introduced by Chow (1997) for continuous models.

280

9. Optimal Consumption by H-J-B Equation and Lagrange Method

9.2.1

Introduction

As we know, the optimal consumption problems for financial models described by stochastic differential equations driven by Brownian motion can be discussed by many methods, e.g. by the martingale method (Karatzas et al., 1991), by the backward stochastic differential equation technique (El Karoui et al., 1997), by the dynamic programming method - Bellman equation (Cox et al., 1985), and others (Merton, 1971). However, for practical use more direct ways which can be easier to evaluate are preferred. In this sense the Lagrange method may be an appropriate one (Chow, 1997). Because by this method the necessary conditions are much easier to derive, and the partial differential equations in necessary conditions for finding the optimal consumption are easier to handle in many cases. The idea of deriving the necessary conditions is rather intuitive and simple: Suppose we know that a utility functional arrives at its maximum at some optimal consumption process, which with its corresponding optimal wealth process are subject to some stochastic differential equation. What are its necessary conditions? Obviously, it is a question like the maximum problem with constraints in the Calculus. Hence we try to introduce a Lagrange multiplier to recast this problem into a maximum problem without constraints. Then, after calculating the directional derivatives of the recast functional at the maximum point, we obtain the necessary conditions for optimal consumption. Here the only difficulty to be overcome by us is to choose an appropriate Lagrange multiplier, to reconstruct a new utility functional without constraints correctly, and to use the Ito formula carefully.

9.2.2

Models

Suppose that in a financial market there is one kind of bond and m kinds of stock, and their price processes satisfy the following stochastic differential equations (SDE in short), respectively (Bardhan and Chao, 1993): i = 1 , 2,...,m; t 2 0 ,

d

m-d

where w(t) = (wl(t), . - ., ~ ~ ( tis)a )d-dimensional ~ standard Brownian motion process (BM), (aT means the transpose of a ) ; G(t) = (&(t), - . ., G,d(t))T is a m - d - dimensional centralized random point process, i.e. dQi(t) = dQj(t) - vi(t, w)dt, where Qi(t) is a random point process with jump density vi(t, w), i.e. E Qi (t) = E J~~ vi (t, w)dt. The Model (9.3) can be explained as follows: Po(t) is the price of the bond a t time t, Pj(t) is the price of the ith stock at

9.2 Optimization for a Financial Market with Jumps by the Lagrange Method

281

time t, r(t) is the instantaneous rate of interest, b(t) = (bl(t),- . .,b,(t)) is the vector of instantaneous appreciation rate for the stocks, and Z(t) = [a(t),p(t)] is the m x m volatility matrix process. In particular, the jump term can be explained as the stochastic jump perturbation caused by, for example, the announcement of some important economic policy to this financial market, the performance of an important decision made by some big company, or the occurrence of some big accident like an earthquake, a big fire and others. Obviously, if one sets m = d, then (9.3) reduces to the usual continuous financial market. Suppose now a small investor with initial capital x > 0 uses his wealth for consumption and investment in the financial market. His investment policy is described by a portfolio process ~ ( t )0, t, an Rm-valued process that represents the dollar investment that the investor maintains in the m stocks. Assume that his consumption at time t 2 0 is C(t) 2 0, with C(0) = 0. Then, after a simple calculation the investor's wealth process will satisfy the following SDE (Bardhan and Chao, 1993): t 2 0,

<

dxt = rtxtdt - dCt

+ x;atdwt + ~

; ~ ~ d = 6 x, ~ , x ~ (9-4)

where dt = ( dlt, . .,6,t)T is a dividend rate process, I=[I,.. -,1IT is the m-dimensional vector with all components equaling 1. The investor would like to get a maximum utility from his investment. Let the utility U be a function of the consumption Ct and the wealth xt, and solve the following optimization problem: Find Ct such that e-Ptu(xt, Ct)dt = sup E

e-@TJ(xt,Ct)dt,

(9.5)

C(.)

where ,LJ > 0 is some discount factor, one can get the optimal consumption process. To solve problem (9.4) and (9.5) let us consider a more general optimal stochastic control problem: Find u(.) E 6such that 00

e-l:flads(i(t, xt, ut)dt = sup- E

e-j:Pads~(t, st,ut)dt, (9.6)

u(.)EU

where ,LJt 2 0 is a non-random and bounded function, the admissible control set is u(.) : u(t, w ) E R' is a St-predictable process such lut12dt < M, for arbitrary 0 T < co that E

< } denote xt = 221, and xt satisfies the following SDE: for all u(.) E 6,

fl

(9.7)

9. Optimal Consumption by H-J-B Equation and Lagrange Method

282

-

-

-

where wt = (wit, w ~is a dl-dimensional ~ ~ ) ~BM, Qt = (Qlt, ..., Qd2t)T is, without loss of generality, a d2-dimensional centralized random point proc_ess satisfying dQit = dQit - dt, 1 i 6 d2, i.e. Qit is a random point process with jump density 1 ( E Qi(t) = ds = t); and xt = (XI,,. - . , ~ , t ) E~ R~,S(') E R ~ ~ ~ ~E ,Rnxd2,and s ( ~ ) we assume that dl d2 = n. . . e l

<

+

9.2.3 Main Theorem and Proof To solve problem (9.6)-(9.8) let us introduce a Lagrange functional as follows: for any u(.) E U

EL

00

L=

{ e - 6 P s d s ~ ( st, t , ut)dt-e- 1;flsdsX(t,w ) T d y t - e - ~ ~ ~ ~ d sy]t), d[~Tl

(9.9) where X(t, w) E Rn is the Lagrange multiplier process, which is assumed to be a right continuous with left limit (RCLL in short) semi-martingale, and t yt = xt-x-J0 f (s, x,, u,)ds-~l s(l)(s,x,, u,)dw,-sot s(~)(s,x,-, u,)dG,, where xt is also a RCLL semi-martingale. Here we denote the quadratic variation process connected with the RCLL semi-martingales Mt and Nt by [ M ~N]t , , which is defined by [MT NIt = (MCT,NC)t Co<sc AM: A Nsl where M C is the continuous martingale part of M, AM, = M, - M,-. Then by Ito's formula (Liptser and Shiryayev, 1989) d(e-1; P s d s ~ ,(W) t Txt) = e-j; P a d s ( - & ~ F ~ t d tXTdxt +x;dXt d[XT,x]t). Hence (9.9) can be rewritten as follows: L = E S;;O e-l;Psds { (U(t,xt,ut) XTf(t,xt,ut) - &XTxt)dt

+

+

+

+ + & s(')(s,

x,, uS)d6,]t +d[X(.IT, .&S(l)(s,x,, u,)dw, +xrdXt ) E(X;X) - lim Ee- 1; Pads$xt, t-+00

+

or

f

L

00

-

+ $f (t, xt, ut) - pt$xt)dt ~ ( ~ ) ( t , x t , u (Xc(.)T, t)d w . ) ~+ ~ ( ~ ) ( t , xu ~~ -) ,~ [ x ~ ( . ) ~ , @ . ] ~

L = E

+x;dXt

e

f;Pads

{ (U(t, xt, ut)

) + E(X;X) - lim Ee-1; PadsXTxt, t++m

(9.10)

where Xc and Xd is the continuous martingale part and the pure discontinuous martingale part of A, respectively, and we make the following assumption: exists and is finite; Assumption A. 1. lim Ee- 1; Pads~Txt t++m 2. U, f , ~ ( ' 1 , s ( ~ have ) all bounded, continuous first and second partial derivatives.

9.2 Optimization for a Financial Market with Jumps by the Lagrange Method

Now after a simple calculation one obt'ains by (9.10) for arbitrary h E

dL

-( u ;h ) lim au(.>

for arbitrary ?I E

h-)L(u) E

E-+O

+ iETdl\t + d[A(.)T,/?I:( as(')(as x;x , ,u s )dw,

a d 2 )( s ,x,- ,u s ) dx

+~

k

6"

-BtXt)dt

+

L(u

283

0

-

dQs)]t)

and by (9.9) for arbitrary

+E

X ~- ~t++m lim O ~ e - J ~ ~ * ~ " l(9.12) \ ~ h ~ ;

x E 6"

Suppose that (uO(.), xO(.),A'(.)) is a maximum point of L defined by (9.9), then from (9.11)-(9.13)one finds that

<

-

&(u0; a4.) h ) 0 , * ( x.)0 ; for a11 ( h ,h,x) E k x

6. x 6" x Gn

h)

< 0 , &(A';@

< 0,

cnx cn. Hence one has that for all (h,jlh,T) E The right side

1

of (9.11)l (9.12) and (9.13) (,(.),x(.),x(.))=(,~(.),x~(.),xO(.))

= 0.

(9.14)

284

9. Optimal Consumption by H-J-B Equation and Lagrange Method

Therefore, by appropriately choosing 6, one has that by (9.13) x! =

+

1 t

f (s, x t , uo)ds

+

1 t

s(l)(s,xo, ut)dw,

+

1 t

s(')(s, xo-, uo)dG,.

(9.151 Now assume that A: = AO(t,x :), AO(t,x) E C112. Moreover, for the more precise expression of the necessary condition on the maximum point of L let us make the following assumption: \

(2) (2) Sij (t, x,u)Sij, (t, x, U) = 0, as j # j',

T

(9.16)

n,dz

where ~ ( ~ ) ( t , x ,=u )[ s $ ) ] ~ ~ = ~ . -.

Then by Ito's formula for AO(t,x:) and the arbitrariness of (h,

onone obtains that a.s. a.e.

x)

E

Er x

From above we have proved the following Theorem 290 . Consider the optimization problem (9.6) - (9.8). Assume that Assumption A.2 holds, (x:, A!) satisfies Assumption A.1. Then we have the following conclusions: 1) If (uO(.),xO(.),A'(.)) is a maximum point of L defined by (9.9), then (9.14) and (9.15) hold. Hence (uO(.),xO(.)) is also a maximum point of (9.6) - (9.8). 2) Under assumption in 1) if (9.16) holds and A; = AO(t,x:), AO(t,x) E C112, then (9.17) holds. Furthermore, we can also show that under appropriate conditions for a maximum point (uO(.),xO(.))of (9.6) - (9.8) there will exist a A'(.) such that they still satisfy (9.17). However, (uO(.),xO(.),A'(.)) may not be a maximum point of L defined by (9.9). We now need to use the conception of a regular point (Curtain and Pritchard, 1977). Assume that X , Y are two Banach spaces, fi c X is an open subset, A : 6 -+ Y is FrBchet

9.2 Optimization for aFinancial Market with Jumps by the Lagrange Method

285

differentiablewith continuous FrBchzt differential dA. xo E 6 is called a regular point of A, i f fdA(xo) maps f2 onto Y.Set (x(.),u ( . ) ): xt is & - adapted, ut is gt - predictable such that E J", e- j;flads(lxt12+ 1utl2)dt< m,

x={

where 0 < 60 < ps < ko is non-random, ko and 60 are constants, and write t ~t = y(x(.),u(.))(t)= xt - X O -So f ( ~ , x s , u s ) d s xs- ,us)dGsl - Jo S (1) ( 8 7 2 s us)dws - J: Y = I ( . ) : yt is gt - adapted such that E J," e-l;@ndslg12 dt < m ) . W e have the following

s(')(~,

I

.

Theorem 291 Assume that Assumption A.2 holds, ( s o ( . )uO(.)) , is a uO(.))is a regular maximum point of (9.6) - (9.8), and assume that (xO(.), point of y(x(.),ti(.)).Then, 1) there exists a zO(.)E Y such that dL(xO(.), uO(.))= 0, where xt, ut) - ~ O ( t ) ~ y t ) d t ; L(x(.),u(.))= E J," e- flsds(U(t, 2) in addition, if there exists a neighborhood q,o(.),,o(.)) of (xO(.), uO(.)) such that as (x(.),u(.)) E q,o(.),,o(.)), JL~I~ ( ef i- @ a d s y(x(.),u ( . ) ) ( ~ ) ~ A : )

P a d S ~ [ Je-; 1; @rdrzO(s)ds I gt], then exists, where A: = T, L(x(.),u(.))= E J;;" e-l;flsds{U(t,xt, ut)dt-Ao(t, w ) T d y t - d [ ( ~ O )y]t) - lim E(e- 1; @sdsy(x(.), u(.))(~)~x:). t--too Therefore, in case 2) if (xf , A:) satisfies Assumption A. 1,(9.16) holds, and A: = X O ( t , xy), AO(t,x)E C112,then (9.17) is still derived. Proof. 1) is derived from Theorem 12.3 of (Curtain and Pritchard, 1977)[~'1and the Riesz's representation theorem for a linear functional, since now Y is a Hilbert space. Let us show 2): By the martingale rep = 5y'Q-predictable resentation theorem (Theorem 103) there exists a 1ctl2)dt< m, for all T < m, processes bt and a ct satisfying EJ:(lbt12 such that E[J; e-lt@adszO(t)dt 15t ] - E J w e- 1; flads~o(t)dt+J: bsdws+J: csdQs,

+

-

- > 0. Write XE) = E[J,"

for all t

st

e - l ~ f l ~ d r z O ( sI )gt]. d s Then

A: = - J: e - J ~ @ ~ d r z O ( s J; ) d se-+ j;@sdszO(t)dt+J, ~ bsdws+J: c,dCj,. Hence L(x(.),u ( . ) )= E ~,"(e--f;@adsU(t, xt, ut)dt yTdA:).

+ By an application o f Ito's formula to y T ~ :the conclusion of 2 ) is derived.

9. Optimal Consumption by H-J-B Equation and Lagrange Method

286

For the practical evaluation one can solve the following equation

I

+

+

T

ax s(l)(t,x , u) %(t,x,u) s ( t , x , ~ ) ~ ltr(aS('i~~+') \ +a d 2 ) ~ , x , u jui ) $ ~ ( ~ ) ( t , x , u )=) O, 1 i r ,

< <

au ( ~ , x , u+) E ( t , x ~ u-)BJi ~ ~+ $ f ( t , x , u )

+

+ ~ t r ( & s ( ' ) ( t , x , u ) s ( l ) ( t x, , u ) ~ )tr(aS('ij::fi)T s ~ ( l ) (x t, u) , a d 2J)Z~i , X), U$ s ( 2 ) ( t , x , u ) ) + ~ + ( X i ( t , x + ~ ( 2 ) ( t , x , u ) ) - X i ( t , x )

+

.

,

From the solution u = u ( t , x ) of (9.18) one solves the following feedback

SDE dxt = f ( t ,x t , u ( t ,xt))dt

+ s-( ' ) ( t ,x t , u ( t ,xt))dwt

+d2)(t, xt-, ~ ( xt t,- ) ) d ~ t ,xo

=x, t

2 0;

(9.19)

to get the solution xt. Then the feedback control u ( t ,st-) may be an optimal control of (9.6) - (9.8). Note that if the coefficients f ( x ,u ) ,s(')(x, u ) and s ( ~ ) ( xU ,) of (9.8) and the utility function U ( x ,u ) do not depend on t , then in 2) of theorem 1 we only need to assume that the optimal Lagrange multiplier A' depends on x , i.e. A' = X O ( x ~ X0 ) , E C 2 . In this case in (9.17) and (9.18) the term $$ disappears and the equation becomes simpler.

9.2.4

Applications

In what follows we are going to give some simple examples based on the financial market with only jump stochastic perturbation, and the optimal consumption problem for special utility functions U ( c ) = c6/6, 6 < 1,6 # 0;and U ( c ) = logc, which are considered in Karatzas, et al. 1991, for a continuous financial market.

.

Example 292 Consider a simple financial market with jumps. In (9.3) assume that bi(t) = r t , i = 1 , 2 , . . .,m; d = 0 , and i n (9.4) assume that St E 0, dCt = ctdt, i.e. assume the wealth process (9.4) satisfies the following I-dimensional SDE: dxt = rtxtdt - ctdt

-

- = x,t 2 0. + T T, ptdQt,xo

(9.20)

where = (&, ...,Qdzt) is a m-dimensional centralized Poisson process such-that dQit = dQit - dt, 1 < i m,

<

9.2 Optimization for a Financial Market with Jumps by the Lagrange Method

287

Jot

that is, Qit is a Poisson process with jump density 1 (EQit = ds = t ) , and we also assume that any two Qit, Q j t , i # .?',haveno common jump time. Naturally, we assume that pt E RmQgm, which is bounded, is deterministic and its inverse matrix p', ezists and is also bounded. Our object is to find SUP(c(,),,(.)),~ E J ; e- 1; f l s d S ~ ( tx, t , ct)dt1 where U ( t ,x , c ) = c6/6,6 < l 16 # 0 , -

k02P,2r~~60>~,P,-~~,2~o>0;

60 is a constant, and the admissible control set is ( c ( t ) ,n(t)): they are 3t-predictable, c ( t ) 2 0 and nl(t),. . .,n,(t)) satisfy that for any 0 T < cx, c(s)')ds < cq such that there exists a solution xt o f (9.20) satisfying that xt 2 0 , for all t 2 0. T o simplify (9.20) let us set - = e- Sot ' a d s x t . xt T h e n (9.20) becomes

<

+

Since U ( s ,Zs, c,) = U ( s ,x , ,?), the problem becomes finding SUP(,(.),,(.)),Bl E S,m e- So B , d r U ( ~Zs, , cS&, where ( ~ ( t n(t)) ) , : they are zt-predictable, c ( t ) 2 0 and nl(t),. . .,nm(t))satisfy that for any 0 T < co c ( s ) ~ )<~ m, s such that there exists a solution Zt o f (9.21) satisfying that Zt 2 O l for all t 3 0 . Now b y (9.18) one finds that i f Pij(t)pijt(t) = 0 , for.?' # j', then

<

+

c6-1 - e- Sot ' q t , x ) = 0 , A l ( t , x ) ~ - ~ S ; C c . = l pioj(t)pij(t)ni= 0 , io = 1,2, ...,m; -PtX(t,x) - Xj(t, x)e-S: 'sdsc X(tl x e-St'sds CG=ln i p i j ( t ) ) - X ( t , x ) - Xk(t,x)e-S;'sdS CGZ1 nifij(t) & ( t , x ) = 0. (9.22) In the special case

+

+

+

we can get a more precise answer. Indeed, solving (9.22), one has that

288

9. Optimal Consumption by H-J-B Equation and Lagrange Method

where ko is any constant. To show that the optimal consumption process c: and optimal portfolio process T: do exist, we need the following proposition 1 even for m 2 1.

.

Proposition 293 Suppose that x 2 0 and a consumption process ct 2 0 satisfying E e- o lt " ~ ~ ~ c ( < t )00 ~ dis tgiven such that E D = E J r e- 5t'sdsCtdt x , where D = e- 5; radsctdt.Then there exists a portfolio process ~t such that the pair (c(.),.rr(.))E 3 for the initial endowment x . Conversely, i f x 2 0 and a consumption process ct 2 0 with a portfolio process ~t are given such that (c(.),T ( . ) ) E U', then E D = E S T e- 5: 'adSctdt< x.

Sr

<

Proof. Let us show the first part. Set Et = E [ S cse-5: ~ PvdVdsI&] ( x - E D ) ,

+ 52. Then Ct 2 0, and Ft

+

where 3, = = { r mt - c.e-5: r u d v d s ) , - E D . By the martingale representation theorem where mt = E [ D I (Theorem 103) there exists a St-predictable process 4, such that mt = 4 T d a s , E Jot 1 4 ~ds1= ~ E lmt12 < E ( D 2 ) < E o l ~ o , e- 5: r * d S ~ ( t ) 2= d sLO. Hence E where we assume that E 14. l2 ds <

st]

Jot

00.

Now let

Then one easily verifies that (c(.),n(.))E 3.T h e inverse conclusion is also easily verified by Zt 2 0 , for all t 2 0. Now we return t o our problem just discussed. Under assumption (9.23) applying Proposition 1, we get that the optimal consumption is given by (9.24), we denote it by cO,where the constant ko satisfies E Soooe - J ~ ' ~ d s ( e S o t ( P s - ' ~ ) d s k O ) - l / ( l - 6 ) d t = z; or

and the optimal portfolio is given by (9.25) determined by &, hence by cO. Moreover, the optimal wealth process is

9.2 Optimization for a Financial Market with Jumps by the Lagrange Method

289

From the discussion above we see that for the case m = 1 the optimal consumption, the optimal portfolio, the optimal wealth, and the optimal value functional are (9.24), (9.25), (9.27) and (9.28), respectively. In particular, under condition (9.23) as rt is non-random, so are the optimal consumption ct and D. Hence by the proof of Proposition 1 the optimal portfolio r t 0. This is reasonable, because in SDE (9.20) the appreciation rate rt for the stock now is the same as the interest rate of the bond. In this case it means that it is not necessary to take a risk to invest money in the stock. However, as rt is random and predictable, the above analysis tells us that it is necessary and possible to take an optimal portfolio rt # 0 like (9.25) with an optimal consumption ct like (9.24) in order to get an optimal value functional like (9.28).

-

zt-

.

Example 294 Consider the same financial market and the same wealth process as (9.20). However, here our object is to find 03 U(t, xt, ct)dt, SUP(c(.),,(.))EfjE SO e 10 where U(t,x,c) = logc, ko 2 PS 2 rs 2 So > 0; So is a constant.

Then one easily sees that in (9.22) only the first equation is changed to the following C-l St rsds), = 0. Hence under assumption (9.23), a conclusion similar to that in example 1 is still valid, oyl: now ~'(t,= ~ ) P=dsko,CO = C"(t,x) = (e.ft(Ps-'a)dskO)-l, where ko satisfies E ~ r ( e . f ~ f l s ~ " k ~ )=- x, ' d tor Ico = xP1E S , e-JiPadsdt. Thus the :ptimal we$th process is xO - e.fo 'sdsx - e.fo~ v d v e- .f; 'udVc:ds e-.f: rudur:psdas] t - e.ftr~dsx'ud"[kgl J; e- .I;Pudvds+ Jo e-j;r,,dv rs o Psdas], where the optimal portfolio rf is determined by formula (9.25) with c! given here, and the optimal value functional is E e- "Padsu(t,xt, ct)dt SUP(c(,),I(,))Efj

,-

[Ji

Jr

+ Si

290

9.2.5

9. Optimal Consumption by H-J-B Equation and Lagrange Method

Concluding Remarks

Finally, let us point out that as in $7.5 of Chow (1997), after a simple modification to the system (9.8) with jumps we can similarly derive the optimal portfolio percentages given in Chow (equation (7.50)) when all securities are risky, or in equation (7.56) when there is one riskless security, and a similar mutual fund theorem that then follows. To economize space, the derivations are omitted. Let us also note that when s ( ~ = 0) (i.e. the system has no jumps), if the discount factor ,6 in (9.6) and the coefficients f (x, u), s(~)(x,u) of (9.8) all do not depend on t, then (9.18) implies (7.21) and (7.22) in Chow (1997).

10 Comparison Theorem and Stochastic Pathwise Control

In a practical case involving the control of a real engineering dynamical system we need to control the system pathwise. So, for a stochastic dynamical system from the practical point of view, pathwise control appears to be more useful. However, one can easily imagine that to minimize (or maximize) a target functional subject to the solutions of some stochastic system through the controls can actually be achieved if we can compare the values taken by the target functional as a result of different controls. So the comparison results are very important and useful for us. In this chapter we will discuss the comparison theorem for solutions of SDE first. Then we will see that they can be used to get the optimal stochastic pathwise Bang-Bang control under some appropriate conditions. Since the optimal trajectory (that is, the solution of the stochastic system when applying the optimal control) should be pathwise (that is, a strong solution). Otherwise, it is hard to apply it from an engineering point of view. So we will also discuss the existence of a strong solution when the coefficient is non-Lipschitzian. (That is, the case that the optimal control makes the system coefficients non-Lipschitzian).

10. Comparison Theorem and Stochastic Pathwise Control

292

10.1 Comparison for Solutions of Stochastic Differential Equations 10.1.1 1-Dimensional Space Case The Tanaka formula is a very useful tool for discussing the comparison of solutions to SDE. Assume that xf, i = l , 2, satisfy the following l-dimensional SDEs

where A t , Mt and Kk(dt,dz) have the same meaning as that in (4.11). We have the following comparison theorem for solutions of these SDEs. Theorem 295 Assume that 1" Ipi(s,w)I ds < co,i = 1,2,W 2 0 , P - a s . 9 there exist b1( t ,x , w) and b2( t ,x , W ) such that b1( t ,x , w) L b2(t,x , w ) , P1(t,w)L b1(t,x2,w),b2(t,x;,w)L P2(t,w), sgnb - Y ) . ( b2(t,x,w)- b2(t,y,w>) I kTN(t)pTN (1% - Y I ) , as 1x1, l ~ 5l N,Vt [O,TI, where N , T > 0 are arbitrarily given, and 0 5 k s ( t ) is non-random such T that So k;(t)dt < co;O 5 &(u) is non-random, strictly increasing in v 1 0 with pN ( 0 )= 0 , and So+du/pN( u ) = co;

Jot

-

3" b ( t , x , w )- 4t,y,w)12 I ~TN(t)PTN(12- Y I ) , as 1x1, l ~ 5l N , t E [O,TI; (u) have the same property as that in P; where k; ( t ) and 40 X > Y x+c(t,x,z,w) L y+c(t,y,z,w). If x; x& then P - a.s. x; 2 x;, v t 2 0 , where xt, i = 1,2, satisfy (10.1), respectively.

pTN

Proof. Define T N =inf { t > 0 : lxil +lx;l > N } . By the Tanaka type formula (Theorem 152) for any given N , T > 0 we have <E I.:,~: ( p 2 ( sW, ) - p1 ( s ,w ) ) d ~ , E(&rN - xt*rN)+ IE Ix;>2: (b2(s,x:) W ) - bl(s,x t )w))dA,

SotATN

I E SetAt

sotATN

sotATN

=

Yt =

~ N , T ( S ) P N , T( ( 2 ; - x;)+) dAs,Vt E

Sot ~ N , T ( s ) ~ +( As),Vt , 2 0, 0, otherwise.

[O, T ].

10.1 Comparison for Solutions of Stochastic Differential Equations

293

Then we have that W 2 0 E I y ( T t ~ ~ , )5l S : PN,T(E I Y ( T ~ A ~ ~ ) I ) ~ ~ . Hence, P - a.s. Y(t)=O,W>O, i.e. P - a s . x i x:, Vt E [O,T). Since T is arbitrary, the desired result is obtained.

>

10.1.2 Component Comparison i n d- Dimensional Space Motivated by the 1-dimensional case we can discuss the comparison of the components of two d-dimensional solutions for two d-dimensional SDEs. Assume that xf,i = 1,2, satisfy the following d-dimensional SDE: j = 1,2,... , d

where At, Mt and fik(dt,dz) have the same meaning as in (4.4). We have the following comparison theorem for the components of solutions fo these SDEs.

10. Comparison Theorem and Stochastic Pathwise Control

294

Proof. By using the notation in the proof o f the previous theorem and applying Theorem 153 one finds that for any given N , T > 0 E(xj,tArN - ~ j , t A r N ) + I ZrL E ~ x ~ , > x( @ : ~p(s,~ - )~ ~ p ( s , w ) ) d ~ p s

SiATN

rn

I c , = , E SFTNI,;s>,;*(b:p(s, 4,w) - bjp(s,x:,w))dAPs ) Vt E [O, TI . 5 Cp=lE ~N,T(s)PN,T - 5 j S ) + dAps,

SetAt =

JiATN

k~,~(s)dC,"=~(A +ps),Vt . s 2 0,

7"

(xjt - xjr)+, as t t IO,T), 0, otherwise. Then we have that W 2 0,Vj = 1 , 2 , - . . ,d I Y ( T j , t ~ 7 ~ ) I Jot PN,T(EIY(T~,SATN)I)~S. Hence, as in the proof of the previous theorem, we have that P - a.s. v j = 1 , 2 , . . . ,d xjt > xjt, Vt E [O,T). Since T is arbitrary the desired result is obtained. rn As an application we give the following example.

qt = Tt =

Example 297 Consider the following SDE with jumps t t X j t = X j o + Jo bj(s,xjsIw)dsf Jo u j k ( s , ~ ) d ~ k ( ~ ) JZ c j ( s , ~ j s , ~ ) ~ kd (~d )js ,,= 1,2,. . . , d. Suppose that sgn(xj - Y j ) . ( b : ( t , ~ j , w-bj(t,Yj,w)) ) I k$(t)&(Ixj - Y j I ) , as b j I , I Y j I I N,Vt E [O,TI, ) ~j+cj(t,~j,z,w)~ xj 2 ~ j x j + c j ( t , x j , ~ , w 2 where k$(t) and p$ (u) satisfy the same conditions as in Theorem 296. If x f , i = 1,2, are two solutions corresponding to the given initial conditions xb, i = l , 2 , respectively, then x1 30 > x 320 , j = 1 , 2 , . . - ,d xjt 2 x ! ~Vt , 2 0, j = 1,2,. . . ,d.

-

+ Ji

*

Notice that in the above example the u j k ( s w) , do not depend on x.

10.1.3 Applications to Existence of Strong Solutions. Weaker Conditions. For a 1-dimensional SDE one can use the comparison theorem (Theorem 295) to derive the existence of a strong solution under weaker condition in some sense. Let us consider the following SDE with jumps in 1-dimensional space.

10.1 Comparison for Solutions of Stochastic Differential Equations

+

c ( s ,x8-, z)iS*(ds, d z ) ,w

295

> 0,

First we establish a theorem on the existence of a pathwise unique strong solution by combinning three results: the existence of a weak solution (through Theorem 175), the validity of the pathwise uniqueness of solutions (through Theorem 170), and then the existence of a pathwise unique strong solution (through Theorem 137).

Theorem 298 Assume that lo b = b ( t , x ) : [O,co) x R~ -+ R d , u = u ( t ,x ) : [O, co) x Rd -+ R d B d , C = c ( t , x , z ) : [O,co) x Rd x Z -+ R d , are Bore1 measurable processes such that P - a s . Ib(t,x)l < ~ l ( t ) (+l Ixl), x , 4124 d Z ) < ~ l ( t ) + ( l1x12), l d t , x)12 + JZ where c l ( t ) is non-negative and non-random such that for each T < oo :J cl (t)dt < co; 2' b(t,x ) and u(t,x ) are continuous i n x ; and limh,0 Jz Ic(t, x + h, Z ) - c(t,x , z)12~ ( d z=) 0 ; S for each N = 1 , 2 , . . . , and each T < co, l u ( t , x d - u(t,x2)12 < $(t)pTN(Ix1 - ~ 2 1 1 , x 2 y ===+ x + c ( t , x , z ) > y + c ( t , y , z ) , as lxjl < N , i = 1,2, t E [O,T];where g $ ( t ) d t < 00;and p$(u) > 0,as u 2 0 , is non-random, strictly increasing, continuous and concave such that So+duIpTN(4 = 00; 4" = Rd - ( 0 ) r and JRd-{0) % ~ ( d z ) < co,

<

5" (x1 - ~ 2 , b ( t , ~-l b(t,x2)) ) cTN(t)p$(Ix1 - x212), as lxil N , i = 1,2, t E [O,T];where c F ( t ) and p$(u) have the same properties as that i n 9. Then (10.3) has a pathwise unique strong solution.

<

Proof. By conditions l o and 2", by applying Theorem 175 one finds that SDE (10.3) has a weak solution. From conditions 3' and 5O, by applying Theorem 170, one finds that pathwise uniqueness holds for SDE (10.3). Therefore applying the Yamada-Watanabe theorem (Theorem 137) there exists a pathwise unique strong solution for SDE (10.3). Now let us use the comparison theorem to establish an existence result on strong solutions under conditions that in some sense are weaker.

10. Comparison Theorem and Stochastic Pathwise Control

296

Theorem 299 In the above theorem (Theorem 298) ifwe remove condition 5" and, in addition, assume that b(t,x) is jointly continuous, then SDE (10.3) still has a strong solution. (But it is not necessary unique). Proof. Since b(t,x ) is jointly continuous, for each n = 1,2,. . . , one can find a jointly Lipschitz continuous function bn(t,x ) such that < bn(t,x) < b(t,x)+ b(t,x) Applying Theorem 298, for coefficients (bn(t,x ) ,a ( t ,x ) ,c(t,x , z ) ) there exists a pathwise unique strong solution xy satisfying the following SDE

i.

+

By the comparison theorem (Theorem 295) one finds that Et 5 x:+l 5 xy ,Vn, ?here Et is the pathwise unique strong solution of the SDE with coefficients (b(t,x ) ,a ( t ,x ) ,c(t,x , z ) ) , where b(t,x ) = -cl ( t ) ( l 1x1). Hence the limit xt = limn,, qn,P - a.s. exists. However, it is not difficultto show that as n -+ m, (1) bn(s,~ : ) d s b(s,xs)ds, (2) 4% 4dws 4% xs)dws, (3) So( j&+, x n ,z)fik(d~,dz) Jz C ( S , X S - , z)Zk(ds,d l ) , in probability. Let us show the first result. In fact, for arbitrary given E > 0, p(l bn(s,x?)ds - b(s,x,)dsl > E ) I P(lx,l > N )

+

Ji

+

J,t

Ji

-+

Jgt

-+

fi

Ji

Ji

+supnP(Ix:( > N ) + $ E $ [ I ~ ( ~ , -xbF( ~) , x s ) l+ i ] I l x : I < N d ~ = I ? + I? Now, for any Z > 0 take a large enough N > 0 such that I? +I? < Notice that Ilxnl 0, P(I Jz ~ ( s , ~ : - , z ) f i k ( d s , dz ) J Z e ( ~ , x , -z)fik(ds,dz)l , > El) 5 p(lxsl > N ) +supn P(lx?( > N )

+ 5. d

+

5.

Jz Ic(s,x:,~) + I? + I F n .

+$ZEJ~

=I?

-C ( S , X S , Z ) I ~ I ~ ~ ~ ~ < N ~ ( ~ Z ) ~ S

By condition 2' as n -+ oo, JZ 14% x:, z ) - 4%xs, 412rIT(dz)-+ 0, since x? -+ x,. On the other hand, Jz 143, x:, 412~ ~ ~ : ~ < I~ cr l ( td) (zl+) N ~ ) . So, by Lebesgue's dominated convergence theorem, for each N as n

-+

cm,

10.1 Comparison for Solutions of Stochastic Differential Equations

297

I?" -+ 0. Therefore, (3) is similarly established. The proof of (2) is even simpler. w By using the method by which Theorem 299 was proved, one can easily obtain a comparison theorem under different conditions for the following SDEs. Suppose that x f , i = 1,2, satisfy the following 1-dimensional SDEs

respectively. Theorem 300 Assume that 1' bi = bi(t,x) : [O,oo) x Rd + R d , a = a ( t , x , w ) : [O,oo) x Rd x 0 + RdBd, c = c-( ~ , x , z , w ) : [ ~ , M ) x R ~ x Z X ~ ~ + R ~ ,

are g?lNk- adapted and measurable processes, (where bi, i = 1,2, only need to be jointly Bore1 measurable) such that P - a s . lbi(t,x)l < s ( t ) ( l +14,i = 1,2, 1 4 4x,w)12 + JZ lc(t,x,z,w)l24 d ~ <) ~ l ( t ) (+l 1x12), where q ( t ) is non-negative and non-random such that for each T < oo cl (t)dt < m; F'a and c satisfy that la(&x,w) - a(t,y,w)12I kZ(t)pTN(Ix - Y I ) , as 1 ~ 1 , 1 ~I1 N , t E [O, TI; limh--rO JZ Ic(t, x + h , z,w) - c(t,x , z,w)12 n ( d z ) = 0 , kG(t)dt < oo, for each given where 0 5 kG(t) is non-random such that T < oo;0 I &(u) is non-random, strictly increasing in u 2 0 with p T N ( ~ ) = 0 , and So+ dulpTN(u) = M ; ' 3 bl(t,x) 2 b2(t,x), b2(t,x) (or b l ( t , x ) ) is jointly continuous such that the pathwise uniqueness for solutions of (10.5) with coeficients (b2,a , c) (or (bl,a , c ) ) holds, 50 x l y x+c(t,x,z,w) 2 y+c(t,y,z,w). If x i 2 x& then P - a.s. x: 2 x:, 'dt 2 0 , where x i , i = 1,2, satisfy (10.5), respectively.

'

Proof. Take Lipschitz continuous functions b2>"such that b l ( t , x ) 2 b2(t,x)> b2(t,x)-& > b2,n(t,x)> b 2 ( t , x ) - A , n = 1 , 2 , . . . Then by Theorem 170 there exists a pathwise unique strong solution satisfying the following SDE x2,n = x; Jot b21n(s,X ? * ~ ) ~ S a ( s , xi^^, w)dw,

+ Ji

+

+ h

2N

J; c(s,x.i , z , w)fik(ds,dz),Vt t 0. Applying the comparison theorem (Theorem 295) one also finds that

.

10. Comparison Theorem and Stochastic Pathwise Control

298 xi

> x2,n 2 x:,n-l t

t -

So the limit ii$= lim,,, x:'" < x: exists. Moreover, as before one easily shows that Zz is a solution of the BSDE with coefficients (b2,a ., c). , Since bv assumption the pathwise uniqueness hold for this SDE, 'coincides with x: . Hence x: 5 xt ,Vt 2 0.

2

10.2 Weak and Pathwise Uniqueness for 1-Dimensional SDE with Jumps By Corollary 140 we know that the pathwise uniqueness o f solutions implies the weak uniqueness of weak solutions for d-dimensional SDEs with jumps. Here for 1-dimensional SDEs with jumps. we will use the local time technique to conclude that under some general conditions the inverse statement is also true. Consider the following 1-dimensional SDEs

Theorem 301 Assume that the coeficients b(t,x ) , a ( t ,x ) , and c(t,x , z ) in (10.6) do not depend on w , d = 1 and lo J' Ic(s, x , z)12r(dz)ds < 00, 2" X L Y x+c(t,x,z) ty+c(t,y,z); '3 L?(xl (.) - x2(.))= 0, where x a ( t ) , i= 1,2, satisfy (10.6) on the same probability space with the same BM wt and the same Poisson martingale measure fik (ds,dz). Then the weak uniqueness of weak solutions to (10.6) implies the pathwise uniqueness of solutions to (10.6).

Ji

*

Proof. Let Y, = x l ( t ) V x2(t)= max(xl ( t ) x2(t)) , = x2(t) max(xl(t)- x2(t),0)= x: + (xt - x:)+. Then by (4.19) Y, = X: JOt I x a - x : > ~ d ( ~ Sx:) . = xo + Jot b ( ~Ys)ds , + Jota(s,Ys)dws Jz C ( S , Ys-7 z ) q ( d ~dz), , where we have applied the fact that > ~- (qs, b x:)) ( ~ , b ( ~ , + ~ ~ : - ~ ~ x:) = b(s,Ys), = Ix:-x:,ob(s,x3 + Ix:-x:
+

+

+

10.2 Weak and Pathwise Uniqueness for 1-Dimensional SDE with Jumps

299

P(supt20(yt - x:) > 0 ) > 0, then it is easy to see that there will exist a rational number r, a positive number E and a t > 0 such that P(yt > r 2 2:) 2 E > 0. Since & 2 x:, and {yt > r ) = {x: > r ) u {yt > r > x:) , we have P { y t > r ) > P { X : > T ) + E >P { X : > r ) . However, by the weak uniqueness o f weak solutions to (10.6) the probability laws of { y t ,t 2 0 ) and {x:, t 2 0 ) should coincide. This is a contradiction. So P(supt2,,(yt - x:) = 0 ) = 1. Similarly, we also have P ( s ~ p , ~ ~-( x:) y t = 0) = 1. Hence = 0 ) = 1. P(suptro That is, the pathwise uniqueness of solutions to (10.6) holds. From the proof of Theorem 301 one immediately sees that the following corollary holds. Corollary 302 Under the same assumption as those in Theorem 301 we have that x ( t ) = max(xl(t),x 2 ( t ) )and ~ ( t=) min(xl(t),x2(t))both satisfy (10.6). Furthermore, we have the following sufficient condition for L!(xl(.) x2(.))= 0. Proposition 303 Assume that the conditions lo in Theorem 301 holds. and assume that Q" for each N = 1,2,. . . there exists a k g ( t ) and &(u) > 0, as t ,u > 0, such that 14,x ) - d t , y)12 5 k%t)pTN (12 - Y I ) , as 1x1, lyl N , t E [O,Tl, where N , T > 0 are arbitrarily given, and 0 5 k$(t) is non-random such that J: kg(t)dt < co;O &(u) is non-random, strictly increasing in u > 0 with pN (0) = 0, and So+du/pN(u)= co. Then L f ( x l ( . )- x2(.))= 0.

<

<

Proof. Applying Theorem 148 one finds that t Ix: - x: = So sqn(x: - xi)d(x: - x:) +J" -:.I[ - x : - + c 1 ( ~ , x : ~ , z , ~ ) - c 2 ( s , x ~ ~1 , z , ~2 ) ~ - ~ x , ~ - x , ~ ~ O z1 -sgn(x,- - xi.-) - (cl(s,xa-,z, w) - c2(s,xi-,z, w))]N.(ds,dz). On the other hand, by Lemma 158 Ixj - x:I = J: sgn(x:- - xi-)d(xf - x i ) + L:(xl(.) - x2(.)) 2 2 1 2 - 2,- +c(s,x:-,z,w) - c(s,x,-,z,w)I - Ix,- - xs-I -sgn(xB- - 5:-)c(s, z, w))N.(ds, dz).

I

+J:J~(IX:-

300

10. Comparison Theorem and Stochastic Pathwise Control

.

Therefore, L:(xl (.) - x2(.)) = 0. Combinning the results of Theorem 301 and Proposition 303 one immediately obtains the following theorem.

Theorem 304 Under conditions 1" and .P in Theorem 301 and the condition 4" in Proposition 303 the conclusion of Theorem 301 holds, that is, the weak uniqueness of weak solutions to (10.6) implies the pathwise uniqueness of solutions to (10.6).

10.3 Strong Solutions for 1-Dimensional SDE with Jumps 10.3.1 Non-Degenerate Case Recall that the Yamada-Watababe type theorem says that the existence of a weak solution the condition that pathwise uniqueness holds +the existence of a pathwise unique strong solution. So first we can use the Girsanov type theorem to find the existence of a weak solution under very weak conditions. Then as applying the result on the weak uniqueness implying the pathwise uniqueness under some general conditions, we can obtain a pathwise unique strong solution for a 1-dimensional SDE under very weak conditions. Consider a 1-dimensional SDE with jumps as follows: Vt 0,

+

>

where xt, b(t, x), a(t, x), c(t, x, z) E R1, wt and fik(dt, dz) are d-dimensional BM and Poisson martingale measure, respectively. Moreover, fik(dt,dz) = Nk(dt ,dz) -n(dz)dt, where Nk(dt, dz) is the Poisson counting measure generated by a St-Poisson point process k(.) and n(dz)dt is its compensator, where n(dz) is a a-finite measure on some measurable space (2,Bz).

Theorem 305 Assume that b, a and c are Bore1 measurablefunctions such that 1" Z = R1 - (0) ; 2" one of the following conditions holds: (2) 144 211 + JZ144 2, z)12d d z ) I ko, where ko 2 0 is a constant, and n(dz) = dz/ 1z12; (ii) lff(t,%)I I ~ l ( t ) (+ l lx12), J, I4t, x, 4124 d . 4 I Cl(t)

10.3 Strong Solutions for 1-Dimensional SDE with Jumps

301

T

where cl(t) 2 0 satisfies that for any T < m, So cl(t)dt < m, moreover, a ( t ,x ) is jointly continuous in ( t , x ) and J" Ic(t hl,x h , z ) - c(t,x , z)12~ ( d z=) 0. limh,h~-rO

+

+

and Jz $ & " ( d ~ ) < m; ' 3 there exzsts a 60 > 0 such that so, 144I>. and for each N = 1,2, . . . ,T < m, there exists a kK(t) and &(u) > 0, such that 2 (1% - Y I ) , as 1x1 1 I Y ~ N , t E [O,T], b ( t , x ) - a ( t l ~ )5~kZ(t)& where N , T > 0 are arbitrarily given, and 0 k s ( t ) is non-random such that :J k$(t)dt < m;0 &(u) is non-random, strictly increasing in u 2 0 with pN (0)= 0, and So+du/pN (u)= m; a: L Y x c(t,x , z ) 2 Y c(t,y, z ) ; 5" ~ b ( t , x5) c1(t)(l+ 1 x 1 ~ gk(x)), where ~ ~ (= x1 +) ln(1 + ln(1 + . . . ln(1 + ~ x [ ~ ~ o ) ) ) ,

>

<

<

* + \

n~

-

<

+ /

k-times

(no is some natural number), and cl(t) 2 0 is non-random such that for cl(t)dt < CQ;furthermore, b(t,x ) is locally bounded in x , each T < CQ, that is, for each r > 0, as 1x1 I r , Ib(t,x)l I k - 1 where k, > 0 is a constant depending only on r . Then there exists a pathwise unique strong solution for (10.7).

Proof. Under the condition (i) of 2", applying Theorem 182 one finds that (10.7) has a weak solution. In the case that the condition (ii) of 2" holds, by Theorem 183 one still finds that (10.7) has a weak solution. Now by Theorem 304 one sees that if the weak uniqueness holds for SDE (10.7) then the pathwiae uniqueness also holds for this SDE. Thus we can, use the Y-W type theorem (Theorem 137) to obtain a pathwise unique strong solution and the conclusion of our theorem is proved. So for the rest o f the result we only need to prove that the weak uniqueness holds for SDE (10.7) on t 2 0. However, this can be reduced to showing that for each T < CQ the weak uniqueness of weak solutions for (10.7)holds on t E [0,TI. In fact, if this is true, then for each T < m we will have a pathwise unique strong solution on t E [0,TI. This immediately yields a pathwise unique strong solution on t > 0. Now let us show that the weak uniqueness of weak solutions for (10.7) holds on t E [O,T].Suppose that xf,i = 1,2, are two weak solutions of SDE (lO.7), defined on probability spaces (ai,p,(5: ) t20 ,Pi),i = 1,2, respectively. That is, xf,i = 1,2, satisfy that Pi - a.s.W 0, xf = xo + Sot b(s,x:)ds + a ( s ,x i ~ ) d w + : Sz C ( S , xi-, Z ) Nki (ds,dz),

2 -

302

10. Comparison Theorem and Stochastic Pathwise Control

where wt and Zki(ds, dz) are the BM and Poisson martingale measure, defined on probability spaces ( n i , { ~ s, ) ~ ,,~ Pi),respectively, such that

fiki(ds, dz) = Nki (ds, dz)- r(dz)dt, that is, the Poisson martingale measures still have the same compensator r(dz)dt. Now for any given T < co as i = 1,2, set t 2 Z: = exp[- ~;(a-lb)(s, xt)dwt - So (aF1b)(s,x:)] ds], d9- Zi d p i ,

I

Wj

T- T = Jot b(s, x:)ds

Then one finds that

+ wi,

F$ - a.s.W E [O,T],

where G i l t E [O,T], is a BM under the new probability measure F$, and Nki((0, t], dz), t E [O, TI, is a Poisson martingale measure with the same compensator r(dz)t, also under the new probability measure F$. However, by Theorem 154 the pathwise uniqueness holds for SDE (10.8). So, applying Corollary 140, one finds that the weak uniqueness holds for SDE (10.8). Furthermore, take any non-negative BDx Bw0x BD-measurable function f : (D x WOx D , B D x Bw0 x B D ) -+ (R1,BR1), where BD,Bw,, and BR1are the Bore1 field in D , Wo and R1, respectively. Moreover, D and Wo are the totality of all RCLL real functions and all real continuous functions f (t) with f (0) = 0, defined on [O,T], respectively. By Corollary 140 one finds that J, f(xl(.), ~ ' ( 9 O >tU E IB ( Z~) , ~ ~ ~ f Nki(dt, dz) = Nki (dt, dz) - n(dz)dt. In particular, take y(xi(.), ci(.)) = f (xi(.)) exp[~?(a-'b)(sl xa)dzt 2 I(a-'b)(s,x:)1 ds a-'(S,X~-)C(S,X~-, ~)N~i(d~,dz)]. One_sees that J, f (x' (.), (.))dF$ = J, .T(x2(.),c2(.))dF$. This is equivalent to s_ayingthat J, f (x1(.))(zk)-ldP+ = J, f (x2(.))(z+)-ld@. Thus, J, f (xl(,))dP$ = J, f (x2(WP$. That is, the weak uniqueness of solutions to (10.8) holds. Let us give an example for Theorem 305 on the existence of a pathwise unique strong solution to a 1-dimensional SDE with jumps and with the coefficients such that b(t, x) is discontinuous and very much greater than

-+

<'

~

~

10.3 Strong Solutions for 1-Dimensional SDE with Jumps

the linear growth in x , u(t,x ) is only Holder continuous with index and c(t,x , z ) is also not Lipschitz continuous in x.

303

i in x ,

Example 306 Set u ( t ,X ) = t - " 1 2 [ k o m k1 lxl] k2, Q < 1, where ko, kl 2 0 and k2 > 0 are constants. Then u ( t ,x ) satisfies all conditions i n Theorem 305. However, a ( t , x ) is only Holder continuous i n x with index so it is not Lipschitz continuous in x . Wrirte b(t,x ) = t-"[kj ki IX+o - k;~'"~+' I ox

+

i,

+

+

Wt + +&xgl(x) + Z Z x g 2 ( ~ + ) " '-f% m ~ f l ~ = l g k ( ~<) ]1,,

where all k:, i = 0,1,2, and ki, i = 0,1,2, . . . ,m, are non-negative constants, no is any natural number, and g k ( x )= 1 + ln(1 ln(1 . . .ln(1 1x1'))).

+

+

v

+

k-times

Obviously, b(t,x ) is not continuous i n x , its absolute value is very much greater than linear growth i n x. However, b(t,x ) satisfies all conditions i n Theorem 305. Let ~ ( 2t ,,z ) = t-a12[(& I Z I ~ ' ) I ~ Zf ~k:~ lzlPa N ~ IIzI>N11 .(% G ' x I ~ ~+ ~% ~' ~N~ ~N I I ~ ~ I@> (Nx~) * I ~ ~ ~ ~+Ik ~ f i I l x l > i ) l i , 0 < P3 < all of them are constants, where a! < 1, Dl > D2 and all k:, i = 0,1,2, and Icy, i = 0,1,2,3,4, are non-negative constants, No, N l , N2 > 0 also are constants. Apparently, c(t,x , z ) may not satisfy the Lipschitz condition: Jz Ic(t,x,z) - c(t,y,z)12a(dz)5 ko lx -?A 2 , as 1x1, IYI < 1. However, it satisfies all conditions i n Theorem 305. For the coeficients b, a , c Theorem 305 applies, so (10.7) has a pathwise unique strong solution.

-

+ -

-+ 4, <

+

-

i,

10.3.2 Degenerate and Partially-Degenerate Case First for the 1-dimensional SDE with jumps we can weaken the condition on the pathwsie uniqueness. So we can obtain an existence theorem on a pathwise unique strong solution for 1-dimensional SDE with jumps under weaker conditions. Theorem 307 Assume that 1' Z = Rd - {O) , and SZ &,(dl) <m 2' b = b ( t , x ) : [O,m)x R - t R, a = a ( t , x ) : [O,m)x R-+ RIBd, c = ~ ( 2, t ,Z ) : [O, m) x Rd x Z --+ R, are jointly Bore1 measurable such that IW,% ) I 5 c1(t)(l+ I X n~:==' , g k c ~ ~ ) l 10(t, 5 kO(1 + 1x1 n:=flk(x)), J, 146 X I 41'4 d z ) ~ l ( t ) ,

.)I'

<

304

10. Comparison Theorem and Stochastic Pathwise Control

where k-times

(no is some natural number), ko 2 0 is a constant, and q ( t ) 2 0 is nonrandom such that for each T < co, J: cl(t)dt < co; furthermore, b(t,x ) is locally bounded for x , that is, for each r > 0, as 1x1 I r, Ib(t,x)l I kr, where k, > 0 is a constant depending only on r; 30 b(t,x ) is continuous in x , a ( t ,x ) is jointly continuous in ( t ,x ) ; and limh,h~--tO J" Ic(t + h', x + h , z ) - c(t,x , z)12~ ( d z=) 0; - Y I ) , as 1x1 ,lYl 5 N , t E [O,TI; 4" la(t,z,w) - 4 t , Y , 4 1 2 where 0 I k;(t) is non-random such that J: k$(t)dt < co, for each given T < co;0 5 &(u) is non-random, strictly increasing in u 2 0 with = 0, and So+ du/&(u) = 0;); 5" sgn ( x - v) . 2 ) - b(t,Y ) ) 5 kTN(t)pTN(Ix- Y I ) , as 1x1, lvl I N,'%Y E R 1 , h E [o,T]; where k s ( t ) and &(u) have the same property as that in 4 O , besides, pTN(u) is concave; 6" one of the followzng conditions is satisfied: (2) Sz I~(t7x , 211 4 d z ) I ~ ( t ) g ( l x I ) , Jz Ic(t,272) - ~ ( Yt ,,z)I 4 d z ) 5 kiXt)pTN(Ix- Y I ) , as 1x1, lvl I N,Qx,y E R1,W E [O,T]; where c(t) and g(u) satisfy the following properties: ~ : c ( t ) d t < co, for each T < oo; g(lx1) is locally bounded, that is 1g(1x1)15 k,, as 1x1 I r , for each r < oo; moreover, k$(t) and &(u) have the same property as that in 4") besides, &(u) is wncave; (ii) x 2 y ===+ x c(t,2 , z ) 2 y c(t,y, z). Then, for any given constant xo E R , (10.7) has a pathwise unique strong solution on t 2 0.

<~TN(~)PTN(Ix

pc(~)

+

+

Proof. First, by Theorem 176 (10.7) has a weak solution. Second, from Theorem 154 the pathwise uniqueness for solutions of (10.7) holds. So we can apply the Y-W theorem (Theorem 137) to show that (10.7) has a pathwise unique strong solution on t 2 0. From the conditions of Theorem 307 one sees that to have a pathwise unique strong solution for the a to be possibly degenerate, in the 1-dimensional case the coefficient a only needs to be Holder-continuous in index f , which is weaker than the condition on a in Theorem ?? for d-dimensional SDEs. Furthermore, we can also have the existence of a strong solution for 1-dimensional SDEs with jumps and with discontinuous coefficients b and with a that may possibly be degenerate. For this let us first introduce a lemma, which is a special case of Lemma 271.

10.3 Strong Solutions for 1-Dimensional SDE with Jumps

305

Lemma 308 Assume that 1" b(t,x , w ) is continuous in x E R1\G, where G c R1 is a Borel measurable set such that x E G + a ( t , x ) > 0,Vt 2 0 , and m l G = 0 , where ml is the Lebesgue measure i n R1; moreover, b(t, x , w ) is a separable process with respect to x , (i.e. there exists a countable set { x i ) z l such that for any Borel set A C R1 and any open set B C R1 the w-sets {w : b(t,x , w ) E A , x E , B ) , {w : b(t,x , w ) E A , xi E B , V i ) only differs a zero-probability w-set; 2' Ib(t,x,w)l ~ l ( t ) ( lIxl), + cl (t)dt < oo. where cl ( t ) 2 0 is non-random such that Let bn(t,x , w ) = inf,(b(t, Y , w ) (nV C I ( t ) )ly - X I ) , n 2 1. Then Qn, bn is still adapted, and 1) bn(t,x,w) T , as n T , V t 2 0, x $ G ; Ibn(t, x,w)I ~ l ( t ) ( l1x1); + 2) Ibn(t,x,w) - b n ( t , ~ , w ) l ( n v c l ( t ) )I Y - X I 3) xn --+ x i n R 1 ~b,(t,xn,w)-+b(t,x,w).Qt~O,x~G.

<

zt-

<

+

<

Now we can get a result on the existence of a strong solution for SDEs with discontinuous b and with partially degenerate a.

Theorem 309 Assume that 1" b = b ( t , x , w ) : [ O , o o ) x R x f l - + R , (T = ( ~ ( t , x , w:)[ 0 , m ) x R X fl + RIBd, c = ~-( xt ,,Z , W ) : [O, m) x R x Z x fl -+ R , are $y'Nk - adapted and measurable processes such that P - a s . Ib(t, x,w)l ~ l ( t ) ( l lxl), 1 4 4.,w_)12 + Jz lc(t,x, Z , 4 1 2 4 d . 4 ~ l ( t ) (+l lx12), where - $y'Nk is the a-field generated by w and fik up to time t, that is,

<

+

<

z y ' N k = u(w,, fik((0,s ] ,U),\JU E B z , s 5 t ) , and cl(t) is non-negative and non-random such that for each T < oo cl (t)dt < m; 2' b(t,x , w ) is continuous in x E R1\G, where G c R1 is a Borel measurable set such that x E G ===+ a ( t ,x , w ) > 0 ,W 2 0,Qw and m l G = 0 , where ml is the Lebesgue measure i n R1; moreover, b(t,x , w ) is a separable process with respect to x , (i.e. there exists a countable set {xi)zO=,such that for any Borel set A C R1 and any open set B C R' the w-sets { w : b(t,x,w) E A , X EB ) , {w : b(t,x , w ) E A , xi E B , V i ) only differs a zero-probability w-set; 30 a ( t ,x , w ) is continuous i n x ; and limh--rOJZ Ic(t, x + h, Z , W ) - ~ ( tx , Z , w)12~ ( d z=) 0;

fl

306

10. Comparison Theorem and Stochastic Pathwise Control

= 1,2, - . . , and each T < m, x1,w) - ~ ( x2,w)12 t, 2 x 1 , 3 , ~ -) c(t,52,z,w)I 4 d ~ <) 4 ( t ) p $ ( I x 1 - x212), + SZ T as [xi1 < N , i = 1,2, t E [O,T];where So g ( t ) d t < m; and p$(u) 2 0,as u 2 0 , is non-random, strictly increasing, continuous and concave such that So+~ u / P $ ( u >= 0 ; 5 • ‹ x L y===+ x + c ( t , x , z , w ) 2 y + c ( t , y , z , w ) . Then (10.7) has a minimal strong solution. ( A solution xt is called minimal, i f all other solutions yt are greater than it, that is yt 2 xt,Vt 2 0 ) . Furthermore, if, i n addition, the following condition also holds: so ( x - Y ) . 3 ,w ) - b(t,Y , w ) ) 5 k$(t)pTN(Ix - y12L as 1x1 , I Y ~ 5 N , V x ,Y E R 1 ,V t E [O, TI; where ts(t)and P$(u)have the same property as that i n 4'; then (10.7) has a pathwise unique strong solution.

40 for each N

Proof. Let b,(t,x,w) = inf,(b(t,y,w) ( n V c ~ ( t ly ) ) - x I ) , n 2 1. Then by Theorem 170 (10.7) has a pathwise unique strong solution xT for coefficients (b,, a , c). Write h ( t ,x , w ) = ~ l ( t ) (+l 1x1). Again, by Theorem 170, there exists a pathwise unique strong solution J t satisfying (10.7) with the coefficients (h,a , c) : . tt = xO+s; h ( s ,E,)~s+.$~ ( S , I , , W ) ~ W , + S ; S~ C ( S , L, Z , W ) & ( ~ S , d ~ ) . Moreover, by the conclusions 1)-3) o f Lemma 308, applying the comparison theorem (Theorem 295) one finds that x r < x:+l < Et,Vn. Hence there exists a limit P - a.s. xt = lim xy,Vt > 0.

+

n+oo

O n the other hand, 1x;12

< lx:12 + ltt12,and by Lemma 114

l2

E sup,,,[lx: + lltl21 I k* < 03. Hence by Lebesgue's dominated convergence theorem one finds that lim E IxT - xtl 2 ds = 0. ,400 From these one can take a subsequence i n k ) o f { n ) , denote it by { n ) again, such that dt x d P - a.e. x y -+ xt, in Rd and 2 E ( r Ix: - x,l ds < 1/2", n = 1,2, ... Hence E J , sup 1x;12 dt < m. NOW notice that n

XT

+

+

-

+AT J O t ~ , ( ~ , x ! , w ) d w S JZ~n(~,xF-,~,w)Nk(ds,dz), where A: = Sot bn(s,x:, w)ds. Hence there exists a limit ( i f necessary, take a subsequence) At = lim AT, = X:

n+cc

10.3 Strong Solutions for 1-Dimensional SDE with Jumps

307

and we have xt = xo At a(s,x,, w)dw, Jz c(s,3,-, z, w)fik(ds,dz), since a is continuious in x , c satisfies condition 3', and they are less than linear growth. Notice that xY, w)I < ~1 ( t ) [ l +S U P ~ ( ~ X ? ~ ) ] . Ibn(~, Hence, one finds that A? is a finite variational process, and so also is At. Let us show that

+ Jot

+ +

~t

=

Jo1

b(s,x., e,ps,w)ds.

(10.9)

Indeed, from m l G = 0 applying the occupation density formula for the local time of semi-martingale (Lemma 159) one finds that 0 = JGLf(x)da= J ~ ~ I ( ~ . EI GU ()S , X S ~ W ) I ~ ~ ~ . Since by assumption la(s,x,,w)12 > 0, as x, f G. Hence m l ( s E [O,t]: x, f G ) = O . Therefore, by conclusion 4 ) of Lemma 308 J; Ib(s,xs,w) - bn(s,xY,w)Jds J ; I ( , ~ ~I ~~()~ , x . , w - )~ & , x : , w ) I ds 0. Thus (10.9) 1s derived. Now suppose that x: is another strong solution of (10.7). By Lemma 308 bn(t,x , w) < b(t,2 , w). Hence applying again Theorem 295 one has that x? < xi,Vn. Letting n -+ oo. It is obtained that xt < xi. Therefore xt is a minimal solution of (10.7). In addition, if 6' holds, then by Lemma 115 the solution is also pathwise unique. w Furthermore, in the case that b, a and c are non-random, then the condition on a can also be weakened. Actually, we will have the following theorem.

c

-+

Theorem 310 Assume that 1' b = b(t,x ) : [0,co) x R -t R , a = a ( t , x ) : [O,co)x R -+ R'@*, c = ~ ( x, t , z ) : [O, co) x R x Z R, are jointly Borel measurable processes such that Ib(t,x)l c ~ l ( t ) (+l 14, 144 4l2+ Jz 144 x , 4124 d z ) c ~ l ( t ) (+l 1x12), where cl(t) is non-negative and non-random such that for each T < co J: cl (t)dt < oo; T' b(t,x) is continuous in x E R1\G, where G c R1 is a Borel measurable set such that x E G ==+ ~ ( xt ), > 0,Vt 2 0, and mlG = 0, where ml is the Lebesgue measure in R1; 3 ' a ( t ,x ) is jointly continuous in ( t ,x ) ; and JZ Jc(t hl,x h, z ) - c(t,x , z)12?r(dz)= 0; limhh~-+O 4' for each N = 1,2,. . . , and each T < co, 14,x1,w) - 4 4 x2,w)12 c c T N ( ~ ) P T N ( I x ~ - X Z I ) ,

+

+

308

10. Comparison Theorem and Stochastic Pathwise Control

as Ixjl < N , i = 1,2, t E [O,T];where g $ ( t ) d t < CO; and &(u) 2 0,as u 2 0, is non-random, strictly increasing, continuous and concave such that So+ dulpTN(u)= 00; 50 x 1 Y x + c ( t , x , z , w ) I y+c(t,y,z,w), k Z = R~ - { 0 ) , JZ &,(dz) < m. Then (10.7) has a minimal stromg solution. Furthermore, zf, in addition, the following condition also holds: 7" sgn(x - Y ) . (b(t,x)- b(t1Y))2 k%)&(Ix - Y I ) , as 1x1 , lvl 2 N,Vx,y E R1,W E [%TI; where k c ( t ) and &(u) have the same property as that in 4'; then (10.7) has a pathwise unique strong solution.

*

Proof. Let bn(t,x ) = infv(b(t,Y ) + (nV ci(t))I Y - $11, n > 1. Then b, is Lipschitz continuous in x. By Theorem 170 SDE

(10.10) has a weak solution xr. Notice that by conditions 4" and 5" the following Tanaka formual holds: Ixrl, - xr2 = sgn(x:l - ~ : ~ ) d ( x :-l xr2), if { x ~ ) , , , ,i = 1,2, are two solutions of (10.10). This means that the pathwisehniqueness of solutions to (10.10) holds, because b, is Lipschitz continuous in x. Hence by the Yamada-Watanabe type theorem (Theorem 137) (10.10) has a pathwise unique strong solution xr. Now the remaining proof follows in exactly the same way as in Theorem 309. rn

1

Ji

Example 311 Set a ( t , x ) = (It#ot-"O So)[ko k l x ] , ~ ( xt ,,Z ) = 1 t ~ ~ t - 1(iFo -+ C1x)co(z), where ai < 1, So > 0, Ici, ki, i = 0,1, kl 2 0 all are constants, ko > 0 and J, 1 . 0 ( z ) 1 2 ~ ( d<~CO. ) Write b(t,X ) = It+ot-""[k; kix k h f i I n f01, where as < 1, k:,i = 0,1,2, all are constants. Then (10.7) has a strong solution for the coeficients (b,a , c). In the case that kb < 0 then (10.7) has a pathwise unique strong solution.

+

+

-

+ +

The results cannot be applied to ODES, because that la(t,x)l > 0 at the discontinuous points x for b(t,x). (However from SDE, returning to an ODE, one needs to put a ( t ,x ) = 0). Furthermore, we can also generalize the result to the case that b has not necessary a less than linear growth. Theorem 312 Assume that

10.3 Strong Solutions for 1-Dimensional SDE with Jumps

1" b = b(t,x ) : [0,co) x R + R, a = a ( t , x ) : [O, co) x R + RIBd, c=c(t,x,z):[O,co)~RxZ+R, are jointly Bore1 measurable processes such that Ib(t, x)l I ~ l ( t ) (+l 1x1 9&)>, ( l1xl2), 14t,x)12 I ~ l ( t ) + Sz Ic(t,x,4124 d z ) < ~ l ( t ) , where g k ( x ) = 1 ln(1 + in(1 . . . ln(1 + I X ~ ~ ~ O )

309

rIL

+

+

) ) ,

7

Ic-times

(no is some natural number), ko > 0 is a constant, and c l ( t ) > 0 is such that for each T < co, :J cl(t)dt < co; .P b(t,x) is continuous in x E R1\F, where F c R1 is a compact set such that m l F = 0 , where ml is the Lebesgue measure in R1; moreover, there exist a So > 0 and a open set G I > G such that x E G1 a ( t ,x ) 2 So > 0 ,V t 0; ? a(t,x ) is jointly continuous i n (t,x ) ; and limhh~+oJz Ic(t + h l ,x h, Z ) - ~ ( Xt , ,z)12~ ( d z=) 0 ; 4' for each N = 1,2, . . . , and each T < co, I . ( t , ~ l l ~ )-ff(t,x2,w)12 < cTN(t),JTN(Ix1 -x21), as lxil < N , i = 1,2, t E [O,T];where c c $ ( t ) d t < co; and p$!(u)> 0,as u > 0 , is non-random, strictly increasing, continuous and concave such that So+d ~ l &(u) = co; 5" x L Y ===+ x + c ( t , x ,z,w) > Y + c(t,y, z,w), a z= RI - ( 0 ) , JZ $ $ l ( d t ) < m. Then (10.7) has a weak solution. Furthermore, i f , i n addition, the following condition also holds: 7" sgn ( x - Y) - (b(t,x) - b(t,y)) I G ( t ) & ( I x - Y I ) , as 1x1 , lyl 5 N , ~ X , fY R1lVt E [O, TI; where k z ( t ) and &(u) have the same property as that i n 4"; then (10.7) has a pathwise unique strong solution.

>

+

Proof. Let bn(t,x ) = b ( t , x ) W n ( x ) , where W n ( x ) is a smooth function such that 0 5 W n ( x ) 5 1, W n ( x ) = 1, as 1x1 5 n; and W n ( x ) = 0, as 1x1 2 n 1. Then by the previous theorem (Theorem 310) for each n there exists a pathwise unique strong solution x: satisfying the following SDE: V t 0 , t xt = xo bn(s,xS)ds SOa ( s ,x,)dws J: JZ C ( S , 2,-, z)%k(ds, d z ) . Now in the same way as in Theorem 176 one finds that "the result of SDE from the Skorohod weak convergence technique" holds. In particular, we have that @in) satisfies the following SDE with Gin and p ( d t , d z ) on -In (6, P), where (-In x t ,wt ,q-n (dt,d z ) ) come from "the result of SDE from

+

+ sot

g,

+

>

+

310

10. Comparison Theorem and Stochastic Pathwise Control

the Skorohod weak convergence technique", etc:

Also in exactly the same way one can prove that as n_ -+ co a(s, ZLn)dz Sot a(,, Z;O)dG;O, in probability P , and S," C(S,ZiY, z)P(ds, dz) -+ C(S,Z i t , z)$(ds, dz), in probability F. Write 2in = 2 0 A; Sot a(s, ZLn)dXn J: SZC(S,5:; ,z)?(ds, dz), where A; = Sot bn(s, ZLn)ds. Hence there exists a limit (in probability) At = lim A;, n-+OO and we have -+

+

+

+

Write .rN = inf { t 0 : IZiOI> N} . So it only remains for us to prove that I~;~'"(bn(~, a;.) - qs,~ ; o ) ) d ~ l

>

=

1~:~'"-

(bn(s, ZLn) - b(s, Zi0))dsl -+ 0, in probability

-

If this can be done, then letting n satisfies: P - a s . Z

=

-+

CTN + CTN l

xo

+

F.

co in (10.11) we can show that ZiO

+

b(s, ZLO)ds

~ATN

a(~,e)dG;~

c(s, ZLy, z ) k ( d s , dz), ~t 2 0.

Notice that TN f co, as N f co. We obtain that ZP is a weak solution of (10.7), as t 2 0. Now observe that for any E > 0 P ( l ~ : ~ ' " - ( b ~ ( s , j ~~ ~b(s, ) ZLO))dsl> E )

I SUP,

-

SUP,^, IZLnI + SUPS<* lZLO1> fi)

Sot A r ~ -Iipec, Isup,i,li~~15fids ~ATN-10 +f E'- So - IikO@G /bn(s,Ztn) - b(s1 2 s )I ~ = + I; + ~ y ,

+?EP

1;yN

nrz1

s u p ~ ~ T \ ~ ~ ~ ~ + s u p ~ ~ ~ l

where kfi = 2(1+ N gk(fi)), and F C Gz C GI, G2 is an bounded open set. Obviously, for arbitrary 2 > 0 one can take a large enough fi

10.3 Strong Solutions for 1-Dimensional SDE with Jumps

-

311

g.

such that I:'~ < (See the proof in Thoerem 176). Notice that by the occupation density formula of local time (Lemma 159) Ifi < 2k- j5 ~ A T N = L$,TN-(iZ1O)da. 2 - 3 E So I ~ Gmds G6,, However,

3 ~SG2'

E lAtATN-l i: E lZ$,,,-l+

+

E 1x01 E

1~:~'"-

O(S,Z~~)~G:~~

~ATN-

+EIJ0 Szc(s,Z~%,z)~(ds,dz)l<~~. Hencelby Lemma 158 as a E ??2 (the c'losure of G2)

i k(N, since is a bounded set. Thus, we can choose a small enough 6, and an bounded open set G2 such that F C G2 c G I , and mlG2

c2

6og2

2k$~k,-2

>0

<-8k,z<

3'

< " - 8.

Hence

Since F is a compact set, we can also choose an open set G3 such that F c G3 c E CG2 ~ CG1. Write 61 = i n f { d ( x , y ) : x ~ ? ? 3 , yE G ~ ) . > 0. Noice that Then F$=Gjn x ~ R ~ : l x l < G } is a compact set, where G j is the complement of Gg;and for each s, b(s, x) is uniformly continuous on x E F i . So for any E > 0 there exists a 61 > q > 0 (q may depend on the given s) such that as lxl - xfll q and x', x" E FA, Ib(s, x') - b(s, xt')l < E. Now take N 2 IS, one finds that as n 2 N,

{

<

lFA

=

f EP

so

~ATN-

IE;otG2

Ib(s, 6 )- b(s, ZiO)l

.IsuP,
Eowever, for each given s ~ ( l b ( s , i Z i~ )b(s,ZLO)IIZboEF;

-

> E)

< F(lZin - ZiOI> q)

+P(IIz;n.,--;E150 I ~ ( sZin) , - b(s,ZL0)I IZ;oEF; > E ) 5 F(lzin - ZLOI > q).. where F$ = ~ ; n {x E R ~1x1 :5 IS}. Hence, for each given s as n -+ co, ~ ( l b ( ~ , i Z :-~ b(s,ZL0)/ ) IE;oEF; > E) 5 F(lZLn - ZiO1> 17) + 0. So applying -Lebesgue's dominated convergence theorem, one finds that as n -t co, I;'~-t 0. Therefore, we have proved that as n -+ co, I ~ i ~ ' " ( b ~ ( s , Z :~ )b(.s,Z~~))dsl 0, in probability

-

B..

312

10. Comparison Theorem and Stochastic Pathwise Control

So that ZiO is a weak solution of (10.7). Finally, if 7' holds, then by the Tanaka type formula the pathwise uniqueness of solutions to (10.7) also holds. Hence by the Y-W theorem (Theorem 137) (10.7) has a pathwise unique strong solution. w

10.4 Stochastic Pathwise Bang-Bang Control for a Non-linear System 10.4.1 Non-Degenerate Case Now let us formulate an existence theorem of the optimal Bang-Bang control for a very non-linear stochastic system with jumps. Let J(u)=E 1x71' dt, where xTis the pathwise unique strong solution of the following 1-dimensional SDE with jumps: W E [O,T],

and the admissible control u = u ( t ,x t ) is such that u ( t ,x ) is jointly measurable, lu(t,x)l 5 1,Vt E [O, TI,x E R1, and it makes that (10.13) has a pathwise unique strong solution. Denote the admissible control set by U. Our object is to find out an optimal control u0 E Usuch that J ( u O )= minUEuJ ( u ) . We have the following theorem.

Theorem 313 Assume that 1' At, i = 0,1,2,3,4,Cf , and Bt all are non-random real continuous functions o f t ; No 2 0 is a constant, k l , k2 and kg are any even natural numbers; and D t ( z ) 2 0 is jointly measurable, non-random such that Jz lW)12 4 d - 4 < 00, Jz IDt+h(z) - Dt(z)12~ ( d z ) 0 , as h 0 , <m; moreover, Z = R - ( 0 ) , JZ $n(dz) ,T A 2 0 ,i = 2,3,4; C i 2 0;Cf 2 60 > 0 , where So is a constant, Bt > CfCj; '3 g2(t,u) is jointly continuous in ( t , u ) E [O,m)x R such that 92(t,0 ) = 0, g2(t, U ) 2 0 , u 2 0; g2(t,x) 2 g z ( t , d , x Y20 4' gl(t, 1x1') is jointly continuous i n ( t , x ) such that gk(x)), gl(tl lx12) I ko(l +

>

*

n:=,

10.4 Stochastic Pathwise Bang-Bang Control for a Non-linear System

313

where k-times

%

(no is some natural number), and exists such that it is uniformly locally 4, as 1x1 5 r , where k, 2 0 bounded in x , that is, for each r < m, is a constant only depending on r. Then an optimal stochastic control exists, which is admissible and BangBang, that is, there exists an admissible control u0 E U such that J ( u O )= minuEUJ(u), uZ) = -sgn x 0t , where the optimal trajectory xZ) is the pathwise unique strong solution of (10.13) with ut = -sgn x. That is, the SDE for the otimal trajectory is: \Jt E IO,Tl,

1 9 1<

Let us explain the meaning o f Theorem 313. I f we consider J ( u ) = T E So 1 ~ dt, ~t o be1 the~ target functional as an energy functional, where xy is a trajectory subject t o a stochastic system like (10.13), controlled by some bounded process ut (its bound is assumed t o be 1 for simplicity), which we call an admissible control, ("admissible" means that it can be applied t o the system and makes the system have a solution), then Theorem 313 tells us that an optimal control exists that makes the energy expended by this system the smallest. T h e idea is, i f the energy reaches zero, thas is, o f course, its smallest value. I f so, then xt r 0,Vt t [0,TI. This is impossible in general, because 0 is not neccessary a solution o f the given system. However, i f the trajectory is closer t o the original central point, then the energy expended is smaller. Thus t o make the enrgy expended as small as possible, we should apply an optimal control as follows: At each time when the trajectory x t , the solution o f the system, departs from the original central point, we should apply a control ut such that it immediately fully pulls back the tracjectory xt directed towards the original central point such that it makes the energy l x ~ smaller. 1~ Obviously, such a control should be ut = -sgn(xt), which is called a Bang-Bang control. Theorem 313 tells us that the Bang-Bang control actually is admissible and optimal. T o prove this theorem we first need t o introduce a lemma. Lemma 314 Let ( x ~ , w ~ , be C ~a ) triple of d-dimensional Tt-adapted P ) such that random processes defined on a probability space (a,5, (5)t,o, wy is a BM, and i f we write

10. Comparison Theorem and Stochastic Pathwise Control

314

p ( d t , d z ) = p""(dt,d z ) - x(dz)dt, and we assume that Z = R 1 - {O) , and S , &a(dz) < m; then p ( d t , d z ) is a Poisson martingale measure with the compensator n(dz)dt. Let (a!, wZ), be a similar triple defined on another space (O', $, (#)t,o,- P'). p) and four d-dimensional Then there exist a probability space (fl,5, St-adapted random processes (Zy,q,i&,Zt) such that the finite probabilty distributions of (Zy,i&, and .iiit,ct) coincide with those of ( x y ,wy, CT) and (xZ),wZ), respectively. Moreover, Gt is a d- dimensional St -BM under the probability f;, and i f we write F((O,tI, U ) = Co<s
c:)

-- -

-

ct)

c:),

(q,

-

w%

Lemma 314 is easily seen from the proof of Theorem 137. Now let us prove Theorem 313. Proof. First let us prove that (10.14) has a pathwise unique strong solution, so uZ) = -sgn x! E U.In fact, if we write

+ +

b(t,x) = A!x AZxgcgl(t, [xi2)- A:xgz(t, 1x1) - A ~ ( x ) ~ ~ ~ + ~ -~;(~)(2ki++1)/(2ka+l) - Bt s g n ( ~ ) , ( ~ ( xt ), = C? Ctl 1x1 , 4 4 x , 2) = D t ( . > ( x q x-l < a + fijyIxI>a), then xb(t, x ) 5 Atx2gl(t,(xi2) AZ)x2 - Btx . sgn(x) 5 Po + IA!11x2(l + n:=1 iik ( X I ) , d t ,x ) 2 60 > 0 , JzIc(t, x , z)12 ~ ( d zI) < 00, x y ===+ x c ( t , x , z ) 2 y c(t,y , z ) . Hence Theorem 305 applies, and (10.14) has a pathwise unique strong solution. Secondly, let us show that J ( u O )5 J ( u ) , V u E U. In fact, for any u E U let x: be the pathwise unique strong solution of (10.13). Write t w: = & [ ~ : + o & + Ix:=oldws = J ; p(x:)dw.. Then E[(w;)~ -(W;)~~ = SE ~[ (] w ~ ~ : ) ~ 1 5=, t]- S . Hence ( ~ 2 "is still ) ~ a ~BM~ by Theorem 97. Moreover, one easily sees that p(x)-l exists, and p(x)-l = 1, as x = 0; p(x)-l = as x # 0. Hence xp(x)-' = 1x1, and we can write dwt = P ( x y ) - l d w ~ So . xy will satisfy (10.13) with p(xy)-ldw: substituting dwt. Similarly, x! will satisfy (10.14) with p(x:)-'dwZ) substituting dwt. Applying Lemma 314 we find a probability\space (0,5,(5t)t,o, - F)

+

>

+

KO

+

y,

--

10.4 Stochastic Pathwise Bang-Bang Control for a Non-linear System

315

and four 1—dimensional $(—adapted random processes (a;™, x\, Wt, Ct) such that the finite probabilty distributions of (x^,wt,(t) and (x%,u>t,Q) coincide with those of (x™,u)",£") and (x®,Wt,(,t), respectively. Moreover, Wt is a 1—dimensional gt—BM under the probability P, and if we write P((0,t],U) = Eo< s < t I 0 *&c.eu> for i > 0,f/ 6 ®(Z), q(dt,dz) = p(dt, dz) — ir{dz)dt, then
,z), xo = x £ R . Applying Ito's formula to \xf\ , one finds that (for simplicity we still write xt for x", and ut for u(t, xf)) yt = y" = \xf\ satisfies the following SDE: P - a.s. Vt € [0, T], dyt = {2(A°tyt + A\ytgi{t,yt) - A2tytg2(t, ^yJVO) - A3t(yt)k3+l

(C?)2 + (

Similarly, by Ito's formula to to |a^| , where a^ is the pathwise unique strong solution for ut = w? = — sgn 0;°, one sees that $ = |aT(| satisfies the following SDE: P - a.s. yt € [0,T], A2ty°g2(t,

Now write b(t, y) = 2(A°ty + A\ygi{t, y) - A2tyg2(t, ( /

f c

) / (

f c

)

Then ^ is a strong solution of the SDE with coefficients (b, a, c). Moreover, since (x-y)(b(t,x)-b(t,y))<2\A°t\\x-y\2 \2

+kN \A\\ \x - y\2 + (CD2 \x - y\2 + \x - y\2 Jz

10. Comparison Theorem and Stochastic Pathwise Control

316

5 ~ T , N12 - y12, as 1x1, lvl 5 N, t E [(),TI, where kT,N is a non-negative constant depending only on T and N. Dividing both sides by I+, 1% - yl, one easily finds that s d x - ~ )' (b(t,x) - b ( t , ~ ) )5 h , N Ix-YI 3 as 1x1, I Y ~ 5 N, t E [O,T]. On the other hand, one also sees that a s 1x1, lyl 5 N, t E [O,T], I.(~I $1 - c(t, y)12 5 k & , N d m . Moreover, as x 2 y, x c(t, x, 2) 2 Y c(t, y, 2). Let P' (4 W)= 2(A!$ + A:$gl(t,Z) - A:~tga(t, J W ) - A,3($)k3+' - ~ ; ( $ ) ( ~ l + " + ~ ) l ( ~ ~ z + lBtZ;ut) ) (C2)2 (Cj)2$ + Dt(2)'(ZImSm +N o I m , & 4 d z ) . +2G'c: Then one finds that P V , w) 2 b(t, $1, because that Z?ut 2 -Zrsgn(Zy) for u(.) E U. Therefore, the comparison theorem for solutions of SDE with jumps (Theorem 295) applies, and we ,Vt E [O,T]. That is, P - a.s. obtain that P - a.s. 1 ~ ? 1=~ $ 2 3 =

+

+

JW 6

+

+

+

-

1xr12 L. 1xy2,Vt E [0,T1, because {Z?),, O,Tl and {x~)tEIO,Tl have the same finite probability distributions. There ore, J ( u ) = E :J 1x?12dt 2 J(uO)= E I X E )dt, ~ ~VU(.) E U.

I

SOT

.

10.4.2 Partially-Degenerate Case Consider the following 1-dimensional stochastic system: Vt E [0,TI,

The admissible control set is: U ={u = u(t, w) : u(t, w) is 3t-adapted, and lu(t, w)l 5 1). In general, if u E U, u is called an open-loop control. However, if u = u(t, x?(w)), where x?(w) is the pathwise unique solution of (10.15) with this ut, and lu(t, xy(w))l 5 1, then u(t, xy(w)) is called a feedback control, or a closed-loop control. We still want to minimize the energy functional J(u) = E 1 ~ 1 1 dt 1 ~ among all u E U.We can have the following theorem.

Theorem 315 Assume that

10.4 Stochastic Pathwise Bang-Bang Control for a Non-linear System

317

- 0 , l ;A: and Bt all are non-random real continuous funct,z tions of t ; and Dt(z) 2 0 is jointly measurable, non-random such that

10

~i ~i

t,

'

Jz Pt(z)124 d z ) < w, Jz IDt+h(z) - Dt(z)12r ( d z ) -,0, as h

0, moreover, Z = R - 10) , Jz &r(dz) < m; .P A:, A: 2 0, ICtI > 0, Bt > C t C j , and 0 5 a < 1 is a constant; 9 gl (t,u ) is jointly continuous in ( t ,u ) E [0,w ) x R such that I ~ I ( ~ , U ) ~5 k ~ , g l ( t , O )= O; 91 (4u ) 2 0, as u 2 0; g1(t,x) 291(t,y). x LY 2 0 Then an optimal stochastic control exists, which is admissible, feedback and Bang-Bang, that is, there exists an admissible control u0 E U such that J(u), U: = -sgn x:, J(uO)= minuELL where x: is the pathwise unique strong solution of (10.15) with ut = -sgn x. +

-

Proof. First let us prove that (10.15) with ut = -sgn x has a pathwise unique strong solution, so u: = -sgn x! E U.In fact, if we write

b(t,x ) = A:x - At xagl ( t ,1x1) - A: &Izt +O - Btsgn(x), a ( t , x ) = C: C; 1x1, ~ ( x, t ,2 ) = Dt(z)x, then xb(t,x ) < A:x2 - Btx . sgn(x) 5 ]A: x2, la(t,0)12 2 60 > 0, Jz l.(t, x , .)12 r ( d z ) L x2 JZ lDt(z)12r ( d z ) < w, x r y ===+ x + c ( t , x , z ) > y + c ( t , y , z ) , and b(t,x ) is discontinuous at x = 0. However, la(t,x)122 ( C ? ) > ~ 0, as x = 0. 2 Moreover, 2(x1 - x z ) ( b ( t , x ~-)b ( t , x d ) L 1A:I Ix - yl , 2 I.(t,x1) - 4 , x 2 ) 1 2 L lGl Ix - Y I , x L y 3 x+c(t,x,z) L y+c(t,y,z). Hence Theorem 310 applies, and (10.15) with ut = -sgn x has a pathwise unique strong solution. Now the proof of the remaining part is similar to that of Theorem 313. Notice that in the above theorem the coefficient b(t,x) has a discontinuous point x = 0, and at this point la(t,0)12 > 0. That is, o ( t , x ) is non-degenerate at the discontinuous points of b(t,x), and the Lebesgue's measure of the set of all discontinuous points is zero. Now let us consider another partially degenerate stochastic system, where its coefficient b(t,x) can be greater than linear growth: W E [O,T],

+

1

318

10. Comparison Theorem and Stochastic Pathwise Control

Theorem 316 Assume that 1' A:, C:, i = 0 , l ;A!, A: and Bt all are non-random real continuous functions of t ; and D t ( z ) 2 0 is jointly measurable, non-random such that Jz l ~ t ( z >4 ld~z ) < 0 , JzIDt+h(z) - Dt(2)12' ~ ~ ( d z 0) , as h 0, moreover, Z = R - { 0 ) , Jz &n(dz) <m ; 2' A:,A: 2 0, C!(Ci)-l < 0 , Bt > C:Ctl, Bt > 0 a n d 0 a! < 1 is a constant; '3 g l ( t ,u) is jointly continuous in ( t ,u ) E [0,m) x R such that 191(t,u)1 I ko, g1(t, 0 ) = 0; gl(t, u ) 2 0, as u 2 0; x 2 Y 2 0 ===+ g1(t,x) 2 g1(t,y); 4' g2(t, 1 x 1 ~is) jointly continuous i n (t,x ) such that

<

where k-times

%

(no is some natural number), and exists such that it is uniformly locally bounded in x , that is, for each r < m, as 1x1 r , where k, 2 0 is a constant only depending on r. Then the optimal stochastic control exists, which is admissible, feedback and Bang-Bang, that is, there exists an admissible control uo E U such that J(uo) = rninUEuJ ( u ) , u2) = -sgn x2), where x2) is the pathwise unique strong solution of (10.16) with ut = -sgn x.

121< +, I

I

<

10.5 Bang-Bang Control for d-Dimensional Non-linear Systems

319

Jz Ic(t, 2, 2)l2r ( d 4 5 No Jz l W ) 1 2 r ( d 4 < 03, x 2 y ==+ x + c ( t , x , z ) 2 y + c ( t , y , z ) , and b(t, x) is discontinuous a t x = 0. Obviously, F = (0) is a compact set in R1. Moreover, the degenerate points of o(t,x) should satisfy that 1x1 = -C,"(C,')-' = ft. Since by assumption -C,"(C,')-I is positive and 61 > 0, as t E [0,TI, where 61 is a continuous, then one should have f t constant, which exists. Write Gal = ( - h , & ) . Then mlGal = 2S1 > 0, and a(t, x) is non-degenerate in x L. Ga1,Vt E [O,T]. Since u(t,x) is jointly continuous, there exists a constant 60 > 0 such that la(t, x)12 2 60, as x t. W E [0,TI. Write G1 = Galla Then G1 satisfies the condition 2O in Thoerem 312. Moreover, as 1x1, lyl 5 N, b - y12 + kN IA?l IX - Y I ;~ (X- y)(b(t,x) - b(t,y)) 5 SO, as 1x1, lvl N, k~ IA?l) Ix - YI . sgn(x1 - xz)(b(t,XI) - b(t, 22)) I F'urthermore, 2 I.(t,x> - 4t,Y)12 I Ix - Yl x 2 y =+- x c(t, x,z) 2 y c(t, y, 2). Hence Theorem 312 applies, and (10.16) with ut = -sgn x has a pathwise unique strong solution. Now the proof of the remaining part is similar to that of Theorem 313. rn

<

(I@/ +

+

ptq

+

10.5 Bang-Bang Control for d-Dimensional Non-linear Systems 10.5.1 Non-Degenerate Case Consider the following d-dimensional stochastic system: Vt E [0,TI,

The admissible control set is: U ={u = u(t,w) : u(t,w) is St-adapted, and lu(t,w)l 5 1). T We still want to minimize the energy functional J ( u ) = E So 1x;l2 dt among all u E U. We have the following theorem. Theorem 317 Assume that 1" Af,i = 0,1,2,3,4, and Bt all are non-random continuous d x d matrices o f t ; No 2 0 is a constant, k1,1c2 and k3 are any even natural numbers;

320

10. Comparison Theorem and Stochastic Pathwise Control

and d x d matrix D t ( z ) 1 0 (non-negative definite) is jointly measurable, non-random and such that Jz I D ~ ( ~z () d~ z~<) m, Jz IDt+h(z) - Dt(z)12~ ( d z ) 0 , as h 0 , moreover, Z = R~ - ( 0 ) , Jz $ $ r ( d z )< m ; 2" d x d matrices At 1 0 , i = 2,3,4; and d x d matrix C: 1 0 , and > 0 is a constant, and C: 2 60Idxd, that is, C: - 60Idxd 2 0, where Idxd is a d x d unit matrix; moreover, Bt > C,OC:, that is, d x d matrix Bt - C:C: is positive definite; 3" g2(t,u ) is real, jointly continuous i n (t,u) E [O, m) x Rd such that exists such that it is unifomly locally bounded i n u, that is, for each r < m, 5 k,., as lul < r , where k,. 2 0 is a constant depending only on r ; moreover, g2(t,0 ) = 0,g2(t,U ) 2 0 , as u 2 0; x 1 Y 1 0 ===+ g2(t,x) 2 ~ 2 ( t , ~ ) ; 4' g l ( t , [ x i 2 )is real, jointly continuous i n ( t , x ) such that d t , 1 x 1 ~I ) ko(l + ck(x)), where & ( x ) is defined i n 4' of Theorem 316. Then an optimal stochastic control exists which is admissible and BangBang, that is, there exists an admissible control u0 E U such that x0 J(uO) = minus J ( u ) , u ! = -i$JIlx:l#o, where the optimal trajectory xZ) is the pathwise unique strong solution of the SDE (10.17) with ut = - -&IlxlZo. -+

+

%

1 2 1

nL

Proof. First let us prove that (10.17) with ut = - ~. .I l l l f .has o a pathwise unique strong solution, so u! = - ~x0I l x g l fEoU.In fact, if we write .

+

-

b ( t , x ) = A!x Atxgl(t, 1 x 1 ~-) A:xg2(t, 1x1) - A:x 1 x 1 ~ ~ ' 1~1~(~1-~2) /Bt ( ~j ~I 2 + ~ ) xl lxlZ0 a ( t ,x ) = C: C: 1x1, c(t,x , z ) = Dt(")("IlXI.fi + J N a ~ I l x ( > m ) , then ( x ,b(t, x ) ) < A! 1 x 1-~ Bt 1x1 + ko / A t [(xi2(1 nF=l & ( x ) ) , lu(t,x)12 1 6: > 0 , J, Ic(t, x , 412.rr(dz) I No Sz lW)12 < 00, and b(t,x ) is discontinuous at x = 0. Moreover, 2 ( x - y,b(t,x) - b(t,y)) I [A![1% - y12 + ~ N , TIx - y12, as 1x1, lyl <- N , t E [O,T],where kN,* 2 0 is a constant depending only on N and T; and we have applied the facts that l92(t, 1x1) - 92(t, I Y ! ) ~ 5 I&92(6 1x1 + ~ ( I Y I - 1 ~ 1 ) 111x1 - 1 ~ 1 15 IGN 12 - 91, as 1x1 , lyl N ; etc. Obviously, 2 I d t , x ) - 4 4 y)12 I Ix - Y I

+

~(w

<

p:l

10.5 Bang-Bang Control for d-Dimensional Non-linear Systems

<

321

2

144 x , 2 ) - 4 4 Y , 4124 d . 4 JZ l ~ t ( 441d ~. 4 12 - Y I . Hence Theorem 185 applies, and (10.15) with ut = - fiIl.I#o has a pathwise unique strong solution. Now for any u E U write the solution of (10.17) corresponding to this u as x:. Let dwr = p(x?)dwt, where p(x) is a d x d orthogonal matrix, that is, p(x) .p*(x) = I , (p*(x)is , as the transpose of p(x)),such that the first row vector is x # 0; and is 1, as x = 0. Then w:, t 2 0, is a BM in the same probability space, which satisfies the conditions that dwt = p-'(x:)dw: = p* ( x ~ ) d w ~ and x: . p-'(x:)dw: = x:=l (2;)' #d(w:)l xg=l ( X : ) ' ~ ~ ~ ( X : ) ~ ( W . -.

J,

(g,&, g),

+. . . + x;=l

+

(~;)jpd~(~:)d(w =:I )X ~ ;~

~ ) ~

~(WY)',

where w: = ( ( w : ) ~.,. . , ( w ; ) ~ )In . fact, VO

<

s 5 t, t E [ ( w-~ w:)* (w? - w:)' IT8]= E Js Gijdr = ( t - s)6ij. Hence by Theorem 97 wp, t 2 0, is a d-dimensional BM. The other conditions for w:, t 2 0 stated above are obviously true. So x: satisfies (10.17) with p ( x ~ ) - l d wsubstituting ~ dwt. Similarly, xE) satisfies (10.17)with p(x!)-'dwE) substituting dwt. B y Lemma 314 we find a probability space (0,5,(xt)t>o, - F ) and four gt-adapted random processes (Z: ,ZE),Gt , where the first three componenets are d-dimensional and the fourth is 1-jimensional, an2 such that the finite probabilty distributions of (Z:, Gt , Ct) and (3, Gt ,Ct ) coincide with those of (x: - ,wr ,C:) and (xE), WE), respectively. Moreover, Gt is a d-dimensional gt-BM under the probability P, and if we write F((O,tI,U)= C ~ < s 5 t I o # a ~ , Efor u , t L 0 , U E B(Z), c(dt,dz) = F(dt, dz) - ~ ( d z ) d t , then g(dt, dz) is a Poisson martingale measure with the same compensator n(dz)dt under the probability P. So we have that Z: sastisfies the following SDE: F - a.s. Vt E [0,TI, dxt = ( A b t g i ( t ,1xt12)- Aixtg2(t, [xtl)- A2(xt)2k3+1 -~;(~~)(2k1+1)/(2kz+ l) AE)xt Btut)dt (C; C,lzt)p(xt)-ldGt + Sz D t ( ~ ) ( x t - I l . , - ~+~~~ ~ I l z , - I > ~ o ) ~ ( d xot , =d xz E) R1. , Applying Ito's formula to 1Z:l2 , one finds that (for simplicity we still write xt for Z:, and ut for u(t,Z:)) ct = $ = )Zy12 satisfies the following SDE: - a.s. Vt E [O,T], , - Aiytgz(t, -A : ( ~ t ) ~ ~ + l dyt = [2(A!yt + A : ~ t g i ( tyt) -~t(~~)(k1+kz+1)/(2kz+l) +Bt ( x t , ~ t ) ) f ( C ~ ) ~ + ( C : ) ~ y t f 2 C , O C , ' d ~

--

rt),

c:),

+

+

+

+

m)

@,Om

t ( ~ ) ~ ( ~ t I+ ~N o, Ifmi> f i ) 4 w l d t + +C:~t)dGt+ SZPDt(z)( ~ t - f i1 + m~ m ~ I m > f i )

+ Sz D

+Dt(~)~(w-I@i-,~ +~NOIW>fi)lF(dt,

dz), YO

=

lxo12 E R1;

322

10. Comparison Theorem and Stochastic Pathwise Control

where Gt = ( G i , . . - ,G t ) , and G i is a 1-dimensional BM, which is the first component of Gt. Notice that the SDE for yt now is a 1-dimensional SDE, and ( x ,u) - ( x , = - 1x1, for all u E Rd such that IuI 1. So the proof of the remaining part follows as in Theorem 313.

>

AI~.~~~)

<

10.5.2 Partially-Degenerate Case Before we discuss the stochastic Bang-Bang control for a system which can be partially degenerate, we first need to establish a theorem on the existence of a strong solution for a SDE with jumps and with discontinuous coefficients and with partially degenerate a in d-dimensional space. Consider the following d-dimensional SDE with jumps as (5.15): V t 0 ,

>

Theorem 318 Assume that &r(dr) <m 1' Z = Rd - ( 0 1 , and 2' b= b(t,x) : [O,co) x Rd -+ R d , a = a ( t ,x ) : [0,co) x Rd RdBd, c = c ( t , x , Z ) : [O, 00)x Rd x Z R d , are jointly Bore1 measurable such that Ib(t, x)l I c l ( t ) ( l f 1x1 nT=l g k ( ~ ) ) , gk(x)), la(t,x)12 I k o ( l + 1 x 1 ~ S, x , 412r ( d 4 < ~ l ( t ) , where

sz

-+

-+

n:~

k-times

(no is some natural number), and c l ( t ) > 0 is non-random such that for each T < co, ~ ; f c l ( t ) d 0, as 1x1 5 r , lb(t,x)l 5 ky, where k, > 0 is a constant depending only on r ; 30 a ( t ,x ) is jointly continuous i n (t,x ) ; and limh,h~-+o SZ ( ~ ( thl,x + h , z ) - c(t,x , z)12~ ( d z=) 0; 4' b(t,x) is continuous i n x E Rd\F, where F c Rd is a compact set such that m d F = 0 , where m d is the Lebesgue measure i n R d ; moreover, there exist a So > 0 and an open set GI > F such that x E G1 + l a ( t , x ) ( 2 60 > 0,Vt > 0. Then for any given constant xo E Rd (10.18) has a weak solution on t > 0. Furthermore, if, i n addztion, the following condition (for the pathwise uniqueness) holds:

+

10.5 Bang-Bang Control for d-Dimensional Non-linear Systems

323

(P WU1) for each N = 1,2,,- . , and each T < co, b(t,~ 2 ) ) 2 ( ( X I - x2),'(b(t,2 1 ) ; + b ( t ,x i ) - ~ ( 2211 t , + Jz 144 X I , z) - ~ ( x2, t , z)12~ ( d z ) < .TN(t)PTN(I.l - x212), as Ixil 6 N , i = 1,2, t E [O,T];where c g ( t ) > 0 such that c G ( t ) d t < co; and p$(u) > 0,as u > 0, is strictly increasing, continuous and concave such that So+ duIpTN(u) = 03; then (10.18) has a pathwise unique strong solution.

Proof. Let us smooth out b(t, x ) and a ( t ,x ) only with respect to x to get bn(t,x ) and un(t,x), respectively. (See Lemma 172 and its proof). Then by Theorem 175 for each n there exists a weak solution x; with a BM w? and a Poisson martingale measure fikn (dt,dz), which has the same compensator ~ ( d z ) d tdefined , on some probability space (Rn,Zn,{ S r ) ,Pn) such that Pn - a.s. Vt > 0, X: = xO+J; bn(s,x:)ds+S; an(s,x:)dw:+Si JzC ( S , 2:-, ~ ) f i k n ( d ~ , d ~ ) . As in the proof of Theorem 176, one finds that "the result os SDE from the Skorohod weak convergence techniquet' holds. In p a r t i c ~ l a r ~Lave ~ e that (Zin)satisfies the following SDE with G;n and p ( d t , dz) on (a,5,P ) , where (Zin,Gln,p ( d t , dz))come from "the result of SDE from the Skorohod weak convergence technique", etc:

Write

>

= inf { t 0 : IZiOI > N } . Let us show that for each N as n

7~

-

-+ co

IPTN (bn(s,ZLn) - b(s,~;'))dsl ATATN= JiATN bn(s,Zin)ds. 1)

For this write

0, in probability

Then there exists a limit (in probability) A~ATN= n--03 lim ATAT,, and we find that

P.

10. Comparison Theorem and Stochastic Pathwise Control

324

+SEP JiATN IE:OQG~ lbn(s,ZLn)- b(s,Z20)l + qN.

=I?

Obviously, for any -

I? <

I.

~ s u-p ~ ~ T l ~- ; ~ l + s u p ~ ~ T ~ ~ ~ ~ l

> 0 one can take an @ large enough such that -

However, the term IFsNrequires more discussion. Notice that

EP +SEP =

J~ATN -

I ; E . : ~ E G ~ I ~ ~ ~ , < ~ ~ E ; ~ ~ ~ ~ ~ ~

JFTN-IE:OQG~ lbn(s,ZLn)- b(s,ZkO)I

~ u p- ~ < T ~ ~ ~ n l + s u l ~ ~ T ~ ~ ; ~ I ~

IE + I;;',

+

n;, (a)),

@ and F c G ~ c G3 C G2 C G I , G2 and G3 are open sets, E3 is the closure of G3, all of which will be satisfies the following SDE: determined below. Notice that yt = Z;:,, vt 2 0, ~t = X O + A ~ A , ~ + J ~~ ~( s , Y ~ ) I ~ ~ , ~ ~ N ~ G JZ L ~C +( SJ, ys-, ~ 4 1 1 y , - I s ~ P (dd ~ ,) . Moreover, Ig(s,Y ) I ~ ~ ~ <JzN I ~C ~( S , Y ~ ~~)~I y~, - ~ s N " ( d
+

zN

&

IE f

5 k ( ~k~, di T )& . 1

(J:

< k'(p,kT,d,~)&

JG2 dads)lip

(md(~2))l".

Thus, we can choose a small enough SN,z> 0 and an open set G2 such that F c G2 c G I , and (md(~2))1'p

<6

~< $-. ,

$/(d+l)

2

~

Since F is a compact set, we can also choose an open set Gg such that

F

c G3 cZ&, C G2 C G I .

Write

61 = inf { d ( x , y ) : x E G3,y E G 2 ) . Then S1 > 0. Noice that ~ $ = ~ j n { x ~ ~ ~ : ~ x l < @ ) is a compact set, where Gj is the complement of Gg; and for each s, b(s,x ) is uniformly continuous on x E FA. So for any E > 0 there exists a d l > 77 > 0 (7 may depend on the given s ) such that as lx' - x''1 q and x', x" E F$,

<

Ib(s,xl)- b(s,x")l < E. Now Qne finds $hat ~ATN&Isl I E ~ QIb(s,i-Ln G ~ - ): - b(s,zLO):O)( s E P SO ~Isup,,Tlzb"l+supB~TIz:oI~z J(F)&ds. However, for each given s as n >

<

10.5 Bang-Bang Control for d-Dimensional Non-linear Systems

325

-

p(lb(s, Zin) - b(s,ZiO)lIE;oeP; > E ) < p([ZLn - ZiOl > q ) +P(IIEpa---qollv Ib(s,Zkn) - b(~,ZkO)lIqoeF; > E ) < F([Zin- ZiO[> q).. where

-

F ; = G ; ~ { X E R ~ : I ~ ~ < Q ) .

Therefore, for each given s as n -+ co, P ( ~ ~ ( s , z :-) b(s,qO)II~~~~~ > E ) I P(1qn - qOI> 7)--+ O. So applying Lebesgue's dominated convergence theorem, one finds that as n -+ m, I$ -+ 0. Therefore, we have proved 1). However, since the conditions o n a and c are the same as those in Theorem 176, so the other terms in (10.18) also have similar limits t o those in Theorem 176. So that ZiO is a weak solution o f (10.18). Finally, i f 7O holds, then by the Tanaka type formula the pathwise uniqueness o f solutions t o (10.18) also holds. Hence by the Y-W theorem (Theorem 137) (10.18) has a pathwise unique strong solution. rn Now let us discuss the stochastic Bang-Bang control for d-dimensional SDE with jumps and with partially degenerate coefficients. Consider the following partially degenerate stochastic system in d-dimensional space, where its coefficient b ( t , x ) can have a greater than linear growth: Vt E [O, TI 1

T h e o r e m 319 Assume that 10 Ait , Cit , z' = 0 , l ; A:, A: and Bt all are non-random d x d continuous matrices of t ; and Dt(z) 2 0 is a jointly measurable, non-random nonZ () d~ z~ <) co, negative definite d x d matrix such that JZ I D ~ ( ~ Jz 1Dt+h(z) - Dt(z)12~ ( d z ) 0 , as h -* 0 , and, moreover, Z = Rd - ( 0 ) , JZ &s(dz) <m; 2' AZ,A: 2 0 , C:(C:)-' < 0 , Bt > CfC:, Bt > 0 a n d 0 5 a < 1 is a constant; 30 gl(t, u ) is real, jointly continuous i n (t,u)E [O,co) x Rd such that exists and is uniformly locally bounded in u, that is, for each r < m, <_ lc,, as lul <_ r , where Ic, 1 0 is a constant depending only on r ; and, moreover, gl ( t ,u) is jointly continuous i n ( t ,u ) E [0,co) x R such that 1 ko, g1(t,0 ) = 0; 191(t,~ ) < gl(t,u) L 0 , as u 2 0 ;

1 %I I

% I

326

10. Comparison Theorem and Stochastic Pathwise Control

x L Y L 0 ===3 g1(t,x) 2 g1(t,y); 4' g2(t, (x12)is jointly continuous in ( t , x ) such that 1g2(t,I X I ~ 5) ~ko(l +

n;, aca

where & ( x ) is defined in 4 O of Theorem 316. Then an optimal stochastic control exists, which is admissible, feedback and Bang-Bang; that is, there exists an admissible control u0 E U such that J ( u O )= m i n u ~ nJ ( u ) , u: = - & ~ ~ ~ # o , where x! is the pathwise unique strong solution of the SDE (10.21) with U t = - fiIX#0.

Proof. First we prove that (10.21) with ut = - f i I X f o has a pathwise

I

- - & I ~ ! # ~is well defined. Since Iu: 5 1, unique strong solution x:, so u0 we have uZ) E U. In fact, if we write b(t,x ) = - A h ( t , 1). - A:+L#o + A:xgz(t, 1x1 ) - &&S~+O, a ( t ,2 ) = C: c: 1x1, , ,~~f ~i I<, ~x - l > a ) , c(t,x , z ) = D ~ ( ~ ) ( X I + then lxlf I b ( t , ~ )5l l~tl+IAfII X I ~ - " + ( [ @ ~ + k0 1x1 kO(l+n:=",,k(~)), 144 0)121 60 > 0, Sz 1 4 4x , z)12~ ( d zI) No SZ l ~ t ( z )~l(~d z<) 00, and b(t,x ) is discontinuous at x = 0.Obviously, F = ( 0 ) is a compact set in Rd. Moreover, the degenerate points of a ( t ,x ) should satisfy that 1x1 Idxd = -CF(Ctl)-l. Let us discuss all situations. 1) I f there is a solution 1x1 = f t , since by assumption -Cf(C;)-l is positive > 0, as t E [O,T], definite and continuous, then one should have f t 2 where is a constant, which exists. Write Gal = { X E Rd : 1x1 < 6 1 ) . Then m&a1 = ~ 6 >; 0, and a ( t ,x ) is non-degenerate in x E Gal,Vt E [O,T].Since o ( t , x ) is jointly continuous, there exists a constant 60 > 0 such that (a(t,x)a*(t,x)A,A) 2 60 1Al2,VA E R d , as x E Ga1j2,VtE [O,T]. Write G1 = Gal/2.Then G1 satisfies the condition 4 O in Thoerem 318. Furthermore, one finds that as Ixl,iyl 5 N , 2 ( ( x - Y ) , (b(t,x)- b ( t , ~ ) I ) Ix-y12 + k IAfl ~ Ix-Y~ 7 where we have applied the facts that ( x - y, = - 1x12-a - I ~ I ~ - ~ 2

+

I&/)

-++ +)

I

- 1x12-a - ly12--"

+1

x y IyI

+ lyll-

+ # + jy? 1x1

= -(lxll-" - lYll-")(lxl - Ivl) i 0, etc. Furthermore, 2 1 4 4 x ) - .(t, y)12 I lC:l lx - Y I , 2 S, 144 x, .) - 4 t , Y , z)124 d ~I) JZ l a ( z ) l 24 d z ) 12 - Yl

10.5 Bang-Bang Control for d-Dimensional Non-linear Systems

327

Hence Theorem 318 applies, and (10.21) with ut = - i I Z Z ohas a pathwise unique strong solution. Now the proof of the remaining part is similar to that of Theorem 313 and Theorem 317. 2) If there is no solution for 1x1I d x d = -C:(Ctl)-l. Then a(t,x) is nondegenerate. In particular, since u(t,x) is jointly continuous, there exists a constant a0 > 0 such that (a@,x)a*(t, x)X, A) 2 60 1 ~ ,VX 1 E~ R ~ as, 1x1 5 N,Vt E [O,T].Write GI = {x E Rd : 1x1 < N) . Then GI also satisfies the condition 4' in Thoerem 318. So the proof still can be completed.

Stochastic Population Control and Reflecting SDE

The components of many important physical quantities can only take nonnegative values, and usually the quantities can be considered to satisfy some dynamic evolution systems. Therefore these situations can be examined by studying the dynamic systems. In the deterministic case one can put some conditions on the coefficients to make all components of the solutions to the dynamic systems non-negative. However, if the system is disturbed by a random process, e.g. a Wiener process, even in the simplest case the components of solutions to stochastic differential equations (SDE) can still change between arbitrary large positive and negative values. This prevents research proceeding any further. To overcome this difficulty we are motivated to examine the reflecting stochastic differential equation (RSDE) with coordinate planes as its boundaries, or with a more general boundary. In this chapter the general theory and applications of RSDE with jumps, which can be used as a stochastic population control model, etc., are systematically studied. In particular, for the case when the coefficients of RSDE are discontinuous, with greater than linear growth, the questions of the existence, uniqueness, comparison, convergence and stability of strong solutions to such RSDE or population RSDE are examined. Applications to the stochastic population control problem is also developed.

330

11. Stochastic Population Control and Reflecting SDE

11.1 Introduction Yu, Guo and Zhu (1987) has considered the following deterministic population control system

where xf is the size of the population with age between [i,i xt = (x:, .... xTm);pt is the specific fertility rate of females, ...... 0 -(I + 771(t)) 0 -(1+v2(t)) 0 i*. 0 1

i

...

o

...

...

......

0 b,,(t)

1 0

,

+ I), and

..

+ vrrn(t))

:b

. . . 0 b,, (t) . . . ... . . . . . . . . . . . . . . . . . . . .. \ (1- pOO(t))ki(t)hi(t)# 0,i = TI, ' . -,T2; 1 < r1 < r 2 < TT,, 0 0

poo(t) is the death rate of babies, vi(t) is the forward death rate by ages, ki(t) and hi(t) are the corresponding sex rate and fertility model, respectively, xf is the size of the population with ages between [i,i I ) , xt = ( x i , . ..,xLm), r, is the largest age of the people, and ,Bt is the specific fertility rate of females. The size of the population should be non-negative, i.e. xf 2 0 , l i r,. Under appropriate consideration Yu, Guo and Zhu showed that this is true. Furthermore, they and others, developed the theory of this model, used @, a s a control, and established a population control theory. By means of their theory, after some practical computations; they were able to make population control suggestions to the Chinese goverment and, finally, such suggestions were considered by the goverment. Hence the population control problem is interesting and useful in both its theory and in its applications. In the practical case, the population control system is disturbed by some stochastic perturbation. So we will consider that the stochastic population control system is a more realistic way. However, in the stochastic case the situation is very different. As the system is disturbed by a random process, e.g. a Wiener process, even in the simplest case the components of solutions to stochastic differential equations (SDE) can still change between arbitrary large positive and negative values. This means that if we consider the following SDE system:

+

< <

+

dxt = (A(t)xt B(t)xtPt)dt xo = x, t E [O,T].

+ atdwt

Then, in general, we cannot have the components x f ,i = 1,.. . ,r, of the solution non-negative.

11.1 Introduction

331

Now, for simplicity, let us explain first how could we get a non-negative solution for a SDE similar to the above in 1-dimensional space. To make xt 0, it is natural to consider the following SDE:

>

+

dxt = dyt d4t dyt = (A(t)xt B(t)xt&)dt x o = ? J o = x 1 4O = O , xt 0, t E [O,T],

+

+ atdwt,

>

where, intuitively, we introduce a process Notice that

4,

to make xt

> 0, t

E [O,T].

So, intuitively, we require that the process 4, has the following properties: 1) 4, is continuous in t such that 4, = 0,4, f , as t f . 2) If at time t, xt > 0, then at time t, we can require that ddt = 0; because A4, = dq5t+o(At) and yt is right continuous in t, so there exists a 6 > 0 such that by (11.1) as 0 < At < 6, xt+nt > 0. 3) If at time t, xt = 0, and there does not exist a 6 > 0 such that as 0 < At < 6, Ayt 2 0, then at time t, we should require that d4, > 0, so there still exists a 6 > 0 such that as 0 < A t < 6, xt+nt > 0. Or, roughly speaking, 4, is an instantly reflecting process such that when xt hits the boundary 0 and wants to become negative, 4t immediatly has a positive increment to make xt remain in the region [O, m). A SDE that has an instantly reflecting process in it, is naturally called a reflecting SDE (RSDE). Let us write it in a more precise form:

+

dxt = dy, d4, dyt = (A(t)xt B(t)xtPt)dt atdwt, 4, s a - adapted, continuous and increasing process such that q50 = 0, xSdds = 0,Vt 0, x o = y o = x 40 = 0 , xt E [O,w), Vt 0. (11.2) Here x,d4, = 0,Vt 0 means that x, > 0 ==+ d4, = 0, and d4, > 0 & x, = 0. This is properties 2) and 3) of 4,, which we required above. In (11.2) if we write a domain Q = (0, m), its closure = [0,w ) , its boundary dQ = (01, and its inner normal vector at the boundary n = 1, then J; x,dd, = 0,Vt 0 can be rewritten as J: xsnsd4, = 0, Vt 2 0, where n, = n = 1. For a solution with non-negative components to a SDE in a 2-dimensional space the situation will be more complicated. First, in this case we want the solution value in a domain O = (0, w ) x (0, w ) . So its closure is O = [O, w ) x [O, w), its boundary is ((0) x [0,w))U{[O, w ) x (0)). For the inner normal vectors at the boundary there are three kinds: n, = (1, O),

zt

+

+

sot

>

> > >

11. Stochastic Population Control and Reflecting SDE

332

as x = ( x I , x ~ )E (0) x ( 0 , ~ ) n, ; = (0, I), as x E (0,m) x (0); and 8, sin B), V8 E [0, $1, as x = (0,O). So one sees that at some spen, = (COS cial boundary point the inner normal vector may be non-unique. Secondly, the reflection process 4t is now also two dimensional, so the properties that we require should become 1) 4t is continuous in t such that 4o = 0,4, is finite variational on any finite interval; that is, I4IT < oo, for each T < m, where 141T is the total variation of 4 on [O,T]; 2) if at time t, xt E 0, then at time t, we can require that d 141, = 0; 3) if at time t, d 141, > 0, then we should require that xt E 8 0 . For a solution value in a general domain Q to a SDE with jumps in a d-dimensional space similar requirements for the reflection process can also be made. See the next section.

11.2 Notation Now let us consider a d-dimensional reflecting stochastic differential equation (RSDE) with Poisson jumps as follows: (As before we write "RCLL" for "right continuous with left limit".) '

+

+

+

dxt = b(t,xt,w)dt a(t, xt,w)dwt 1, c(t,xt-, z,w)fik(dt, dz) d&, xo=xEG, t r o , xt E G, t 2 0, 4, is a Rd - valued Zt - adapted RCLL process with finite variation 141, on each finite interval [O,t] such that do = 0, and I4lt = J: 188(xs)d 141s 4t = n(s)d 141,, n(t) E N,,, as xt E dQ, (11.3)

Jgt

where - wt is a d-dimensional standard Brownian motion process (BM), Nk(dt,dz) is a Poisson martingale measure generated by a Tt -Poisson point process Ic(-) with a compensator x(dz)dt, x(.) is a a-finite measure on a measurable space (2, %(Z)) such that Nk(dt,dz) = Nk(dt,dz) - x(dz)dt, and N, = u,+ON,,,, N,,, = {n E Rd : In1 = 1, B(x - n r , r ) n O = 0) , B(xo,E) = {y E R~ : ( Y - 201 < E ) , and Q is a given convex domain in Rd, dQ is its boundary, G is its closure, where we assume that Q satisfies the so-called uniform inner normal vector positive projection condition: if there exists a unit vector e E Rd and a constant c > 0 such that e . n = (e, n ) 1 C,Vn E u ~ E ~ Q N ~ . It isobvious that Q = R $ = { x = ( x l , . . . , x d ) E Rd : x i > 0 , 1 < i s d )

-

11.2 Notation

333

obviously satisfies this condition with e = (-73, • • • > T7=;)J an< i co = -73Condition |0| t = JQ Id&(xs)d\cf)\s in (11.3) obviously means that \\s increases only when xs £ dQ, and the condition t = Jo n(s)d \cf>\s means that the reflection d<j)t happens along the direction of the inner normal vector n(t), so we may call such a reflection a normal reflection. The geometrical meaning of Afx is that it is the set of all inner normal vectors at x, when a; £ dQ. Obviously, when x £ 6, by definition Mx = $ (the empty set). Actually, in the case that i t € 6 no refelction is needed, so we do not need an inner normal vector.

Remark 320 1° In the case 9 = Rd one can take t one easily sees that in (11.3) conditions 3): \\t = /0* Iae(xs)d\4>\s , and 4): (f>t = /„* n(s)d \<j>\a , are equivalent to 3'): for all f: 0 —* Rd, bounded and continuous such that f | a Q = 0, and for allt>0

fZf(xs)-d4a=0;

and 4') for all {/(•) e Dd([0,oo), 6), where D d ([0,oo),e) is the totality of RCLL functions f : [0, 00) —* 9 , J0(ys — xs) • dcj)s is increasing, as t is increasing.

Let us show 2° in Remark 320. In fact, 3) and 4) can be rewritten as d\\t = Ide{xt)d\<j>\t,

or 0 = IQ(xt)d\ct>\t,

and <% = n(t)d\(f>\t.

_

So if they are true, then V/ £ Cb(Q; Rd), with / \dQ_= 0, where Cb(Q; Rd is the totality of bounded continuous functions / : 6 —• Rd, t

hence fi(xt)ni(t)d\(j)\t = 0, • • • , fd(xt)nd(t)d\(f)\t = 0,V£ > 0; that is, f(xt) • dcj>t = 0. So 3') is true. Furthermore, since 0 is a convex domain, and d(pt = n{t)d \cj>\t, (yt - xt) • dcj>t = (yt - xt) • n{t)d\<j>\t > 0, Vj/(-) e Dd([0,00),

6).

Hence 4') is true. Conversely, suppose that 3') and 4') are true. Since 9 is a convex domain, and (yt - xt) • d<\>t > 0, Vy(-) £ D d ([0,oo),e),_# t = n(t)d | 0 | t . That is, 4) is true. Furthermore, V/ G Cb{Q;Rd), with / | a e = 0, by f(xt) • d(f>t = 0, f(xt) • n(t)d\tf>\t = f(xt) • dt = 0. From this one has that =0. Iae(xt)d\^\t So 3) holds true. The proof of 2° in Remark 320 is complete.

334

11. Stochastic Population Control and Reflecting SDE

Obviously, according to (11.3), the population SDE discussed in the introduction should be presented as follows:

+

+

+

dxt = ( A ( t ) x t B(t)xtPt)dt ~ ( xt)dwt t, Jz ~ ( xt-, t , z)fik(dt,d z ) -4. +dq!~,,t 2 0,xo = x , xt E R+,for a11 t 0 ,

>

whereR$={~=(x~,~~~,x~)~~~:x~>0,1
~o'

mt

Remark 321 For ( i i . 4 ) , by definition as x E Rd+, Nz = @ (empty set). I f x E 8 ~ $t h, e n n = (nl,nz,..., n d ) EN,, iff,a s x i = 0 , xj # 0 , j # i , ni = 1, nj = 0 ,j # i; and as xi = x j = 0 , xk # 0, k # i,j, ni = cos 8, nj = sin 8, 8 E [O, $1, n k = O , V k # i , j; and as xi, = x i , = x i k = 0, x j # 0 ,j # i l , i 2 , . . . ,ik, nil = s i n 8 c o ~ ~ , c o s ~ ~ . . .cos . . .&-2, ~.. ni, = sin 8 cos q51 cos d2 . . .cos 4k-3sin q5k-2,

. .. . .. . ..

nik-, = sin 0 sin d l , nib = ~ 0 ~80E ,[o:$1, 4 1 , 4 2 . . . , , 4 k E [o,$1, nj = 0 , as j # i 1 , z 2 , - . - ,ik,IC 2 3. Since allni > O , i = 1 , 2 , - . .,d. So all di(t),i = 1 , 2 , . . . ,d, i n (11.4) are increasing, where

4(t) =

&dt),

' -' 7

4d(t)).

For the solution of RSDE (11.3) we introduce the following definition.

Definition 322 1) We say that (st,4,) is a strong solution of (11.3), i f it is RCLL and 5y1'*-adapted, where

5ygfik = 0 ( w s ,Z ~ ( (sO ) , u); , s 5 t ,U E % ( Z ) ) , and it satisfies (11.3). 2) W e say that (11.3) has a weak solution, i f there exist a probability space (0,5,(&), P ) and two 5t-adapted RCLL processes ( x t ,4,) with a BM wt and a Poisson martingale measure (dt,d z ) which has the same compensator n(dz)dt as the given one i n (11.3) defined on it such that ( x t ,d t , w t , %,v(dt, d z ) ) satisfy (11.3) on (0,5,(&), P). 3) W e say that the pathwise uniqueness for solutions of (11.3) holds, zf for any two solutions ( x i ,4f),i = 1,2, of (11.3), which are defined on the same probability space (0,5,( g t ) ,P ) with the same BM wt and Poisson martingale measure fik ( d t ,d z ) ,

11.3 Skorohod's Problem and its Solutions

335

I

xBI

1 2 - 4t = 0 ) = 1. P(suptto / x i = 0 ) = 1, P(supt>o 4) W e say that the uniqueness i n (pro%ability) law for solutions of (11.3) holds, if for any two solutions (x:, &), i = 1,2,of (11.3), which may be defined on different probability spaces (Oi, P i ) , with different BM urt and Poisson martingale measures fiki(dt, d z ) , i = 1,2, respectively, but with Poisson martingale measures Nki (dt,d z ) , i = 1,2, which have the same compensator n(dz)dt as the given one, the probability law of { ( x i ,q$), t 2 0 ) and { ( x i ,4:),t 2 0 ) are the same.

zi,(s),

11.3 Skorohod's Problem and its Solutions T o find the solution o f (11.3) we may discuss the simplest deterministic case first. That is the so-called the Skorohod problem. Now let us introduce the Skorohod's problem. Suppose we are given a convex domain Q in R d ;denote its boundary by d Q , and its closure by g. W e introduce the following Definition 323 For a given Y E D and Yo E g, i f there exists an X E D ( O ) = { X E D : Xt E g,t 2 0 ) and a q5 E D such that (X,q5) satisfies

x,=~,+~~,t2o,xo=~o€'i5, X , E 8,t 2 0 , 4, is a R~ - valued - adapted RCLL function with finite variation 141, on each finite interval [0,t ] such that 4o = 0 , and I4lt = Ias(Xs)d Ids, 4, = n(s)d 14lS, , n ( t ) E Nxt, as Xt E 80, (11.5) '

zt

s;

where

zt = n=l n a ( X s ;s 5 t +

s;

n-l);

then ( X , 4 ) is called a solution o f

Skorohod's problem (11.5) with given (Y,Yo).(For space D see Appendix B). Obviously, the last three conditions in (11.5) indicate that 141, increases only when xt takes values on the boundary, and dr$, has the same direction as that o f nt. Besides, in the case that Nx(t)has only one element, then nt is the inner normal vector at xt E 6'0;in the case that Nx,,, has more than one element, then nt is the one belonging t o Nx(t),and it Gists. Furthermore, it should be noticed that the reflection 4, here can have jumps, but i f its jump increment is not equal t o zero, then it only happens at such times t, that xt takes values on the boundary. For the existence o f solutions o f the above Skorohod's problem we need t o introoduce conditions on the domain in which the solutions take their values. W e have already introduced the uniform inner normal vector positive projection condition in the "Notatation" section. Let us restate it here.

336

11. Stochastic Population Control and Reflecting SDE

Definition 324 We will say that the domain O in R~ satisfies the unif o r m inner normal vector positive projection condition, if there = 1 and a constant co > 0 such that e n = exists an e E Rd with (eln) 2 Q J , V ~E U X E ~ ~ J V X -

.

lei

The geometric meaning of the uniform inner normal vector positive projection condition actually means that all inner normal vectors on the boundary have a uniform positive projection: There exists a vector e E R~ with [el = 1, amd there exists a positive constant co > 0 such that all inner normal vectors n at the boundary have a uniform positive projection on this vector e; that is n . e 2 co > 0. Let us explain why we need the condition that 0 is convex and satisfies the uniform inner normal vector positive projection condition for the solution to Skorohod's problem. Notice that Remark 320 still holds true for (11.5). In particular, by means of the convexity of O one has that if (xt, 4t) is a solution of (11.5), then for all y(.) E Dd([O,oo),g ) , Sot (ys - x,) . d4, is increasing, as t is increasing; or equivalently, for all y (.) E D~([0, co),g ) ,

Such a property will be used many times in the discussion of the solutions. For example, we can use it to prove the uniqueness of the solutions to (11.5): Suppose that (xt,4,) and (xi, 4:) are two solutions of (11.5) with a given Yo,Yt,Vt 2 0 and Yd,E;',Vt 2 0, respectively. We want to prove that (xt14J = (xi,&, V t 2 0, when Yo = yd, and Yt = Y,',Vt 2 0. Naturally, we discuss the difference (Xt - x;12. By means of the elementary inequalit la b] I (la1 lb1)2 5 2[la12 lb12], Va, b E Rd, we have IXt - x;125 2( IYt + 14, - d l 2 ) . Notice that as 4, and 4: are finite variational, we can use Ito's formula to find 14, - dl2= 2S,"cos- - 4 2 . 4 4 , - 4 3 t 2 2 +Co<slt[14s- - 4)s- + n4, - 14,- - 41,-1 -2(A- - 4 L ) * (4% = 2 J,'(ms- - 4:-) . d(ms - 4:) + xO<s6t lams- am:12 = 2 J,'k- 4:) . 4 4 s - 4 3 - 2 S3hS- 4s- - (4: - 4;-N . 4 4 : - 4 3

+ T

+

+

w2

-&,I

WI

=CE1

- CO<s
c:=~

11.3 Skorohod's Problem and its Solutions

337

I2 = -2 CZI S{t,,(4S - 4 s - - (4: - 4',-)).4 4 : - 4:) = 0, where {ti)zl is the set of all discontinuous points of 4; and/or 4: such that 0 5 ti t. Finally, we apply the fact that O is a convex domain, so by (??) Il= ~ J ; ( X , XL) - d + , + ~ J ~ ( xX L , ) .d4',

<

-2 J; (Y, - Y,') ed(4, - 4;) I 0 - 2 Hence we have proved that

(ys- Y,'). d ( 4 , - 4;).

>

Therefore, when ( Y o , x )= (Yd,k;'),Vt 0; we have that Xt = X,',#, = q$,Vt 2 0. So the uniqueness of the solution to (11.5) is proved under assumption that @ is a convex domain. Similarly, if ( X t , d t ) is a solution of (11.5) with a given Yo,Yt,W 2 0, one has that Ixt - x,l2 I 2( IYt - ys12 I4t - 4,12). By Ito's formula and again by (??) as s t

+

<

I4t - 4,12 = 2Sst(du- - 4,) . d4, + CSiu5t lWu12 I 2 JSt(4,- 4,) . d 4 , - C,<,
sst

Hence under the assumption that @ is a convex domain one also has that for s 5 t

Now let us explain why we need the uniform inner normal vector positve projection condition. Notice that this condition says that there exists a vector e E R~ with lei = 1 and a positive constant co > 0 such that all inner normal vector n at the boundary a@of the domain have a positive projection greater than Q at vector e; that is,

Recall that in the discussion of the existence of the solution to an SDE when the coefficients satisfy the Lipschitzian condition and the less than linear growth condition, we can always use an iteration technique to find the approximate solutions. After that we can take the limit to find the solution. However, now for the RSDE or for the special case - the Skorohod's problem - even if we can get approximate solutions, they are a sequence of paired functions ( X t , d t ) .So to discuss their convergenc we need some estimate

338

11. Stochastic Population Control and Reflecting SDE

which derived from the uniform inner normal vector positve projection condition. More precisely, we have the following lemma.

Lemma 325 Assume that O is convex and satisfies the uniform inner normal vector positive projection condition, and ( X , 4 ) is a solution of (2.1) with (Y,Yo).Then as s t

<

141t - 141, 5 G114, - dsl IXt -&I + (141,- 141,) 2 h s u p S + g I K 1

where ko

- KI

1

> 0 is a constant depending on q,onlg. In particular,

141, < 2ko SUP,,,

.

IYSI

Proof. For s < t let

+F

1

2

I X , - x,12 5 + 2 AF (141, - 141,)). Now by using the uniform inner normal positive projection condition we can estimate the second term in the right side of the above inequality. In fact, by (11.9) and (11.5) 141, - 141, 5 Cole. SStnJ141, = Cole. (h- 4,). One finds that

,

l4lt - 141, 5 6' l4t -

and

+I& 161, - 141, 5 c o l ( l x t So applying the elementary inequality 4ab 5 ;a2 +8b2, Va,b 0, one has that 2 1 x , - x , 1 ~ 5 2 1 a ~ 1+ 4 ~ 1 a : : ( ~ x t - x , l + l a - ~ , l )

>

la::12+

+

I a:: I 2 +

2-1 1x - x,12 1scg2 2-l IY,- y,12. 2 Hence 2 1 lXt -x,12 5 l6co2) 1Llr.l . From now on let us always assume that O is convex and satisfies the uniform inner normal vector positive projection condition. In the following let us show that for any given step function Y (11.5) has a unique solution.

(q +

Lemma 326 Let (Y,Yo)E D x has a unique solution.

g , and Y is a step function, then

(2.1)

Proof. Write [ x ] E~ g for the unique vector satisfying the following equality Ix - [x]al= M i n {Ix - yI : y E g ) , i.e. - [ x ]is~the point which attains at the shortest distance from x E R~ to O. We will use a step by step constructive procedure to find the solution

11.3 Skorohod's Problem and its Solutions

339

(Xt, 4,). Notice that Y is a step function. So for any given T < oo, as t E [O,T], Yt only takes finite many different vector values. Moreover, by assumption Yo E g , and Y is is RCLL. So, there should be a smallest time TI such that YTl $ g . (If TI = oo, then we immediately get a solution Xt = Yt, 4, = 0, as t E [O,oo). The construction of a solution is completed at this first step. The same remark holds true for the 2nd step and so on. we We do not repeat it anymore). To construct an Xt always lives in may define X t = Y t , & = 0 , a s t E [O,Tl), and XT1 = [YTlla E a@, = XTI - YTI = [YTI]a - YTI . By this definition one sees that Xt = rt +4, E G , as t E [O,Tl], and d4, > 0 only when Xt E 8 0 , (here it happens at t = TI); moreover, (by the definition of c $ ~ ~Now ) . starting from TI and noticing d4, = ntd that Yt + 4Tl, t 2 TI is still a step function, we can do the same thing to Yt + cPTl, t 2 TI to find a Tz and then to define Xt and 4, on t E [Tn-1, Tn] such that (Xt,4,) is a solution of (11.5) for t E [Tn-l,Tn]. Such solution can be constructed by induction. More precisely, let To = 0, 40 = 0, T, = inf t > T,-~ : ~t +4Tn-1 8),n t I,

+

{

Xt = Yt +#JT,-,,~st E [Tn-l,Tn); xt = [YT, + 4Tn-l]a,as t = Tn; 4t = 4 ~ , , - t~E, [Tn-l,Tn); ~ 4t = [YT, 4 ~ , - ~ ]-a YT,,~st = Tn. Then one sees that (X, 4) is a solution of (11.5) with (Y, Yo). Moreover, it is already known that if the solution of (11.5) with (Y, Yo) exists, then it must be unique under assumption that Q is convex. rn

+

Lemma 327 Assume that (Yn, Yon) E D X g, and (Xn, 4") is the solution of (11.5) with (Yn, Yon). If there exists a Y E D and a Yo E such that limn-+ooSUpt<~ lytn - Ytl = 0, for any T > 0; then (11.5) h a s a unique solution (X, 4) with (Y, Yo), and for any T > 0 it satisfies limn,, supt~T lXtn - Xtl = 0, limn-+ooSUPtg 14; - 4tl = 0. Proof, Notice that Y E D, so x , t E [O,T] is bounded for any given < oo. By assumption we also have that SUPn SUPtg Iqnl ICT < 03, where bT is a constant depending only on T. So by (11.7) SUPt<~ lXp - xyl25 ~ S U P Iqn ~ ~ -Tqm12 4 ~ - ~lqn ~- t <' ~ '(l$nlT f 14mlT)5 2 s u ~ t gIqn ~ - qm12 8 k ~UP,
T

<

+

+

-+

11. Stochastic Population Control and Reflecting SDE

340

14: - 4~1'-+ 0. SuPts~ Hence there exists a (X, 4) E D(Q) x D, such that for any T limn,, (suptgT IX," - Xt I + S u P t g ~14: - dt 1) = 0. Now let us show that (X, 4) is a solution of (11.5). By

>0

x,"= Y," + 4:,

obviously, letting n + m , one has that xt = Yt + & . Now, let us show that 4, is a finite variational function. For this, and for any t > 0, let us make the subdivision in [0, t]: 0 = to < t l < ... < t k = t.

I

+

5 SUPn 1 4 ~+ 1 2,Ej=l t 14tj - 4:t . I SUP, IVlt E . Hence 141t I sup, 14"lt 5 kt, for all t 2 0. So, 4, is finite variational. Furthermore, let us show that 3') and 4') in Remark 320 hold. If this is done, then (X,4) is a solution of (11.5). We will use the step function approximation to complete the proof. Let us show 4'). For any given i? D(Q) we know that for any s 5 t J,"(r-~:).dq5;20,asn21, because (Xn, dn) is a solution of (11.5) with (Yn, T). Now take step functions .ZN E D such that sups
+ 1-

+I-

Iss

J s t ( z - xT- 2;).

dd:l

+ 122.". 44,

- 4:)/

JSt(xr - x:) . d4:I = z:=l I:. Obviously, as n -+ m , N --+ co, I: +I," I sups<, IX, - Xs - Z,Nj [Idlt + 14nltl +sup,<, I& - XI: IPIt I 2kt(sup&t IX,- Xs - Z,NI SUPSltIXS - X:l) 0. = 0, SO for Notice that ,ZN is a step function and limn,, supt,* - 14: each N as n -+ co, 12 = IjSt Z; . d(mr - 4:) + 0. Thus, for arbitrary E > 0, one can choose a krge enough N such that I: + It2 < $. Then for this N there exists an N such that when n 2 fi, 4 It3 + I," < 5. So as n + co, ,EIZ1 I: -+ 0, and we have that for any s 5 t,

c",=, +

+

I

t

-

Js(XT - Xr) .d4, 2 0. 4') is proved. 3') can be shown in a similar way. In fact, we can choose step functions .ZN E D such that

11.3 Skorohod's Problem and its Solutions

Ix,

341

I:z

sup,lt + 0 , as N + oo, for any 0 5 t < oo. For any f : 0" + Rd, which is bounded and continuous and such that f las = 0 , one has that

I~,'f(x.) . d ~ l =I J , ' ~ ( x-dms ~ ) - S,'~(XP). d 5 I s : ( f ( x s ) - f(Z.N)) 4 . 1+ ls,'(f(xs) -

+ Is,' m:).4 4 , -

~ l

fcm4~1

~ ) l +ls,'(f(xS)r(xa))dm:I. -

I I,'

So we can similarly show that f (X,) . d4.l = 0. Therefore 3') is derived. T h e proof is complete. Now we are in a position t o establish an existence and uniqueness theorem for the solution o f (11.5). (For Skorohod's topology see Appendix B). Theorem 328 Assume that Q is convex and satisfies the uniform inner normal vector positive projection condition, and Y E D with E g, then (1 1.5) has a unique solution ( X ,4) with (Y,Y o )If we write this solution corresponding to the given Y E D as ( X ( Y ) ,4(Y))E D x D , then this is a continuous map from D to D x D , that is, i f Yn -+ Y (in D under Skorohod topology), then X ( Y n ) + X ( Y ) E D (in D under Skorohod topology), and 4(Yn) -+ 4(Y) E D (in D under Skorohod topology). Moreover, I4(Y>ltI k b ~ ~IKI, s ~ t where the constant k&> 0 depends only on co. Therefore SUP,$, IXs(Y)I 5 (% + 1) supsit K s l . Proof. First, for any given (Y,Yo)E D x 8, by Lemma 381 one can take step functions Y n E D such that SUp,g IYs - YFl -+ 0 , as n + oo, for any O T < oo. Now by Lemma 326 (11.5) has a unique solution ( X n , with ( Y n ,Yon) for each n. By Lemma 325 14nI~ ~ ~ O S U P IYFI. S ~ T Hence S U P , Idn/, 5 2ko(supSi~IYsI + suPnsuPs<~ IT - YsI) I k ~ , where kT 2 0 depends on Q (introduced in the uniform inner normal vector positive projection condition), Y and T only. Therefore applying Lemma 327, one finds that (2.1) has a unique solution ( X ,4) with (Y,Yo)such that limn,, SUP,
<

<

I

mn)

342

11. Stochastic Population Control and Reflecting SDE

----7L

etc. and from that (X,

I I

,4, ) is the solution of the Skorohod's problem

x=Y,+x, IF, (

--n

I I

and the other statements of (11.5) hold true for (X, ,4, ), * and also from that limn-,supslr - Y, = 0, so we find that

by means of Lemma 327. These mean that exactly X(Y,) -+ X ( Y ) E D and 4(Yn) -+ 4(Y) E D (both in D under Skorohod topology).

11.4 Moment Estimates and Uniqueness of Solutions to RSDE Now let us return to discuss the reflecting stochastic differential equations (RSDE) presented as (11.3) in the "Notation" section. Because we will use the results from the previous section to discuss the existence of solutions and so on, so we will always assume that the domain G satisfies the follwing condition: (HI) O is a convex domain and it satisfies the uniform inner normal vector positive projection condition. An important property of the solution to (11.3) should be mentioned first. See the following lemma. Lemma 329 If (xt,4,) is a solution of (11.3), then for each t > 0 ~ ( txt-, , Z, w)Nk({t}, dz) E 8. xtA4, = 4, - 4,- = 0

+ Sz

Proof. Indeed, n x t = c(t, xt-, 2, w ) ~ ~ ( { t dz) }, ~ 4 , . Hence, by xt E 8, if Adt = 0, then xt- Jz c(t,xt-,~,w)Nk({t},dz) = st- Axt = xt E G. The implication "===+"is proved. Conversely, notice that the RSDE in (11.3) can be rewritten as dxt = b(t, xt, w)dt a(t, st,w)dwt J, c(t, s t - , z, w)fik(dt, dz) d4, = dyt d4t1 or xt = Yt + 4 t . Moreover, from xt = xt- Ayt A& one finds that xt - (st- a y t ) = ~ 4 , . So, if stAyt = xt- Jz c(t, xt-, z, w)fik({t}, dz) E 8 ,

+

J,

+

+

+

+

+

+

+

+

+

+

+

11.4 Moment Estimates and Uniqueness of Solutions to RSDE

343

then, by 4') in Remark 320 one finds that l&t12 = a d , [xt - (xt&)I I 0. Thus A4, = 0. So the inverse implication "+=" is true. rn From now on, for simplicity, let us also always assume that (Hz) x + c ( t , x , z , w ) E G , f o r a l l t > 0 , z E ~ , w E S 1 a n d x ~ ~ . Obviously, by assumption (Hz) it is always true that xt- Jz c(t, at-, z, w)Nk({t), dz) E G. Now, under assumptions (HI) and (H2) the RSDE (11.3) can be rewritten as follows:

-

+

+

+

+

, z, w)fik(dt, dz) dxt = b(t, xt, w)dt a(t, xt, w)dwt Jz ~ ( txt-, +d4,, ~ > o , x ~ = x E G xt, EG, LO, 4, is a continuous R~ - valued - adapted RCLL process with finite variation 141, on each finite interval [0, t] such that 4o = 0, and 141t = Iae(xs)d 141,, 4t = J'n(s)d 141,~ n(t) E N,, , as xt E 80. (11.10) The difference between (11.3) and (11.10) is that in (11.3) 4, is RCLL, but here in (11.10) we can require that 4, is continuous by means of assumption (Hz). For the solutions of (11.10) we have the following a priori estimate.

zt

Jl

Theorem 330 If (xt,&) is a solution of (11.10) with E 1x01~ < co, and l lx12), lb(t,x,w)l2 + ll~(t,x,w)1lZ JZlc(t,x, z,w)12r(dz) I ~ l ( t ) (+ where cl(t) 2 0 is non-random such that for each 0 5 T < co :J cl(t)dt < co, then for any 0 5 T < co kT ~ , E ~ ~ Plxt12 o +~ 141; ~ ~I

+

where 0 5 kT is a constant depending only on J:'cl(t)dt,co Assumption (HI) ), E lxo12 and T.

(appeared in

Proof. By Ito's formula

Since 4, is continuous in t, and xt is RCLL in t , it has at most countable many discontinuous points in t. So by 4') in Remark 320 one finds that

344

11. Stochastic Population Control and Reflecting SDE

+

~ - x~s ) . d4, ( ~2 Jst(xr~ - x,) . d4, I 0. s,t = 2 J Again by the continuity of 4, one finds that as 0 I s 5 t 5 T , EC,t = E CS<,.jt lnxr12 = E S , ~Jz I C ( ~ , X , - , z,w)12Nk(dr, dz) = E Jst Jz Ic(r,xr, z, W )l2 ~ ( d z ) dIr ko [2Jst ci (u)du +4Jst cl(r)E 12, - xo12 dr 4E 1x01~ Jst cl(u)du]. Let s = 0. By the martingale inequality, Gronwall's inequality, and Fatou's lemma (introduce stopping times T N = inf { t E [O,T]: lxtl > N ) , N = 1,2, ..., if necessary), from (11.11) it is easily seen that 2 s u ~ o l t l ~lxt - X O ~ 5 k'$. Hence 2 2 E ~ u P o < ~Jxtl < T I 2 E s u p , g j ~lxt - xo12 + 2E 1x01 I k ~ . Now by Lemma 325

+

-

where yt = XO+J; b(r,x,, w)dr+$ a(r,x,, w)dw,+2 J; Jz c(r,x,-, z, w)Nk(dr,dz). Hence ~141;I k&. Again by (11.11) one finds that as 0 I s 5 t I T J: C I ( U ) ~ U . E s u ~ ~Ixrj -~xs12 ~ It Applying (11.12) again one also has that E(141t - 141s)2< k? Sst cl(u)du,as 0 I s I t I T . The proof is complete. rn For the uniqueness o f solutions to (11.3) we have the following theorem. Theorem 331 Assume that (xi,f ), i = 1, 2, satisfy (11.3) with the same initial value and with the same BM and the same Poisson martingale measure on the same probability space, and assume that the conditions in Theorem 330 hold. Moreover, assume that for each T < m and N = 1,2,. . . 2(x1 - x2) . (b(t,xl,w)- b(t,x2,w)) Ila(t,xl,w) - a(t,x2,w)112

z, w )I

+

2

+ Jz Ic(t1x l , z, W ) - ~ ( x2, t, 4 d z ) I ~ N , T ( ~ ) P N , T ( I-x x2I2), ~ as t E [O,T],x1,x2 E R~ and l x l l , 1x21 I N ; where k N , ~ ( t2) 0 is nonrandom such that S,T ~ N , T ( s )<~m; s and pN,T(u) is non-random, concave and strictly increasing in u 0 such that pN,T(0)= 0, and So+ du/pN(u)= co. Then P(su~t~ lxZ0 - s f + SUPt>0 14: - 4: = 0 ) = 1.

>

1

I

Proof. For arbitrary given T < co write T N = i n f { t E [O,T]: > N). By Ito's formula one finds that as t < T 2 2 E - x L T N I K O E J ; " ~~ ~ N , T ( s ) P N , T(I~ ) d ~ .

IX~~+(X:[

I

x?I

11.5 Solutions for RSDE with Jumps and with Continuous Coefficients

345

>

2

Hence one easily sees that E lx: - x:l = 0,Vt 0 and P - a s . x: = x:, W 1 0. Now again by Ito's formula and after an application of the martingale inequality one finds that P(su~t,[O,T]1x2 = 0) = From this and (11.3), and again by means of the martingale inequality it is also seen that P(su~t,[O,,] (4: - 43 = 0) = 1.

xI:

11.5 Solutions for RSDE with Jumps and with Continuous Coefficients For RSDE (11.10) we discuss the case for non-random coefficients. Recall that we always make the assumptions (HI) and (H2) from the previous section; that is, we always assume that (HI): O is a convex domain, and there exists a constant q > 0 and a vector e E Rd, lei = 1 such that e . n q > 0, for all n E UZGasN,. (Hz): x c(t, x, z, w) E g, for all t 1 0, a E Z, w E R and x E 8. We have the following theorem.

>

+

Theorem 332 Assume that for t 2 0,x E G,z E Z 1' b(t, x), a(t, x), c(t, x, z) are jointly measurable such that there exists a constant Ico 2 0 lb12 + lla112 Jz lc12 ~ ( d z 5 ) ko, 2' b(t, x), a(t, x) are jointly continuous, and as lx - yl 4 0, It - sl + 0 JZ Ic(t, x,z) - 4% Y,4124 d z ) -,0, ' 3 ~ ( d z= ) dz/ ~ z l ~ " ,(2 = Rd - (0)). Then (11.10) has a weak solution.

+

Proof. We will use the step coefficient approximation technique to show this theorem. Let hn(0) = 0, hn(t) = (Ic - 1)2-", as (k - 1)2-" < t 5 k2-". Then by Theorem 328 there exists a unique solution (x:, 4:) satisfying

+ J:

+

bn(s, x2)ds J: an(s, xy)dws +So" .fz cn(%SF-, z)q(ds, dz) 4:, and the other statements in (11.3) hold for (x? ,q5;), x:

-

= xo

+

(11.13)

where for simplification of the notation in this proof we write q(ds, dz) = Nk(ds, dz), p(ds, dz) = Nk(ds, dz), and bn ( 3 , ~ s= ) b(hn (s), xh, (s)) etc.

346

11. Stochastic Population Control and Reflecting SDE

In fact, once x? is obtained for 0 5 t 5 k2-", then (x?, 4;) is uniquely determined as the solution of the Skorohod problem:

+

+

x? = xE2-, b(k2-n,xE2-,)(t - k2+) a(k2-n,xE2-n) .(w(t) - ~ ( k 2 - ~ ) )JZ c(k2-", xE2-., ,~ ) q ( ( k 2 - t], ~ ,dz) 4:, and the other statements in (11.3) hold for (xy, 4): on k2-" 5 t 5 (k 1)2-".

+

+

+

(11.14) Notice that in (11.13) the reflection process 4; is RCLL, but not necessary continuous. Now we show that for arbitrary E > 0 and T > 0

> C) = 0,

(11.15)

P(1$ - r]r(> E ) = 0,

(11.16)

lim supsup P(l$1 n t$T

C-*m

lim sup C-*m

n

sup t,s
hold for qy = s y ,421,wt ,Ct , where Ct = So' J;zlsl " d d ~d") , + So( J;zl>lzp(ds1dz). However, by the proof of Theorem 180, it can be shown in the same way that (11.15) and (11.16) are true for T$ = wt and Ct. Notice that by (11.8) one has that

where

+ So( an(s, x;)dws + Jot JZ cn(s, xy-,

+

Y? = 50 lo'bn(s, x!)ds Moreover, by Theorem 328

z)q(ds, dz).

where the constant Ic:, 2 0 depends only on q which come from assumption (HI). Now by the condition l o , and the martingale inequality, Esup,
On the other hand, by Lemma 325 as s 5 t MnIt -

ldnIs 5

14;

- 4:l-

So by (11.17) one easily sees that P ~- y,"l2. ~ U ~ E(Mnlt - 14nls)2I ~ T E ~ U IY," Therefore, again by (11.17), as s t

<

11.5 Solutions for RSDE with Jumps and with Continuous Coefficients

347

From (11.19) and (11.20) one easily sees that (11.15) and (11.16) still hold true for @ = x? and 4:. Therefore Skorohod's theorem (Theorem 398) applies: There exist a probability space ( Q 5 , P ) and four RCLL processes ( Zr, $,; G r , defined on it with the same finite probability distributions -0 as that of (x;, @, wt, C,), and there exist (i?$',$:,Gy, Ct ), where Gy is a BM on ( 0 , 5 , P ) , and there also exists a subsequence {nk) of {n) denoted by { n )again, such that as n -+ CCI -n T -I ij: -+ $, in probability, as ij: = Zr, q5t I Gy, Ct ,n= 0,1,2, ... Now set P ( d t , dz) = C s E d t I(oZa<:Edz)(~), P(&dz) = P ( d t , dz) - n(dz)dt, n = 0,1,2, ... Then P ( d t , dz) is a Poisson random counting measure with the compensator n(dz)dt for each n = 0,1,2, ..., and it satisfies the condition = J;z,ll zGD(ds,dz) J;II>l z$n(ds,d4,n = 0,112, (See the proof in Theorem 180). By the coincidence of finite probability distributions one easily sees that (Z?, satisfies (11.13) with 6;and p ( d t , dz) on ( 0 , 5 , P ) ; that is, Z? = f$ where f$ = xo+Ji b n ( s , Z ~ ) d s + J ~ a n ( s , Z ~ ) d G ~ + ~ ~ J z c n ( s , Z ~ - , z ) ~ ( d s , d z ) , and (557, J;) also satisfy the other statements in (11.3). Now notice that as n -+ m , Vt 2 0, x E R ~ , 2) -+ b(s, x), Ibn(s, .)I I ko, Ian(%x)I I: k01 a n ( %5) +x), Jz Icn(s, a, z) - C(S,2, z)12~ ( d z ) 0, JZIcn(s, 2, z)12 4 d z ) I: ko. So, similar to the proof of Theorem 180, one finds that as n -+ m IJ'(bn(s,Z:) - b ( s , ~ ) ) d s -+ l 0, in probability.

---

2)

---

T:

+ Jgt

Jgt

4;)

--++;,

+

+

Jot

-+ a(s, q)dG:, in probability, Jz C(S,Zt-, z)GO(ds,dz), in probability, cn(s, 2:-, z)Gn(ds,dz) Jz Icn(s, Z,: z) - c(s, Z:, ,)I2 n(dz)ds -+ 0, in probability. Therefore, letting n -+ cm one finds that

an(s, Zy)dGr

Sot

JZ

-+

q = +$:, where

+

+

+

i$= xo Jot b(s, e ) d s J,a(s1Z!)dG! Jz C(S,$-, z)?(ds, dz). In the rest of the proof one only needs to show that the other statements in (11.3) hold true for (%,$:). However, by the above, one also finds that aSn-+co su~tE[O,~l -+ 0. Hence one can take a subsequence of {n)denoted again by {n)such that P - a.s., as n --+ cq

a12

1%

1%

s u p t ~ [ ~ , ~-]

%)I2

-+

O.

348

11. Stochastic Population Control and Reflecting SDE

Thus, applying Lemma 327 P - a s . , as n -+ m, -n--oz S U P t ~ [Ixt ~ , ~ ]xt --t O ,

1

13;

sUPte[o,~]

-a:('

-

0,

and the other statements in (11.3) hold true for -0

(3$,$:). Furthermore, by -0

assumption ( H z ) & is also continuous. So (3$,qlt) is a weak solution of (11.10). The proof is complete. To obtain a strong solution of (11.10) we need a Yamada-Watanabe theorem as in the SDE case. Since the proof is almost the same as that of Theorem 137, so we will omit the proof.

Theorem 333 (Yamada- Watanabe type theorem). Assume that the coeficients b(t,x ) , ~ ( xt ),, and c(t, x , z ) i n (11.10) do not depend on w,and make the assumption:

Then we have the following two assertions. 1) If (11.10) has a weak solution and the pathwise uniqueness holds for (1 1. l o ) , then (11.10) has a pathwise unique strong solution ( x t ,&), t 2 0. 2) If (11.10) has two weak solutions ( x i ,?f,we, qi(dt,d z ) ) ,i = 1,2, defined on two different probability spaces (Oi, p,($), P i ) , i = 1,2, respectively, where w f is a Pi-BM, qi(dt,dz) is a Pi- Poisson martingale measure with the same compensator r ( d z ) d t ,i = 1,2, then there exist a probability space

(Q5,( Z t ) ,P ) and four 3t-adapted RCLL processes (iZ;,$f),i= 1,2, with a BM Gt and a Poisson martingale measure @(dt,d z ) having the same compensator r ( d z ) d t as the given one defined on it such that ( ~ i , $ : )= , i 1,2, are adapted to 7jt, and the probability law of (3, J:, 6,R d t , d z ) ) coincides with that of (21,q$, w f , qi(dt, d z ) ) ,i = 1,2. Moreover, (Z:, G t , c(dt,d z ) ) satisfy (12.10) on the same probability space (Q3,( S t ) ,P ) ,i = 1,2.

$1,

Applying the previous two theorems and the uniqueness theorem (Theorem 331) we immediately obtain the following theorem. From now on let us always make the following assumption in this chapter: (H3) r ( d z ) = d z / 1zld+l, (2= Rd - ( 0 ) ) .

Theorem 334 Under the assumption of Theorem 332 and the assumption (H3), i f for t > 0 ,x E g, z E Z 2 ( x - Y ) . (b(t,x ) - b(t, Y ) ) + Ilff(t, x ) - f f ( tY)112 , X , z ) ~ ( Y t , , z)12 4 d z ) 5 ~ N , T ( ~ ) P N , T( Iy12), x + Jz as 14,l ~ 5l N , t E [O, TI, where for each T < m, 0 5 k ~ , ~ ( t )k ,~ g, ~ ( t ) ( t )
11.6 Solutions for RSDE with Jumps and with Discontinuous Coefficients

349

SO+dul pN,T(u)= m, for each N = 1,2, ..., then (11.10) has a pathwise unique strong solution. Now let us relax the bounded condition for coefficients b and a to the less than linear growth condition. Theorem 335 Assume that for t 2 0, x E 8,z E Z lo b(t,x ) , a ( t ,x ) ,c(t,x , z ) are jointly measurable, and there exists a constant ko > 0 such that lb12 lla112 ko(l+ 1x12),JZ lc12.lr(dz)I ko, 2' b(t,x ) , a ( t , x ) are jointly continuous, and as lx - yl + 0, It - sl -+0 JZ l ~ ( t , X z, ) - 4% Y , 412.lr(dz)-,0, 3" assumption (HI)- (H3) hold, 4" 2(x - Y ) . (b(t,x ) - b(t1Y ) ) + 1144 x ) - 4 t , Y)112 Jz b(t,2 , z ) - ~ ( Yt ,,z)12~ ( d zI) ~ N , T ( ~ ) P N , T-(y12), ~ as lxl,Ivl 5 N , t E [O,Tl, t and pN,T(u) is where for each T < m, 0 I k ~ , ~ (ct k) ,~ , ~ ( t )
+

<

+

>

SO+

Proof. Let b N ( t , x )= b(t,x),as 1x1 5 . N ; and bN(t,x)= b(t,NxlIxI), as 1x1 > N ; a N is similarly defined. Then by the previous thoerem there exists a pathwise unique strong solution ( x y ,&), where 42 is continuous, satisfying (11.10) with the coefficientsbN,a N and c. Now set T N = inf { t 0: >N), N x,, = xt , 4t = 42, as 0 t 5 7 - N . By pathwise uniqueness the above definition is well posed, and X ~ =A xo ~ ~ b(s,xS)ds B ( S , xs)dws J~~~~~J~ C ( S , 2,-, z)Gk(ds,dz) 4tATN) for a11 t 2 0. Notice that by the a priori estimate (Theorem 330) for any T < m N 2 P ( ~ N < T ) < E I x T A T2 N
> (XFI

+

+

<

JrN

+ SFTN +

11.6 Solutions for RSDE with Jumps and with Discontinuous Coefficients By means of the Girsanov type theorems we can establish the existence o f solutions for RSDE with jumps and with discontinuous coefficients. One key assumption here is that the diffusion coefficientsa ( t ,x ) cannot be de-

350

11. Stochastic Population Control and Reflecting SDE

generate. So the results obtained in this section cannot refer back t o the RODE (reflecting ordinary differential equation) case. Theorem 336 Assume that ( H I ) - (H3) hold true and for t 2 0 , x -

E

O , ~ E Z 1" b(t,x ) , a ( t ,x ) , c(t,x , z ) are jointly measurable, and there exists a constant ko > 0 such that lb12 + lld2+ Jz lc12 4 d z ) I ko, B a ( t , x ) is jointly continuous, and as lx - yl 4 0,It - sl + 0 Ic(t,x , 4 - 4%Y , z)12 4 d . 4 0, 3" there exists a constant 5 > 0 such that as ( t ,x ) E [0,T ]x (4 A) 1 6 1A12, where A = 2-'aa*. Then (I 1.10) has a weak solution. Furthermore, if the following conditions also hold: 4" 2(x - Y ) . (b(t,x , w ) - Y ,w ) ) + IMt, x , w) - d t ,y,w)112 Ic(t,x,z,w) - c ( t , y , z , w ) I 2 4 d ~I) ~ N ( ~ ) P N ( I x - yI2), N , where 0 I k N ( t ) , k ~ ( t ) d< t co, for each T < co, and as 1x1, Iyl I pN(u) i s concave and strictly increasing i n u 1 0 such that pN(0) = 0 , and So+du/ pN(u) = co, for each N = 1,2, ...; but i f condition 1" is weaken to 1"' lla112 k o ( l + [ X I ' ) , JZ lc12 ~ ( d z<) ko, and ( x ,b(t,2 ) ) cl ( t ) ( l + 1 x 1 ~ gk(x)), where g k ( x ) = 1 ln(1 ln(1 . . . ln(l (xlZn0))),

Jgt

+

w,

+Jot

fl

<

nr=l

<

+v + + + k-times

(no is some natural number), and c l ( t ) 2 0 is non-random such that for cl(t)dt < co; furthermore, b(t,x ) is locally bounded for x , each T < co, that is, for each r > 0 , as 1x1 5 r , lb(t,x)l I kr, where k , > 0 is a constant depending only on r ; then (11.10) has a pathwise unique strong solution. Proof. Case 1. Assume that lbl + 1 .1 ~ Jz 1c124 d z > I %, and assume that conditions 2" - 3" hold. By Theorem 332 there exists a weak solution ( x t ,4,) defined o n some probability space ( R , z ) (without loss o f any generality we may assume that (R,$) is a standard measurable space) satisfying the following RSDE xs)dws Jz c(s,xs-, z ) G d d s , d z ) + d, - a.s. xt = xo and the other statements in (11.3) hold for (xt ,A), Let dFt = ztdP, where t t zt = exp[Jo0, . dws los)2 ds],Ot = a-l ( s ,x,)b(t, x,).

+

{

+ Ji

+ 5:

So

11.6 Solutions for RSDE with Jumps and with Discontinuous Coefficients

351

defined on Then by Theorem 124 there exists a probability measure (Q,5)such that P IzT = &,for each 0 < T < co;moreover, t 1) w; = w t - J o e s d s , O ~ t , is a-BM under probability P; 2) Nk(dt,dz) = Nk(dt,dz) - a(dz)dt is still a Poisson random martingale measure with the same compensator n(dz)dt under the probability P. Therefore we find that (xi,4,) satisfies the following RSDE xt = xo b(s,xs)ds u(s,xs)dwi & C ( S , X S - , z)Kk.(ds,dz) +&, and the other statements in (11.3) hold for (xt,&). P - a.s. Now assume that conditions 4" and 5" also hold. Then by the same reasoning as in Theorem 334 we find that (11.10) has a pathwise unique strong solution. Case 2. (General case). In the case that lo' is satisfied, let bN(t,x ) = b(t,x ) , as 1x1 < N ; and bN(t,x ) = b(t,N x l lxl), as 1x1 > N ; u N is similarly defined. Then by the case 1 there exists a pathwise unique where C@ is continuous, satisfying (11.10) with strong solution (x:, the coefficientsbN,u N and c. Set now T N = inf { t 2 0 : Ix,NI > N ) , xt =x?, 4t as 0 2 t < 7-N. By the pathwise uniqueness the above definition is well posed, and ~ATN b(s,xS)ds JotArN U ( S , xs)dwS X ~ A T N = 20 + JO JZ C ( S , xs-, z)fik(ds,dz) &.,rN, for a11 t 2 0. Applying Ito's formula to g(xt) = g,+l(xt) we have that as 0 5 t T ~ATNI 0 ~ ( x ~ A T N=) ~ ( x o+)SO g ( x s ) b ( ~ , x s ) d ~ gl(xs)u(s,xs)dws JZ gl(xs-)c(s,xs- z ) ~ ( kd ~d , ~ ) Jz[g(xS- +c(s,xs-, z ) )- - ~ ( x s - -gl(xs-)c(s, ) X S - , z)]Nk(ds,dz) +2 JotATNgl(xs)- d$, = Ti, IjTN - a.s. where 2nox; x ' % I sm+~ ( % ) = l - I L 1 9 i 1(x)+. . . a gl(x) = grad g(x) = (=g(x), . . . , and we write go(x)= 1. Notice that ~ATNI so s ( 2 s ) .d4s 2no 2 s = SotATN(l-ILig i l ( 4 s ) )-&&$pT(xi(s)- xi(O))d4i( s )

{

+ Sot

+ Sot

+ Sot

&),

=@rl

+ +

+ JiATN

< + sotATN + &,

EL + CL of^^ (n;=,

<

+ JotArN '&

&w,

2no x s

gil(~(~))-&&$pT-~i(o)d~i~s)

= 161 + 1 6 2 < 162. So as t < T , E JotATN g1(xS). d4, I E I I ~ I C~~ . Now discussing the other terms li,as in the proof of Theorem 133, one finds that for any T < oo

352

11. Stochastic Population Control and Reflecting SDE

s(N)P(~N < T)I E ( ~ ( x T A ~ N ) I ~ NI
11.7 Solutions to Population SDE and Their Properties Recall that the population SDE can be given as follows (See (11.4) in the section "Notation"):

+

+

+ J'

dxt = (A(t)xt B(t)xt&)dt a ( t ,xt)dwt ~ ( xt-, t , z)fi:rc(dt,dz) -4 +dq5,, t 2 0, xo = x; xt E R+, for all t 1 0, where R$ = { x = ( x l , . . . , x d )E R ~ : >x0 ,~1 I i s d ) , 0, is a Rd - valued Bt - adapted continuous process with finite variation on any finite interval such that q50 = 0, and

Idt = la'I ~ R :(X.P 101, , mt

n ( t ) EN,,, as xt E d ~ $ .

=

~ o nt

wI

.

~I,

(11.22) Now we easily see that i f &-the specific fertility rate of females is bounded, and coefficientsb and a satisfy the conditions in Theorem 335, then (11.22) has a unique,solution. That is, i f in (11.22) we denote by x: the size of the population with ages between [i,i + I ) , and write xt = (22,. . .,x!), where d is the largest age of the people, then for any given &-the specific fertility rate of females, we can find the population size vector xt with xf 2 0,tli = 1,2, . . . ,d. More precisely, we have the following theorem. -4

Theorem 337 Assume that for t 2 0 , x t: R+, z E Z 1" A(t) and B ( t ) are bounded and non-random, Pt(x), a ( t ,x ) , and c(t,x , z ) are also non-random and jointly measurable, and there exists a constant ko > 0 such that l ~ t ( x ) l Jz ~ lc12 4 d . 4 I b,lla112 5 ko(l+ 1 x 1 ~ ) ; P Pt ( x ) and a(t,x ) are jointly continuous, and as Ix - yl -+ 0, It - sl -+ 0 J' 144 2 , ~-) 4%Y , 4124 d z ) 0; --d a ' 3 x + c ( t , x , z ) E R+,Vt 0 , x E R+,z E 2; 4" ISt(x) - &(y)12+ Il4t.x) - 4t,y)I12 + Jz Idt,X , z ) - c(t,Y , z)l 4 d z ) I ~ N , T ( ~ ) P N (-~y12), x as 1x1 I IYI N , t E IO,TI,

+

+

>

<

11.7 Solutions to Population SDE and Their Properties

353

where 0 5 k ~ , ~ (istnon-random, ) k ~ , ~ ( t )
where ko 2 0 is a constant, c t ) ,i = 1, . . . ,d, and Eo are constant Rd-vectors, (i.e. each component in a vector is a constant), c r ) , i = 1,. . . ,d, and El are constant d x d matrices, (i.e. each element in a matrix is a constant) and, in addition, E ~ ) , E 2~ 0,Vl , ~ ~ 5 i,j 5 d, where E l = ( ~ , j ~ ) &and = ~f,( z ) > 0 is such that JRd-(,) s d z < m. Then all conditions of Theorem 337 are satisfied, and hence the population SDE has a unique solution. Now let us discuss the properties of solutions to the population SDE. 1. The Convergence Property. Let us consider a sequence of solutions ( x r ,4:) to the population SDEs: dxT = (An(t)xT+ Bn(t)xT/3y)dt+ un(t,xT)dwt Jz cn(t,xr-, z)fik(dt,dz) d4:, -4 x$ = x g E R+,t 2 0, xt E $, for all t 2 0, and (x?, 4;) satisfies the other statements as in (11.22). n = 0 , 1 , 2 ,..... W e have the following convergence theorem.

I

+

+

Theorem 338 Assume that lo there exists a constant ko 2 0, for all n = 0,1,2, ... IAn(t)l IBn(t)l + PZ" ko, where PY 2 0 are non-random such that they do not depend on x , and I I ~ ~ ( ~+, Jz x )Icn(t,x, I I ~ 412.rr(dz)I ko(l+ 1x1~1, P IlaO(t,x)- U O ( t jy)1I2 + Jz IcO(t,x,4 - cO(t,y,4I2 4 d . 4 5 k(t)p(lx- yl2L where k ( t ) > 0, p(u) is wntinuous, strictly increasing, concave, p(0) = 0, and they are non-random such that for any 0 < T < m k(t)dt < m, So+ dul p(u) = 0; ? ( a ) li-,, - ~ O ( t )+ l - ~O(t)l]= d t0,

+

<

g

T

li%-+, Jo

C [ I A ~ ( ~ ) IB"(~)

SUP^,^ [lon(%x ) - oO(s,x)jl2

11. Stochastic Population Control and Reflecting SDE

354

Proof. By Ito's formula 2 Ix; - XE) = Ixg - 28 + 2 Sgt(x: - x:) . (bn(s, x:) - bO(s,x:))ds 2 Ilan(s,x:) - aO(s,x:)II ds+2~~(x~-x~)~(a~(s,x~)-a~(s,x~))dw, +2 J;(x: - x:) . J,(cn(s, 2:-, t ) - cO(s,x:-, ~))@k(ds, dz) 2 +2 J;(x;-x:).d(4:-4:)+J; Jz Icn(s, a&,2) - cO(s,x:-, 211 Nk(ds, dz) = C;=I I;@), where we have written bn(t, x;) = An(t)x; Bn(t)xFp:, n = 0,1,2, . . - . Notice that I?(t) = 2 JOt(x: - x:) . d(4: - 49) = 2 J;(x: - x:) . d4: +2 - x:) - d& I 0, and by the condition 2O that E[Ig(t) IF(t)] I 2 k(s)p(E 12: - ~ : 1 ~ ) d s Jp(t), where t J?(t) = 2 6 [supxEE; \Iun(s,x) - uO(s,x)l12

+sot

1'

1

+

+

+

2 + suprneRd Jz Icn(s, x, Z) - cO(s,x, Z)I r(dz)]ds +

-+ 0, as n -+ m. Moreover, by Theorem 330 k ~for , all n = 011,2, .-. E(suPo
<

$

+ EG

1%:

1%: Ix:~ 1%: +koEJ,t p: - ~ 9 1 Ix: d ~ . Notice that by the condition 3O, as n -+ oa EJ,~ IA: - A:J Ix: Ix: - x:] ds

4

I S,t /A: - ds[3EsuPs
1%:

11.7 Solutions to Population SDE and Their Properties

355

So, W E [O,T], 2 limn,,E 1x7 - xI! = 0. Making similar observation, one also finds that 0 2 limn,, supsst E lx: - xsl = 0. Therefore by the martingale inequality and by Ito's formula above we find that lim,,,~ supso - 2 I k' [Jt k(r)p(Kn+& sups5, 1x7 - x:l 2 )dr

Ix:

1%:

+ J, hmn,,E t-

sups5, lx: - x:

1

O2 O

dr].

Hence W E .[0,T]. 2 limn,,E supslt 1x2 - x:l = 0. so, W 0, limn,, E supsst - x,0 2 = 0. Furthermore, observe that

>

1%:

1

+: .1

14; - dy12 r ~ ( I x ;-

+ lJi(bn(s, 5:)

- bO(s,x:))dsI

- x8I2

2

- [G

+ Ifi

+ l~,'(a"(s,x:)

2

- uo(s,x~))dwsl

J~(C~(~,X:-,z) - co(s,x:-, z))fik(dSI dz)12). We also notice that as n oo IA2 - A:/ lx!1 dsI2 5 1x:I2 I k t [ ~ i - A:[ ds12 -+0, etc. Hence by the martingale inequality, as in the above, one also easily finds that as n -+ w, E S U P . ~ ~14: 0Theorem 338 actually tells us the following facts: 1) For a given stochastic population dynamics and a given specific fertility rate of females, if an initial population size vector xg "in closes to" a known initial population size vector x; "in the mean square", then the population size vector x r corresponding to x; will "uniformly close to" the population size vector x! corresponding to xo on any given interval [0,TI "in the mean square"; that is, for a given A(t), B(t), u(t, x), c(t, x, z) and

E[X

IA:

41'

Pt ,

2

-

Ix; I!x

E 12: - 281 + 0 ==+ E S U P , , ~ -+ 0. 2) For given stochastic popula%on dynamics, and a given initial population size vector, if a specific fertility rate of females Py "is close to" a known specific fertility rate of females @ Itin L1([O, TI)", then the population size vector xa corresponding to /3? will be "uniformly close to" the population size vector x! corresponding to @ on any given interval [0, T] "in the mean square"; that is, for a given A(t), B(t), a ( t , x), c(t, x, z) and xo, 2 C I p n ( t ) - p o ( t ) l d t - + O =+ E s u P , ~ ~ ~ x ; -'0.x ~ ) ~ 3) If the population dynamics can only be "approximately calculated in some sense", then for a given initial population size vector, and a given specific fertility rate of females, the corresponding population size vector will 2

356

11. Stochastic Population Control and Reflecting SDE

also be "obtained approximatey uniformly on [O,T]in the mean squarett; that is, for a given xo and p,, - AO(t) P ( t )- BO (t)l]dt

r[I~"(t)1 + I + I.T Cupx,z

jlon(s, x ) - oO(s,x )

112

+ E s u P , ~~X~Y - X ! ~2 4 0 . Furthermore, we have the following approximate error estimate calculation: Corollary 339 Assume that all conditions in Theorem 338 hold, and V n = 0,1,2,... u y t , x ) = u ( t ,x ) , c n ( t , x , z )= c ( t , x , z ) , f ( z ) =I&), r ( U ) = 1, where a ( t ,x ) and c(t,x , z ) are defined by (11.E?), together with the properties there.

where

+

+

h = ~ f &lcf)12 l 6 l 2 ( k ; 1). and the constant ko comes from the condition l o in Theorem 338, kT comes from Theorem 330. Actually, here we can get that 2 E ~ u Plx:l, ~ 5 kT,

<

. that $ ) , c f ) , 1 5 i d and we assume that Eixgli I & , ~ n (Recall wme from a ( t ,x ) , and &,Z1 come from c(t,x , z). See (11.23). Moreover, c t ) ,G, 1 L i I d are Rd-vectors, and c f ) ,E l , 1 I i 5 d are RdBd-matrices). Proof. By Ito's formula, as in the proof o f Theorem 338, one finds that 2 1x11 - xy = Ixg - x; 2 - x:) . (bn(s,x:) - bO(s,x:))ds 2 Ilun(s,x:) - U . ~ ( S , X : ) ~d~ ~ + 2 ~ ( x ~ - x ~ ) ~ ( u ~ ( s , x ~ ) - u ~ ( s , x ~ ) ) d w ~ +2J:Jz(xy - 2 : ) . (cn(s,x:-,z) - cO(s,x:-,z))Gk(ds,dz)

1 +

l2

+S,"JZ

Icn(s,x:-,z) - C ~ ( S , X ~ , Z )~I k~( d s , d z ) +2 Jot(x; - 2:) . d(4; - 4:) = c:=, I?(t) I c!=~ I?(t), where bn(t,xy ) = An(t)xF Bn(t)xFP;, n = 0,1,2, . . . . Hence 2 2 E I ? ( ~5 ) x - x ds,

+

E X xt1Icf)l

E I ; ( ~ )I E

Sot le112(k; + 1) (x: - x ~ I ds, 2

11.7 Solutions to Population SDE and Their Properties

x;12 + 4 J: E Ix> 1:.-2 - = EL,Icfj"12+ 1sl2 (4+ 1).

E (x: - x where

~

~E 2 Is$ -

ds

357

+ EI;(~),

Cl

However, by the proof in Theorem 338 one sees that as t 5 T 2 EIF(t) 5 2(ko k i ) ~ :E 12: -a![ ds

+

+ IB~

ko - BtI +ko +5ktS,t[lA: -A:/ 2 5 2 ( k o + k ~ ) ~ i E 1 x : - x ! 1 ds

-&/]d~

+

+

-A![ ko + 5 k ~J[:~A: - BtI ko IS: - p:l]d~. Notice that from Theorem 330 one finds that E supsgT Ix:l 2 5 kT, V n = 0,1,2,. . . . Hence, by the above Ito formula 2 2 ESU~,~, 5 E 12: E IX:: - X:I ds

1%:

xI:

$;I2 +z1

+EIF(t) + 6 E ~ u p-, <12 ~: - x:l (J:

2

~ f 1cf)I = Ix:~ - x:12ds)l12,

pp69 where we have used the following If M E M2>loc (i.e. Mt is a locally square integrable martingale) with Mo = 0 , then for any T < w a s p € (0,2)

358

11. Stochastic Population Control and Reflecting SDE

<

Since E 1 x 8 1 ~ ko and E

:,1

- x812 -+ 0, we may estimate (assume) that

Now by the above Ito formula, and by applying the martingale inequality, one easily finds that 2

E supslt lx:12

< E lx:12 + 2(k0 + k: + EL,1cf)l ) J,' E 1x:12

ds

In the previous cbnveigence results we have assumed that pt -the specific fertility rate of females, does not depend on xt-the population size vector. However, when we consider some population control problems and use Pt as a control, then in some cases we need to use pt as a feedback control, that is, pt = &(xt) must depend on the instant xt. Because, from the state (at) situation at the presemt time, we want to adjust it instantly by using a control (P,). In such a case we must use could we obtain a convergence theorem for x r ? The following theorem answers this question. Theorem 340 Under assumptions lo,3"(a), and 4" of Theorem 338, but now with the assumption 2' is weakened to IIgO(t,x)- ao(t,~)112 + J, IcO(t,x,z) - cO(t,y,2)l2 4 d z ) 5 ~ N , T ( ~ ) P N (-I xy12), as t E lO,Tl, 1x1, IYI 5 N , where pN(u) is C O ~ ~ ~ ~ U Ostrictly U S , increasing, concave, pN(0) = 0, and k ~ , ~ (2t 0, ) and they are non-random such that k ~ , ~ ( t )
11.7 Solutions to Population SDE and Their Properties

359

3" (b)' pn(t, x ) , n = 0,1,2, .- . all depend on x in such a way that T X ) dt = 0, li%-+, suPx,~; lpn(t,X ) ( ~Y Ix) , as t E [O,Tl, 1x1, IYI I N , IpO(t,x)- pO(t,y)I 5 ~ N , T ( ~ ) P N where k N ,( t~) and pN (u) have the same properties as that in 2"'; then for any E > 0 and t 2 0 P(supslt Ix: - x!l > E ) -+ 0, and P(sups5t (4: - 4: > E ) 0.

1

L

I

--+

Notice that limn,, P(supsst 12: - x!1 > E ) = 0 is equivalent to E ) = 1. limn-.+, P(supslt 12: - x: So, roughly speaking, the physical meaning of the conclusion is that with a very good chance (the possibility P is close to loo%),x: can be sufficiently and uniformly close to x, on [0,t ] .

I<

Remark 341 1) If we take pN(u) = U , k ~ , ~ = ( tkg, ) where ko 2 0 is a constant, then the condition 2"' actually means that ao(t,x ) satisfies a local Lipschitz condition in x , and cO(t,x,z ) also satisfies a local Lipschitz-type condition in x. 2) ~f we let pn(t,x) = pO(t,x)= xI1n1 E ) -+ 0 is weaker 2 than the convergence in mean square: EsupSct - Ix; - x!l -+ 0. In fact,

+

p(supSlt

Ix:

- x:1 > E ) I $EsupSlt lx:

- 281

.

Proof. Notice that now bn(t,xF) = An(t)xF B n ( t ) x ~ ~ ; ( x ~ ) . Set rN = inf { t : 1xf1 > N } , rnlE= inf t : x? -xfI > E } , r F ( n ) = T A rn*€. Then by Ito's formula, as in the proof of Theorem 338, we have

+

II

1

2

i nt x A However, for all t 2 0

1

n-x

A

n = 0.

2

E IX;,,$'(~) -X;~,:(~) 2 ~ ~ p (5 ir N~A 't).~ Hence limn,, P(rn>€5 rN A t ) = 0. W e have that P(supS5,NAt Ix: - x:l > 2E) < p(rn" I rN ~ t-+ )0, as n -+ m. Notice that 12T : - x:l > €1 P(~uP,<,NAT >E) P(SUP~< +P(suP,NAT<. E ) I P(SUP~<,NAT 12: - 4 1 > €1 + p ( r N AT < T ) . Since

1%:

1%:

360

11. Stochastic Population Control and Reflecting SDE

limN.-ta, rN= co. Hence limN+ooP ( T /\~ T < T ) = 0. Therefore the first conclusion is established. Now, by (1.1) P ( S U P ~14 <: ~> 4 ~ I) P(sups5t lx; - 41 > e) +P(suprSt bn(s, x;)ds - J; bO(s,x:)dsl > E) +P(supr5t IS; on(s, xF)dws - S : go(., z:)dwsI > E) +P(su~,,t I .f,' Jz F ( %x&, z)Gk (d%dz) - J,' Jz cO(s,x:-, z)fik(ds, dz)( > e) = Ii. However, it is already known that as n 4 co Il -+ 0. Let us show that as n + co I4-t 0. In fact, for any > 0, by the martingale inequality, I 4 I P ( s u P O + ~Ix; ~ - x q > $1

1s:

41

zE1

2

+ 4 e - 2 ~ ( s k(s)p(sup,5* , Ix: - 4 1 )ds)Is,,<s
+ supzEq Jz Icn(s,x, z) - cO(s,x, z) 1 2 s(dz)]ds = 3 I;. Obviously, by the result just obtained, and by the assumption as n -+ 1 ; + IZ3 0.

co,

-+

However, for is small enough, IF2 I kte-2p($) k(s)ds < E. So, it is seen that as n --+ co I44 0. The proof that I3+ 0,as n -+ co is obtained in a similar fashion. Let us show that as n --+ co, I2--, 0. In fact, notice that pn(t)x;PZl(xZ") - BO(~)X:P:(X~ = IBn(t)xZ"PZ"(xZ") r BO(t)x?PZ"(xZ") r~O(t)x?&(xZ")r B0(t)x?P!(x:) - ~O(t)x:P!(x?)l i ko p n ( t ) - BO(t)llx?l+ ko IPZ"(x?) - P%?)I k: [x? - x:/ , +ko 1xq IP%z") - P!(x!)l and E sup,lT Ix;I 2 I k ~Vn , = 0,1,2, - .. . The last inequality implies that 'dE > 0 as N 4 co, su~np(su~ 1x:1~ 2 ~>T N ) ~ T / N 0. So, in similar fashion, one can similarly prove that VE > 0, as n -+ co, P ( I B ~ ( ~ ) x ~ P : ( x, " BO(~)X:P;(X:)~ > > E) -+ 0. From this, one easily finds that as n co, Iz-,0. Therefore, we arrive at the second conclusion. rn 2. The Stability Property Now we are going to discuss the stability of solutions to (11.22). We have the following stability property for the solutions of the population dynamical system.

Jl

1

+

<

-+

11.7 Solutions to Population SDE and Their Properties

361

Theorem 342 Assume that coeflcients b, a and c in (11.221, where b(t.x) = A(t)x B(t)x@,,satisfy all conditions in Theorem 337, and that a@,0 ) = 0, c(t,0 , z ) = 0. Then 1) (0,O) is a pathwise unique strong solution of (11.22) with initial condition xo = 0. 2) For any given xo E R$ (11.22) has a pathwise unique strong solution (xt,4t),t 2 0. 3) Furthermore, if there exists a positive constant kl 2: 0 such that 2%. b(t,x ) Ila(t,x)112+ SZ 144 x , 2))' 4 d z ) I -kl 1x1' , then 2 E Ixtl 5 E lx01~ e-klt, for all t 2: 0. and there exists a constant I c ~ such that E 14,12 I k4E 1 x 0 1(1~ e-"lt), for all t 2 0. Furthermore, write 4, = lim,, 4,, then ~ ( 1 4 -~1 14 ~~1I ~k)" ' ~lx01~e-klt. 4) In the case that there exists a positive constant kg > 0 such that 2x . b(t,2 ) lla(t,x)112+ SZ 1 4 4x , 2)l2.rr(dz)2 1x1' , and lb(t,412+ Ila(t,4112+ .IZ Ic@,2, 4124 d z ) I ko(l+ 1x12), then E 1q122 E 1 ~ 0 ekst, 1 ~ for a11 t 2 0. Hence lina,,E lxtl 2 = m.

+

+

+

+

This theorem tells us that under appropriate conditions the solutions of the population dynamical system are exponentially stable. Roughly speaking, this means that, if 0 is a solution of the dynamical system with the initial condition 0, then under appropriate conditions any solution xt even if it has a initial condition xo # 0, will "be close to" the solution 0 as time tends to infinity. Moreover, the approximation speed is a negative exponencial. (See the conclusion 3)). Proof. 1) and 2) are true by Theorem 337. Let us show 3): By Ito's formula lxt12 = 1x01~ t ( 2 x s - b(s,2,) Ila(s, ~,)11~)ds 2 x, . ~ ( sx,)dw, , , Sgt & Ic(s,xs-7 z12 Nk(ds, dz), +2Sgt ,fZ 2 s . c(s,xs-, z ) f i k ( d ~dz) where we have used the fact that 2 ~ ; x ~ . d q=5O .~ Set T N = inf { t 2 0 : lxlt > N ) . Then E 1xtArN12= E 1x01~ E ~ y ~ ( . b(s, 2 x5,)~ I I ~ ( SX , ) I [ ~ Sz x s , ~ ( d z ) ) d s . Hence 2 % E IQATNI = E(2xt~rN. b(t A T N , xt~?-N) I l ~ (At T N , ~ t ~ ~ ~ ) + Sz Ic(t A 7 ~ X ~ 1A T N ,z12 ~ ( d z )I) - h E 1xtATN1 2 , a.e. t 2 0.

+

+

+

I.(.,

+

+ S:

+

+

+

1 1 ~

11. Stochastic Population Control and Reflecting SDE

362

Therefore 2 E lxth,, 1 5 E 1x012e-klt, for all t 2 0. Letting N 1oo, by Fatouls lemma one finds that E lxtl 2 5 E lxo12e-klt, for all t 2 0. On the other hand, by Ito's formula T T 1 4 ~-114t12=2Jt ~ d,-&, = 2 J t 4,.(dxS - b ( ~ , x ~ ) d ~ - a ( ~ , x ~ ) d w , - JZ C ( S , xs-, z)fik(ds,dz)) = 2(4T ' X T - 4, ' xt) T -2Jt 4, . (b(s,xs)ds+ a ( s ,xS)dws+Jz c(s,xS--,z ) f i k ( d ~d,~ ) ) , where we have again used the result that JtT x, . d4, = 0. Hence, by assumption we get that Ib(t,x)l (ko k:) 1x1 , and E 1 ~ d~ ~ E 11xTl2) ~ ~ ( 1 4 ~-114~1') ' I 1xt12+ J~~ k'E 1 x ~(e-klt 1~ e-klT). is Letting t = 0, the second conclusion of 3) then follows. Now, since increasing in t for each i, (see Remark 321), 4, exists, and ~ ( l $ ~ lq5t12) 1 ~ 5 k f f f Elxo12e-"lt. 3) is proved. Finally, let us discuss the unstable case 4). In the above discussion, and by assumption, we have xd E lxt12 = E(2xt+(t1xt>+lla(t,xt)l12+Jz Ic(t,xt, z12 s(dz))ds1 ksE lxt12. Hence ekst, for all t 2 0. rn E lxt12 2 E 1x01~

<

+

+

+

<

Corollary 343 Let a(t,x ) and c(t,x , z ) be defined by (11.23) with Eo = 0 C(i)= 0 , l 5 i 5 d, and with the properties required there. Moreover, let , o f ( z ) = I v ( z ) ,s ( U ) = 1. Suppose that A(t) and B ( t ) are as given in the ftIntroductiontfto this Chapter, and that 0 5 P 5 Po. Let =

mint>~(i +

'$ + r ) d ( t ) ) 1

r ~ l ( ~ ) , r ] Z ( ~ ) ', '' , ~ d - l ( ~ ) ,

b~ = matro(brl (t),, . . ,b,, ( t ) ) , where d = r,-the largest age in the population. Then as 2 61 = 260 lcj')l - ~1~ - p 0 b rna(r2 ~ - r l , 1) > 0,

xtl

I

t

E lxt12 5 E 1 ~ 0 e-*lt, 1 ~ for all t 1 0. That is, the population dynamics of (11.22) is exponentially stable in the mean square under the above assumptions. In particular, for a given constant a > 0 we have that t2& l n v ~ l x 5~a. l ~ This corollary actually tells us that if the stochastic perturbation is not

xf=, + 2

too large (that is, lcj')l [ E l l 2 is very small), and the forward death rate is greater than a positive constant (that is, 60 > 0 ) ,then the population dynamics (11.22) can be exponentially stable "in the mean square" i f the fertility rate of females is small enough. Furthermore, if we have a target

11.8 Comparison of Solutions and Stochastic Population Control

363

a > 0, we can find out when the population size vector "in the mean square" can be less than this target a. Proof. By calculation 2 2 . b(t, X) = -2 ~ : = ~ ( 1qi)x: + 2 C:L: xixi+l+ 2(C& bixix1)Ptr

+

2

II~(~,X =) P

where 60 = min(4 d

1c1"1 1 x 1 ~

. .

+ ql, q2,. . . ,~ d - 4~ +, qd). Thus as (i)

I

61 = 260 - Xi=1 lcl - 1E1l2 - & b max(r2 ~ - r ~1), > 0, the population dynamics (11.22) is exponentially stable in the mean square, i.e. by 3) of Theorem 342 E lxt12 E 1xol2e-61t, for all t 1 0. Finally, noticing that E lxo12e-6lt 5 a u t 2 b ; l n w . We arrive at the final conclusion.

<

11.8 Comparison of Solutions and Stochastic Population Control 1. The Tanaka type Formula and Comparison Theorems In this section we will discuss the comparison of solutions to the stochastic population dynamics equations and optimal stochastic population control. However, to show a comparison theorem for the solutions of a d-dimensional RSDE we need the help of a Tanaka type formula for such a d-dimensional RSDE. So we first present the following Tanaka type formula. Consider two d-dimensional RSDEs, which is a little more general than (11.4): i = 1,2,

11. Stochastic Population Control and Reflecting SDE

364

+

+

dxf = bi(t,xf)dt a ( t ,xf)dwt Jz c(t,xi-, z)fik(dt,dz) + d&, -4 x i = x i E R+, t 2 0, x: t $, t 2 0, and the other stantements in (11.4) also holds for ( x i ,4:). (11.24) Theorem 344 (Tanaka type formula). Assume that

lo bl, b2,a and J, lc12n(dz) are locally bounded, i.e. for each r = 1,2, ... Ih(t,x)l < k,., as 1x1 < r, h = b1,b2,a and J , l ~ 1 ~ n ( d z ) ; where 0 < k, is a constant depending on r only; 2' a ( t ,x ) = (CTik(t, x ) d) k-l ~ satisfies the condition that i 2 Iuik(t,x) - ~ i k ( t , ~ ) 0,as u > 0; pN,T(0) = 0; and pN,T(u) is strictly increasing in u and such that So+~ u / P ~ , =~ co, ( u for ) N = 1,2, .... and T < CXJ; 3' c(t,x , z ) = (q(t,x , z ) d) ~ satisfies ,~ conditions that -4 -4 x + c ( t , x , z ) E R+, Vt 2 0 , x E R+,z E 2; X i LYi X i + ~ ( t , x , z )L y i + c i ( t , y , z ) , --d Vt 2 0, x , y E R+, z E 2, where x = ( X I , . . . , x ~ ) , Y = (vl,... ,yd). If ( x i , 4:) satisfies (11.24),j = 1,2, then Vi = 1,2, ...,d, ( x i @ )- x?(t))+= ( x f(0)- xf(O))++ J: I(xt(s-),x:(s-)d(xt ( s )- +)). Furthermore, in detail, Vi = 1,2, ..., d, ( x i( t )- x? (t))+= (xf (0) - x?(0))+ I(xt(s)>zl(s)(br(s, ~ ' ( -4 b?(s,x2(s)))ds ) + I(x:(S)>x:(S) ~ i =( al i k ( ~x1, ( s ) )- a i k ( ~x2(s)))dwk(s) , + Jz I(,~(,-)>~~~-)(G(~,X~(~-), z ) - ci(91x2(s-), z))Nk(&dz) + 1 ( ~ : ( ~ ) > ~ ~ ( ~ ) )-d 4%)). (4,l(s)

fl

*

+So"

Jl So"

Remark 345 An example in which the above assumptions 20 and 30 are satisfied is ~ j k ( t X, ) = C; + cikxi, ci(t,x , z ) = 'Ei0 ++xi, where all c ~ , c i k , % , q , i , k= 1,2,... ,d are constants; and 1 , 2 , - . . ,d. Now let us establish the theorem.

i$,% 2 O,i =

11.8 Comparison of Solutions and Stochastic Population Control

365

Proof. By Theorem 153 one finds that the first and second formulas are true. However, by the definition of the solution one finds that qzt(.5)>$ (,))d4%) = 0. Because d4t(s) > 0 only when x:(s) = 0, and now x:(s) 2 0. Hence the last formula follows. Now let us derive some comparison theorems for solutions to the stochastic population dynamics (11.22). For conveneince let us write out it again as follows:

+

+

dxt = (A(t)xt B(t)xtPt)dt a(t, xt)dwt +d& xo = 2, t 2 0, for all t 2 0, xt E and the other statements hold for (xt, 4t),

+ Sz ~ ( txt-, , z)Ek(dt, dz)

Tid+,

(11.25) where x: is the size of the population with age between [i,i l ) , and xt = (22, ...,2:); while P, is the specific fertility rate of females,

+

bi(t) = (1 - po0(t))ki(t)hi(t)# 0, i = rl,. . -,r2; 1 < rl < r2 < d, poo(t) is the death rate of babies, qi(t) is the forward death rate by ages, ki(t) and hi(t) are the corresponding sex rate and fertility models, respectively, xi is the size of the population with ages between [i,i I), xt = (xt,. . .,x2), d = r, is the largest age of people. Because of the physical meaning, we may assume that all qi(t), bj (t), pt, i = 1,2, . . . ,d; j = rl, ...,rz are nonnegative and bounded. To consider the comparison of solution to (11.25) first we write out all its component equations as follows:

+

dx:

+ q:)x: + C;LIIb$$3t]dt + x:=Id k ( t , x t ) d w t + Sz cl(tlSt-, z)Nk(dt,dz) + d4:, dx: = [-(I + qi)xf + xE-']dt + ai"t, xt)dwf + Jz ci(t, xt-, z)@k(dt,dz) + d4:, i = 2,. . - ,d.

= [-(I

(11.28)

Now assume that conditions 2" and 3" in Theorem 344 hold, and all coefficients qf,b:,& in the above population dynamics are bounded, non-

366

11. Stochastic Population Control and Reflecting SDE

) less than linear growth negative and that la(t,x )'1 ,JZ [ ~ (x t, 2, )l2 ~ ( d zhave in 1x12 ; that is, 0 rll bi + & 5 ko, Vi = 1,2,... ,d;j = r1,e.a ,r2; 14,x>12 JZ 1 4 4x , 4124d-4 I ko(1 + 1x12), where kg 2 0 is a constant. We also assume that there is another solution (zt,3,)satisfying the same stochastic population dynamics (11.25) but with another initial value Eo and another fertility rate for females p,. Then by the Tanaka type formula (Theorem 344) one finds that (x: - ?$)+ = ( x i - zk)+ S," Ix:>q [-(I T&(xf - zt)

< +

+

+

+ E L ,bt(x:P, - e K ) ] d s+ M: - J,' 5 ( x i - 3$)+ + & ( I +

+

~,:>,:d$:

&(x: - $)+

+ EZ,, bt((x: - z ) ++~$.(p,, - p,))]ds + M:, where M: is a martingale. Similarly, (x: - $)+ 5 ( x i - $)+ + J,'[-(l + Q ~ ) ( X $- 4 x s )+ +

+(xi-' - ?i$')+]ds M;, i = 2,. - . ,d, where M i , i = 2,. - . ,d are martingales. Furthermore, now assume that < -0,P V i = l , . . . ,d; 20 Pt I Pt,Vt 0. Then 0 5 yt = E E;=,(X: - ?$)+ I JOt[k~ysy,]ds. Hence yt = 0,Vt 2 0. This implies that P - a.s. xf 5 3 , Vt 2 0 , V i = I , . . . ,d. Thus we arrive at the following theorem.

>

+

Theorem 346 Assume that lo 0 5 ~ f , g , / 3 I, k o , V i = 1 , 2 , . . . , d ; j = r l , . . . ,r2; 14, x)12 JZ I&, x , 4124 d 4 I ko(l + 1 x 1 ~ ) ; 2' the same as 2' in Theorem 344; 30 the same as 30 in Theorem 344. If (xt,4,) and ( F ~&,) are solutions of the stochastic population dynamics (11.25) with the initial value xo, the fertility rate of females Pt and Zo,Bt, respectively; then xb 2 3 , V i = I , . . . ,d; and Pt 2 Pt,W 0 implies that P - a s . xf 2 3 , Vt 2 0 , V i = I , . . - ,d.

+

>

The comparison theorem (Theorem 346) actually tells us the following facts: 1) In a stochastic population dynamic if the initial size of the population takes a larger value, then the size of the population will also take larger values forever, as time evolves; that is, with the same rlf, >~0 , V i = l , . . . ,d a xf > ~ : , V t > ~ , V i = l , . . . ,.d xo -

11.8 Comparison of Solutions and Stochastic Population Control

367

2) In a stochastic population dynamic i f the fertility rate o f females always takes larger values, then the size o f the population will, also take larger values forever, as time evolves; that is, with the same 4,q,x i = xi > $ , V t 2 0 , V i = I , . . . ,d. Pt 2 p t , V t 2 0 Furthermore, the proof o f Theorem 346 also motivates the following more general theorem. Consider two more general RSDEs as (11.24). More precisely, consider the following two RSDEs: i = 1,2,

g,

+

+

dxp) = b(i)(t,xp))dt a ( t ,xp))dwt S, c(t,x ji),z ) E k ( d t ,d z ) + d4p), 4 2:) = x ( ~E) R+, t 2 0 , Xj') € t 2 0, and the other stanternents i n (11.4) also holds for (xii),q5p)). (11.29) W e have the following comparison theorem.

z;,

Theorem 347 Assume that all conditions lo- 3' in Theorem 344 hold, and assume that one of the b(i)(t,x ) , i = 1,2; say, b(')(t,x ) satisfies the following condition: 4" ~ ~ i (b('li(t, > ~ iX ) - b(')i(t,y ) ) 5 c:=~ - yk)+), as t E [O,T];1x1,lyl < N , for eachT < m, a n d N = 1,2,.-- , 4 V i = 1,2,... ,d; V x = ( X I , . . . , x d ) , y = (y',... , y d ) E R+; where b(') (t,X ) = @('I1( t ,x ) , . . . ,b(' ld(t,x ) ) , and c$fT(t) 2 O,Vi,k are such that ~ : c $ : ~ ( t ) d t < m; and p$:T(u) 2 0 ,Vi, k,V u 2 0 are strictly increasing, continuous and concave such that PL:T(o) o and So+ d u -- 00.

cL!~(~)~$!~((x~

pzT(u)

If ,& and ( x r ) ,&)) are solutions of (11.29) with the initial values s f ) and s f ) , respectively; then x t ) i 5 ~ f ) ~ ,=v1i, . . . ,d; and b(')'(t,x) b ( 2 ) i ( t , x ) = , ~I , . . ,d;W 2 0,Vx G E: implies that P - a.s. x p i 5 x j 2 y vt 2 0 , vi = 1,. . - ,d. (1) (st

( 1 ))

<

For each T < co and N = 1,2, . . - Write -Proof. C N , T ( ~=) c$:T(t)(t), W 2 0 ,

-

d

PN,T(U)= Ci,k=l P ~ : T ( ~ ) , 2 E ~ , ~ ( t )
&

368

11. Stochastic Population Control and Reflecting S D E

+ J x x * (( 1 )( s ,~s( 1 )) - bi( 2 )( s ,xL2)))ds + J: l(xy)*>xp)*)~ kd = ~ ( u i k ~( s 2, )- )u i k ( 3 , xi2)))dWk(S) + J: .fz I ( z y p > x p ) ( ~ i (xS-, s(,1 ) 2 ) - ~ i ( sas-, ,( 2 ) z))Nk(ds,dz) - r," 1 ( ~ ~ l ) ~ , ~ ~=) c:=~ i , d 4 I~f < If. However, I 5J

x * , x ( i

4 S,t Now let

(1)

(1)

( 8 , ~ s)

C%:~(S)~$~~((X$')"

(1)

- bi

( 8 , x?))ds

~i~)~)+)d~.

T N = inf{t 2 0 : Ix!l)l+ l x p ) l > N } . Then one easily finds that V t 0, yt = E ~ t = , ( x j ; y , - x:t',i,)+ 5 S," ~ ( s ) p ( ~ ) d s . Hence yt = 0,Vt > 0. Since limN,, T N = m. These imply that P - a.s. xp)" > p i , vt 2 0 , vi = I,... ,d. Analyzing the component equations o f the stochastic population dynamics (11.28) one finds that the fertility rate o f females Pt only influnces the popuation aged in [ r l ,r 2 ](See the first term in (11.28)). This is reasonable, because only females with age in some interval can have babies. However, one also may think that for a better control o f the population, Pt itself may also depend on the population size xt. Is it still true that in such a case the comparison result still holds? T h e following theorem gives a partial answer t o this question.

>

.

e,

Theorem 348 Assume that all u and c satisfy all conditions lo -3' i n Theorem 346, moreover, assume that 4" Pt = & ( x ) satisfies the following condition: V x = ( x l , . . . ,x d ) E 4 R+, & ( x ) = &' + C t l c3xiIoiXiika + %Ix*,ra), where 0 5 py, ci ,. . . ,cf 5 ko, they do not depend on x , and ko is a constnut; besides, k i , . . - ,kg 2 0 are also constants. NOW letVx = ( X I , . . . , x d ) E X : , -

-0

+

+

Pt(x) = Bt ~ $ c1 ~ ( x k~ i ~ I~ t > ~ ~ ;~) . ~ ~ ~ ~ ~ Then the conclusion of Theorem 346 still holds, that is, if ( x t ,4,) and (zt,&) are solutions of (11.25) with xo, Pt and %, p,, respectively; then x i < $ , , V i = l , . . . ,d; and 4

& ( x ) < &(x),Vt 2 O , ~ Xf R+ implies that P - a s . xf 5 ?iVt $,0 , V i = I , . . . ,d.

>

Remark 349 Theorem 348 implies Theorem 346. I n fact, let c; = 0 , i = 1,2,. . . ,d. Then from Theorem 348 we obtain Theorem 346. Now let us establish the above thoerem. Proof. Let

11.8 Comparison of Solutions and Stochastic Population Control

369

+

bcl)(t, x) = A(t)x B(t)x&(x), b(2)(t,x) = A(t)x ~ ( t ) x p ~ ( x ) , where A(t) and B(t) are defined by (11.26) and (11.27), respectively. Then by assumption b(2)"t,x),~i= 1, ,d;Vt 2 0,Vx E b(l)i(t,x) I So if we want to apply Theorem 347, we only need to check that condition 4" in Theorem 347 holds. In fact, ~ ~ i > ~ ~ ( bx)( ~ b(lI1(t, ) ~ ( t y)) , = Ixl>yl[-(l V;)(X' - yl) CF=rl b!(xkDt(x) - ykPt(y))l Ixl>yi C;=,., b!(xkPt(x) - ykPt(y))l = Ixl>yl [b!(xk - ~ ~ ) P t (+ x b!yk(Pt(x) ) -Pt(~))l k,Z CF=, (xk - Yk)+ + Ixl>yl CF=, b:Yk. C ~ [ X- ' I-~ < ~k$Ixi>kk ~ < ~ &- yiIos.iikA - k$y*>kl] I kg C;=rl (xk - yk)+ Izi>yl C?=rl b!yk C;=, ct[(xi - yi) V 0] d ki CF=rl(xk - yk)+ (rz - r1)kiN Ci=l(xi - yi)+ k ; x;=,(xi ~ - pi)+, N; x = (xl, ... ,xd), = ( y l , . - . ,yd) E as 1x1,lyl I More easily, Vi = 2, . . . ,dl ~ = i >(b(lIi(t, ~ i X) - b('Ii(t, 9)) = Ixi>yi[-(I vf)(xi - yi) ~ ~ 5( (xi-I x ~- yi-l)+. - ~ +@*-I - yi-l)] 5 ~ ~ ~ >- y2-l) Therefore, condition 4" in Theorem 347 holds, and Theorem 347 applies.

+

z:.

+

<

< - xtl

+

<

+

+ +

?id+.

+

2. The Optimal Stochastic Population Control Now let us discuss an optimal stochastic population control problem for RSDE (11.25). Denote the admissible control set by : P = pt(x) is jointly continuous, 5 5 pO, G = { Pand p satisfies the condition 4" in Theorem 348 where 0 < & and Do are constants. Suppose we want to minimize the following functional ~ ( t , +~ ( & ) l , J(P) = among all j3 E 6, where 0 T < m is an arbitrarily given constant, and (xf, 4 f ) is the pathwise unique strong solution of (11.25) corresponding to the given /3 E 6; F(t, x) and G(x) are jointly Borel measurable and Borel measurable, repectively. Then we have the following theorem.

Po

E[S,T

4%

<

Theorem 350 Assume that conditions lo-4" in Theorem 348 hold, and assume that 5" Jz Ic(t,x, z)12~ ( d z I ) kg, and as It - sl -,0, Ix - yl -,0, x,z) 4% Y, 4 12d d . 4 0; Jz 6" a(t, x) is jointly continuous. firthemore, assume that F ( t , x ) and G(x) are jointly Borel measurable -4 -4 functions defined on (t, x) E [0,T] x R+ and x E R+,respectively, such -4 that as xi I yi, i = 1,2, ..,d; x, y E R+, W E [0, TI, -+

370

11. Stochastic Population Control and Reflecting SDE

G(x) I G(Y), F(t,x) I F(t, Y). Then the smallest constant Po is an admissible optimal control for the minimization of the functional J(P); that is, 1)

Po2 6,

2) J(P0) = info& J(P> Furthermore, Assume that . . where n ( U ) = 1,ko 2 0 is a constant, c f ) , i = I , . . . ,d, and fi are constant d x d matrices, moreover, 0,Vl I i, j I d, where 11= ( ~ ~ , ~ j ) & = , , and let 60 = mintlo($ 171(t), 172(t), . . . r]d--l(t),$ + rld(t)), b~ =maxt>~(brl(t),...,br,(t)), largest age of the population. where d = r,-the Then as 2 a, = 280 Icf'l - bl12- & b ~ ( r 2 -TI) > O,

>

+

)

where xfO is the so-called optimal trajectory corresponding to the optimal control Po. This theorem actually tells us the following facts: 1) If a target functional monotonically depends on the population size, for example, the energy, the consumption, and so on, spent by the population, then to minimize this target functional we should control the fertility rate of females to take a value as small as possible. 2) It is also possible to make the optimal population size to be exponentially stable in mean square, if we can take the fertility rate of females small enough and when the stochastic perturbation is not too large. To show the truth of this theorem one only needs to apply the existence theorem, the comparison theorem, and the stability theorem of solutions to the stochatic population dynamics. (See Theorem 337, Theorem 348 and Corollary 343). We leave this to the reader. 3. Some ~xblanationsand Conclusions on Population RSDE Finally, by using the results obtained here, the following conclusions can be drawn: The RSDE (11.22) is a suitable model for the stochastic population control system. In fact, by this model one sees that - x i M -(I E ; & ( l - ~Oo(t))ki(t)hi(t)xE~~ +(a stochastic perturbation between [t, t 11) - 42, - xi M -(I+ rl:)x: xi-' +(a stochastic perturbation between [t,'t 11) 4;+l - 42, Vi = 2 , . . . , r m .

+T ~ ) x Z + +

+ +

+ +

11.8 Comparison of Solutions and Stochastic Population Control

371

The second expression tells us intuitively that when i 2 2, xf -the size of the population with ages between [i,i+l), will have an increment ~ f + ~ - xwhen f the time t increases from t to t 1. However, this increment is contributed by two terms when we do not consider the stochastic perturbation. (In the deterministic case 4, = 0). A positive term due to people with ages in [i - 1,i) at time t, who will be in ages [i,i I), when time goes to t 1. That is, the term "+xf-lt'. A negative term is due to the fact that people may die with a death rate caused by a disease or some other cause when the time evolves from t to t 1; moreover, the people with ages in [i,i 1) at time t, will also arrive in ages [i 1,i 2), when time goes to t 1. So from time t to t 1 the size of population with ages in [i,i 1) will lose in all the amount (1 ~ f ) x fThat . is the term "-(I &xitt. However, for the first expression above, the only difference is that x i is the size of the population with ages between [I, 2). So when time goes to t 1 the positive contribution term can only be the number of babies born during this time interval [t, t 11, since women, who can have babies, can only be in some age interval [rl,rz],and a baby if born, may die with a death rate poo(t). Furthermore, the number of babies born and living, also depends on the size of the population x f , the fertility rate of females P,, the sex model lci(t), and the fertility model hi(t) at time t and aged i E [rl,rz]. That is, the positive contribution term from time t to time t 1 is " C ~ ~(1,-. ,~00(t))ki(t)hi(t)~fPt". Now if the system is disturbed by a continuous and a jump type stochastic perturbation, then as we said in the "Introduction" section, we need a reflecting SDE to discribe the population dynamics to keep all population sizes non-negative, that is, xf 2 0, V i = 1,2, .- - ,r,. So we need the stochastic population control system (11.22). In this model we have shown the following facts: 1) The size of the population depends continuously in some sense (Theorem 338 and 340) on the initial size of the population, the fertility rate of females, and the coefficients in the stochastic population dynamics system. Moreover, the resultant error can be calculated in some cases (Corollary 339). 2) If the initial size of the population or the fertility rate of females takes larger value, then so does the size of the population forever as time evolves (Theorem 348). 3) If the stochastic perturbation is not large, and the forward death rate is greater than zero, then it is possible to take a fertility rate of females that makes the system exponentially stable in some sense, and the time when the population size can in some sense be less than a given level, can also be calculated. (Corollary 343). 4) If a payoff value functional (or say, a target functional) depends monotonically on the size of the population, then the payoff value will take the smallest value, as does the fertility rate of females (Theorem 350).

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

372

11. Stochastic Population Control and Reflecting SDE

11.9 Caculation of Solutions to Population RSDE Now let us discuss how to calculate, or say, to construct a practical solution of the stochastic population dynamics. For simplicity we will discuss the population RSDE without jumps. Suppose we have already got stochastic population dynamics as in (11.25) with c(t, x, z) = 0. To calculate its solution practically, first let us see what will happen if (xt, d,) is a solution of (11.25) with c(t, x, z) = 0. For this purpose we now give a lemma for a more general RSDE. Lemma 351 If (xt, 4,) satisfies -d

R+, t 2 0 , xt E ,:?i t 2 0, and the other stantements in (11.4) also hold for (xt, dt), XO=XE

(11.30)

Proof. If (xt, 4,) solves (11.30), set di(t> = s u ~ o ~((-~i(s)) s ~ t V O), xi(t) = yi (t) & (t) then it is obvious that 0 5 &(t) is increasing, continuous, &(0) = 0, and 2 0, for all i = 1,2, ...,d. Moreover, Ti(t) di(t) = J,"I(~;~s)=o,d$i(s), $i(t) = J,"ni(s)d (31, -d where n(t) -= (nl(t), . - . ,nd(t)) E &(,,, as T(t) E dR+, and $(t) = (&(t), . . - ,d d ( t ) ) , ~ ( t= ) (f l(t), . . - ,&(t)). SO (f(t),?(t)) solves &(t) = dy (t) d$(t), -d xo = x E R+, t 2 0, t 0, x, € ,:?i and the other stantements in (11.4) hold for (~(t),$(t)). In particular, (Ti(t), (t)) satisfies -

+

I?/

I

+

>

&

11.9 Caculation of Solutions to Population RSDE

373

i.e. it is a solution of the Skorohod's problem (11.31)in 1-dimensional space. Since (xi( t ) ,q5i(t)) also satisfies (11.31); by the uniqueness of solution to the Skorohod problem we have that Vi = 1,2,. . . ,d, ( x i ( t ) 4i ( t ) )= (%( t ),?i ( t ) ) . This lemma motivates us to use the Picard's iteration technique to calculate the unknown solution (xi(t),q5i(t)), i = 1,2, . . . ,d o f the RSDE (11.30). In fact, we may let Y: ( t ) = xi(())+ bi(s,~ ( 0 ) ) d+s ffik(s,x ( o ) ) c I w ~ ( s ) , i = l , 2 , ...d. By this y: ( t ) we can get s)) 4:(t) = ~ u ~ o < ~ g t ( ( - ~V: (01, x: ( t ) = y: ( t ) 4: (t). By the proof of the previous lemma ( x t ( t ) ,4f ( t ) )satisfies a Skorohod problem similar t o (11.31) but with the given y:(t), and the same initial value xi(0). By using the induction we may construct

Sot

+

+

@ ( t ) = xi(O) Sgt bi(s1xn-' (s))ds )> 41(t) = s u ~ ~ g , ~ t ( ( - y l (Vs 01, x l ( t ) = Y X t ) + q5l(t),

S,t ffik(s,xn-'(S))dwk(~),

(11.32) and we see that ( x l ( t ) q5l(t)) , still satisfies a Skorohod problem similar to (11.31) but with the given Il,"(t), and the same initial value xi(0). Thus, after some calculation, we have a sequence o f (xn(t),q5"(t)),n= 1,2,. . . , which we may call the approximated solutions of (11.30). Can we show that they actually "converge" to the solution of (11.30)? The following theorem answers this question. Theorem 352 Assume that for all t 2 0 w E a,z E 2,x E 1" lb(t,x)12+ 1 1 4x)112 t , 5 ko(l 1x12), where ko 2 0 is a constant, 2 2•‹ 1% 2 ) - b(t,y)12 + Ilff(t,2 ) - 4,d1125 ko lx - Y I . Then we have the following conclusions: 1) (11.30) has a pathwise unique strong solution ( x t ,&). 2) For i = 1,2, . . . ,d the sequence (xp(t),q52(t)),n = 1,2, . . . constructed by (11.32) satisfies that ~ T - q5(t)12] = 0, limn-+,[Esu~t 0, let N, be such that CT n 4 ( 1 2 2 k ~ ~ ) n - ' /( nI ) ! < 77, where CT = 8E 1 x 0 1 ~(30k0 J O ~ O[xo12)T, E VE> 0, let N, be such that 1/n2 < € 7 then as n 2 max(N,,, N,),

+

+

zzN,

x:='%,,

+

+

374

11. Stochastic Population Control and Reflecting SDE

In this theorem the conclusion 1) and 2) mean that a uniform limit xt exists, which is just the unique solution of the RSDE (11.30) and conclusion 3) tells us that with a greater than (1 - g)% possibility, the error of the uniform approximation is less than a given number E > 0, if n is large enough. So the approximae solutions and the approximate error can all be calculated. Proof. 1) is true by Theorem 335. Now let us show 2). By assumption n = 1'2, ... x; = 2 0 b(s, x:-l, w)ds a(s, xy-', w)dw, 4: = yr 4:, 51;.= 50, Applying Ito's formula, one finds that ; . 1 - xt12 I 2 S~(X:- x,) . (b(s, x:-)' - b(s, xS))ds +2 S~(X: - x:-l) - (a(s, x:-') - a(s, x:-~))~w, J,"Ila(s, x:-') - a(s, x:-~) ds. It follows by the martingale inequality that as 0 t T I? = sup,^, 12; - 2;-'12 i: ilo I;-'~S 5 ~ t ( $ t ) ~ - ' / ( n - I)!,

+ Ji

+ Ji

+

11

+

+

< <

Ji

~l~ = E S U P , ~ 1~4: - 4:-'12 5 d ( g t ) n - l / ( n - I)!, where ct and 4 are constants depending on t only.

Therefore for any T < ca

Now, by the Borel-Cantelli type lemma below, there exist a (&,&) such that P - a s . -xt = limn-+a, z;, -g5t = limn+oo4:, uniformly in t E [0,TI, for any given T < w. Since by Theorem 330 E s u P1&12~ ~5~k'T,Vn. EsupSlt 1@12 Hence by Fatou's lemma

+ ESUP.~, 1%12+ EsupSst 1&12

5k ~ . On the other hand, by Theorem 328 there exists a unique solution (Zt, &) solving t Zt = x0 b(s,Z3, w)ds Joa(s, E,, w)dw, = yt &, and the other statements in (11.30) for (&, hold. Now by the Lipschitz condition one easily shows that lyt - y;l2 = 0. Hence one can take a subsequence of {n) , denoted again by {n) , such that P - as., as n -+ oo, suPt,[O,~]lyt - y y 2 --' 0. Thus, applying Lemma 327 P - a s . , as n -+ oo,

+ Ji

+

3,)

+&

+

11.9 Caculation of Solutions to Population RSDE S U P t c [ O , ~ 1xT ] - zt12

375

0,

&I 2

s U P t ~ [ O ,147 ~] 0, By the uniqueness of a limit P - a.s.Vt E [0,T ] - = T t , 4t = &. So (?Et,&) is a solution of (11.30). Since we already know that ( x t ,4,) is - solution of (11.30), we have P - a.s.Vt E [0,T ] the pathwise unique xt = Tt, 4t = 4t. Now let us show 3). For this we only need to calculate the & and CT in (11.33). Observe that by Ito's formula

zt

xn-l

12

I? = Esups,t 12; - 8 5 ~ [ 2Ix; - x;-'1 Ib(s,z;-') - b(s, ds +2 sup,,, IJl(xr - xr-')(a(s, xy-') - a(s,X ; - ~ ) ) ~ W , S,' Ilu(s,x;-') - U ( S , X Z - ~ ) ds]. Notice that by the martingale inequality E sup,<, IJi(x; - x;-')(a(s, 2;-') - a(s,x ; - ~ ) ) ~ w , 5 3 E [ C 12; - x;-'12 Iu(s, x;-I) - u ( s , x ; - ~ ds]'I2. )~~ So applying the elementary inequality a b s $+;,Va,b>O, and by virtue of the Lipschitz assumption one finds that I," 5 2ko[6 54 11 1,"-'ds = 122ko 1,"-'ds 5 (1221c,)~ J; I ~ - ~ d r d5s (122k0)~ Ji(t - r ) I ~ - ~ d r 2 - .- 2 (122ko)"' S ~ d 5 (122k0)"-'-1: r

Ji

x;-~)I

I

112

+

I

+ +

Ji

~i

5 ~(122k~)~-l-. However, 2 It = Esup,,, Ix: - x:1 5 2E lxo12 2EsupS<, 2~ 1x01~ 6E[1xo12 4 !:(I+ 1xo12)ds 4k0 1x01~)ds) 8~ 1x01~ (30ko 30koE lxo12)t= ct. Now the conclusion 3) follows from the lemma below. w Finally, let us give the following Borel-Cantelli type lemma.

+

< <

+

+

+

+

+ Ji(l+

Lemma 353 (Borel-Cantelli type lemma). Assume that {x:),"=~ is a sequence of random processes such that ' /I() n! , V n = 1,2,. . . . P(supSsT Ix; - x;-ll > l / n 2 ) n 4 c ~ ( z ~ ~ ) n - where k~ 2 0 is a constant. Then 1) there exists a random process xt such that P - a s . limn,, S U P , g 1%: - xsI = 0; 2) Vq > 0, let N, be such that CT n4(1221c~~)n-1/(n - I ) ! < q, VE> 0, let N, be such that

<

zz==N,

Cz=NE

< &,

then as n 2 max(N,, N,),

376

11. Stochastic Population Control and Reflecting SDE

P ( S U P , <~xF-' _ ~ - x8I < E ) 2 1 - 7. Proof. 1): I f we can show P - a.s. that VE > 0 there exists a N such that as n 2 N SUPslT ix;+p - x;I < E , vp = 1,2, ; then P - a s . { x ~ ) is ~ a=uniform ~ Cauchy sequence on t E [0,TI,and 1) is proved. However, by assumption { S U P ~ ~Ix; T - x;-'l > lln2)) P ( ~ F =U;=N , xn- 1 > lln2) i Cr=N P ( S U P , ~(x: T < CrZNn 4 @ ( & ~ ) " - ' / ( n- I ) ! -+ 0, as N -+ co. %n-1 > l / n 2 ) )= 0, or So P(nFZ1uFzN SUP,<^ P(UF=' nzN {supslT [x: - x:-'l l / n 2 ) )= 1. This means that there ex~stsa A E 5 w ~ t hP(A) = 0 such that a = (uF=, nF=N {supslT - x ; - ' l i l / n 2 } )u A. NOW fix any w E U ~ = ~ ~ ~ 1~ 2; { x:-'[ S Ui P l / n,2 )~.VE~> O take a N, such that CrZNe l / n 2 < E. By assumption there also exists a N2 such that w E n;=N2 SUP,^^ 1x7 - x:-l i l / n 2 ). Let N = max {N,, N2) . Then as k 2 N , I~+P SUP, lln2 5 oEmNnn 4 ( & ~ ) " - ' / ( n- I ) ! < r), or P(nF=N, SUP,<^ Ix: - x:-ll i l / n 2 ) )2 1 - rl. Hence as k 2 max(Nq,N,), and w E n;=, {supslT 12; - x:-l[ lln2}, (w) i Cr=~c SUP,
I

I

1%:

5

1%;

1

1~:+~

1%:

Ix:

1

1

Ix,(w)

1%:

1)

<

I

1

1

<

Ix;(~)

x~-'(w)I

12 Maximum Principle for Stochastic Systems with Jumps

In this chapter we will establish the maximum principle for stochastic systems with jumps (that is, the necessary conditions for a control to be optimal for controlling the system).

12.1 Introduction In the optimal control problem for a dynamical system one important subject concerns finding an optimal control when we know one exists. It is well known in the Calculus that if a function f(x) attains at its maximum at the point x = xo, and f(x) is differentiable, then fl(xo) = 0. So one can find the maximum point xo by solving the equation fl(xo) = 0 (the neces sary condition for the maximum point). However, to find out a necessary condition for an optimal control to be satisfied is more complicated. This is because now the target we treat is a functional of the control (for example, J(v) = E 1xg2dt), subject to a SDE (xz satisfies a SDE depending on the given control v(.)). The statement of necessary conditions for an optimal control is usually called the maximum principle. To establish such maximum principle we need to work analysis. Consider a d-dimensional stochastic systems with Poisson jumps as follows:

12. Maximum Principle for Stochastic Systems with Jumps

378

where wt is a d-dimensional BM, fik(ds, d z ) is a centralized Poisson martingale - measure with compensator x(dz)ds such that N k ( d s ,d z ) = N k ( d s ,d z ) - x(dz)ds, and T(.) is a a-finite measure on the measurable space (Z,B(Z)). Let U c RT be a non-empty subset. Set

&d

=

{

v ( . ) : v ( t ) is & - adapted and U - valued such that

E

)

sup l v ( t ) ~ <~ co, ~ k = 1,2,3,4 (tE[O,T]

(12.2)

Uad is called an admissible control set. Under appropriate conditions (12.1) will have a unique solution yt for each v(.) E &d. Let J ( v ( ' ) )= l ( t ,~ tvt)dt , ~(YT)]. If u(.)E &d is such that

E[C

+

J(u(.)) =

inf

J(v(-))

(12.3)

v(.)ELd

then u(.) is called an optimal control, and J ( u ( - ) )an optimal value functional. Under much stronger conditions the optimal value functional for the optimal control problem (12.1) - (12.3) will be a Sobolev solution of a Hamilton-Jacobi-Bellman equation. (See [651). This book will not discuss Sobolev solutions.

12.2 Basic Assumption and Notation

12.3 Maximum Principle and Adjoint Equation as BSDE with Jumps

379

Under assumption ( A ) ,by Theorem 117 (12.1) has a unique strong solution. For simplicity we use the following notation: a , = the derivative of the matrix a E Rdmdwith respect to x; E ~ d @ xd . . . x ~ d @ = d (~d@d)d, .. . ax =

(up),

v

Cjka;fyjdwk ...

Cika2tyjdwk "

)

Rd,

where y* = ( y l ,- - . ,yd), W: = (wtl,.. . ,w!), and y* is the transpose of y. Since we will discuss the d-dimensional SDE, it is natural to assume that b(t,x,v) : [O,T]x Rd x U - R d , a ( t , x , v ) : [O,T]x Rd x U RdBd, ~ ( xt ,,Z ) : [0,T ] x Rd x Z HR ~ , are all jointly Bore1 measurable

12.3 Maximum Principle and Adjoint Equation as BSDE with Jumps In the following we always assume that u(.) E U,d is an optimal control for (12.1) - (l2.3), and x (.) is the unique solution of (12.1) corresponding to ti(.). Introduce a Hamiltonian function as follows: H : [o,T]x Rd X U x Rd x RdBd1I+ R1, H ( t ,x , v , P, Q ) = l(t,x, v ) (P,b(t,x , v ) ) traceQa*(t,x, v ) . We have the following Maximum Principle.

+

+

Theorem 354 There exists a unique (P(.),Q(.),N.1) E L i ( R d ) x L%(RdBd) X G(Rd), and there extts a unique (F(.),Q(.), R(.)) E L ; ( R ~ @x ~()L ; ( R ~ @x~32 )5)(~~d~@ satisfying the following respective BSDE

d )

d

+ j=1 C a c ( t , x t , u t ) ~+i SZ c ~ ( t , x t , z ) R t ( z ) ~ ( d z ) +L(t,xt, at)]&- Qtdwt - SZ ~ t ( z ) f i k ( ddz), s, -dPt = [ b z ( t , x t , u t ) ~ t

PT = hx ( X T ) ,Qt E [O, TI; (12.4)

380

12. Maximum Principle for Stochastic Systems with Jumps

where u:ft =

dl

a g ~ Ei RdBd,p(t,x , i ,f i ) = ( f i , ~ ( st, t , z ) ) for f i E

j=1

Rd such that V v E U, a.e., a.s. 1 H ( t ,x t , v , Pt, Qt - Fta(t,x t , ut)) -trace 2 (aa*(t,xt,v ) F t ) (12.6) 1 2 H ( t , x t , u t , Pt, Qt - F t a ( t , x t , u t ) ) -trace 2 ( a 6 ( t , x t , u t ) F t ).

+

+

Equations (12.4) and (12.5) are called adjoint equations. Obviously, they are BSDEs with jumps. However, they are linear. So by the theory of BSDEs with jumps (Theorem 4?34) they have a unique solution. Actually, we can solve the first one, then the second one.

12.4 A Simple Example To illustrate how to use the above Maximum Principle let us give a simple example as follows: For simplicity let us assume that all processes and coefficients appearing below are in 1-dimensional space. Consider the following linear stochastic system:

where v E Ldand Uad is defined by (12.2) such that E

0

k-,< w , V v E Uad and the target functional is

+

sup [v(t)l , 2,

<

J ( v (.)I = E { ~ [ Z Y Hfi lvs121ds) , and yz with yz = x is the unique solution of (12.7) for all 0 s T and we assume that fi > 0, and all A , B, C , F, E, fi are real valued. Suppose that u(.) E Had is an optimal control, and xs is the optimal trajectory satisfying (12.7) corresponding to this control u(.),and we also assume that in the definition of Uad,U a clgsed and convex set. Notice that now we have H ( t , y, v , P, Q ) = R y N [v12 P ( A y B v ) QCy.

+

< <

+

+

+

12.5 Intuitive thinking on the Maximum Principle

381

Applying Theorem 354, by (12.6) we have that u(-) and x, satisfy the following variational inequality: Vv E U

where by (12.4) and by the uniqueness of solutions P(.) E L $ ( R ~ )satisfy

+

+

Let f (t, v) = k lv12 PtBv. Then fL(t, v) = 2 k v PtB, and one sees that fL(t,ut) = 0, as ut = 2N fL(t, v) < 0, as v < ut; fL(t,v) > 0, as v > ut; where Pt satisfies (12.9). So we have that such ut satisfies (12.8). However, solving (12.9) one finds that Pt = 1: e-A(t-~)&s. From these results one finds that for such a ut, (12.7) has a unique solution. Since by the approximation to the infimum, (see Lemma below), one easily finds that there should exist an optimal control u(-) E hdsuch that J (u (-)) = i n f V ( . p a dJ (v (-)) . So the ut obtained above must be an optimal control.

-=;

Lemma 355 There exists an optimal control u(.) E &d such that J (u (.)) infv(.)€uadJ (v (.I) . Proof. Take vn E Uad such that J (v, (-)) for n = 1 , 2 , . . . Since E

=

< infvc.,EuadJ (V(.)) + &, ..

-< ko < co, there exists a subse-

quence of {v,) denoted again by {v,) a i d there also exists a u E kdsuch that v, 2 u in L$(R1). Hence there also exists a subsequence denoted again by {v,) such that its arithmetic means from this sequence converge strongly -+ u, in L$(R1). However, one easily sees to u in L$(R1), that is, "l+'i+"~ that V ~ + ~ ; + V ~ and J ( u I + ~ ; + v ~5) J(vl)+...+J(vn) n 5 Hence by letting n -+ co, one easily obtains that J (u (.)) = infvc.,,u,d J (V(-)) .

ad

i.

12.5 Intuitive thinking on the Maximum Principle In Calculus if a real-valued function with d variables f (x), x E Rd, attains its minimum at the point $0, then we will have f (x) 2 f (xo),Vx E Rd; or equivalently, f(xo y) 2 f (xo),Vy E Rd. TO explain clearly the intuitive derivation of a Maximum Principle for a functional, let us simplify the T notation. Consider a non-random functional J(v) = So 1(x:, vt)dt h(x$), where all functions are real-valued, xt is a solution of a real ODE

+

+

12. Maximum Principle for Stochastic Systems with Jumps

382

for v(.) E U, and where vt takes values in a non-empty set U c R1. If J(v) attains at its minimu at v = u E U, then obviously we will have ~, 1(x?, ut))dt h(x$) J(v) 2 J(u),Vv E U. Or, equivalently, ~ ( l ( xvt) h(x$) 2 0, Vv 'v U. By Taylor's formula we will have, approximately, that vv € U

+

+o( sup IxY - 2211)

2 0,

tE[O,Tl

where 1, is the derivative of 1 with respect to x, provided that 1 is smooth enough. Actually, all kinds of maximum principle are only trying to transform this inequality into a more applicable statement. The way here by which to derive a necessary condition for the inequality (12.11) is by using the following two techniques: 1) We use a "spike variation" ue(t) of the optimal control to obtain a first order variational equation dx:'" such o(c2)dt, where uE(t) = v, as t E [to,to 6); that d(x?' - x?) = dx:.' ue(t) = u(t), otherwise; and xyE is the solution of (12.10) corresponding to the control uE(.) provided that uE(.) E U. In fact by Taylor's formula, and under appropriate conditions, one has that d(x?' - x?) = (b(xyE,vt) - b(x7, ut))dt = bS(x?,ut) . (x? - x?)dt (b(x?,vt) - b(x?, ut))dt o(E). So we may introduce a first order variational equation:

+

+

+

dx;'" = b,(xy, vt) . x:"dt

+

+ (b(xy, vt) - b(xy, ut))dt,

= 0.

(12.12)

2) By using integration by parts we have

SOT

T

- So (dpt)xt1"=

ptdx;" d(Ptx:'E), and we can introduce a function Pt such that T 1 E Mx?, ut)@dt hx(xU(~))(x>") = - J, (xt>)dpt ~ , ( X ~ ( T ) ) ( X > ~ ) -:J (x:lE) b,(x;, ut)Ptdt = :J pt(qx;, u:) - qx;,~ ~ ) ) d t , if

+

J,T

+

This means that (12.11) can be rewritten as that T 0 5 So (1(xY,uB)- ~(xY,us) Ps(b(xY,ue) - b(xY,uS))ds o(E). By the definition of ue one sees that 0<1 V)- I!(%:, us) P,(~(X;,V)- b(xY, u,))ds - B to (~(XY, Letting E 4 0, one obtains the following variational inequality for the optimal control u ( - ):

+

p+"

+

+

I(xE",v)+ Ptb(x;, v) 2 l(x;, ut)

+ 9.

+ Ptb(x;, ut), a.e.t,Vv E U.

(12.14)

12.6 Some Lemmas

383

This is the Maximum Principle. In what follows, by using such idea we will make rigorous the proof of the Maximum Principle for a stochastic system with jumps. That is, we will arrive at Theorem 354.

12.6 Some Lemmas For any to E [0,TI, E

> 0, set

uE(t) =

v, astE[to,to+~), u(t), otherwise,

where v(.) E kdis arbitrary. It is easily seen that ue(-) E Uad. Hence there corresponds a unique trajectory x" (.) (the solution of (12.1)). Let us introduce the first and second order Rd-valued variational equations, respectively, as follows:

and

where

+ +

1 E l e $bxx(s,xSluS)xs9 5.' (bx(s, xs, u:) - bx(s, xs, us)) x:,' lll: = p x x ( s , xs, %)xs'l e X,'l E (ax(s, x s , u 3 - ax(%xs, us)) x p , x ~ ( z )= $cxx(s,x,, z ) x ~ , ~ x ~ ~ ~ , where we will write fX:xiyiy', for f = b, 0, c, 1, h. fxxyy = By Theorem 117 it is known that (12.16) has a unique strong solution and hence again, by Theorem 117, (12.17) also has a unique strong solution. We have the following lemmas.

v:

=

c$=~

Lemma 356 lo For p = 1,2,3,4 E s u ~ t q o , qlxt12' G k ~ ,

12. Maximum Principle for Stochastic Systems with Jumps

384

where k~ = koTP(l+

SUP

1ut lap), and ko 2 0 is a wnstant;

t€lO.Tl

Remark 357 In [184 there are some errors: In (2.10) there is the estimate forp > 1

EIR

2P

JzxIp

S cp IIPI~-'

gO(s,z)Sk(ds,dz)l

IJz

+)lP

E SIP lgo(s,z)12 ds. This estimate is used in [184 to get an estimate similar to 2'. However, the above inequality is not correct. To see this, i f we take I p = [ s , t ] , and p = 3, then by the Kolmogorov continuous trajectory version theorem J: Jz go(s,z)Nk(ds,dz) will have a continuous (in t ) version. This is obviously incorrect. Hence the proof that the general Maximum Principle still holds for stochastic systems with jumps such that the jump weficients also depend on controls in [184 is at the very least incomplete. Actually, by the Burkholder-Gundy-Davis i n e q u a ~ i t ~one [ ~ qcan only get that for p > 1

1~;

E JzxIPgo(%z)fidds,dz)l but not that E 11;

J~~~~go(%z ) k ( d s ldz)l

2P

C kpE 2P

IS;

6 k p E 1.l;

&,

1go(s,z)12~ii(ds,dz)IP,

JZXIrI S O ( S ,z)12~ ( d z ) d s I ~ .

Now let us show Lemma 356. Proof. 1" can be proved by Ito's formula and Gronwall's inequality. Let us use Gronwall's inequality to show 2' by lo.In fact, by assumption and by (12.16) y:aEsup 1 x i . " 1< ~ ~S a p [ % $ ytds ss t +E(J,t Ib(s, xs, u 3 - b(s,xs, us)l dsI2P 2P

+ +

+E(IJ:(o(s, xs1u:) - ~ ( sxs, ' us))dws)l = 52p[Ij I: I:]. However, lo(s,x,, uz) - o ( s ,x., u,)12 d s ) ~ It < L"E(J;:+' 0 ~ ds . )sp-' 1C 2k;klsP. ~ ~ Io(s1z S ,u:) - u(s1x S ,~ S k;E Similar estimate holds for I:. Hence by Gronwall's inequality we obtain the first estimate in 2". 3" is proved by the first estimate o f 2". Finally by (12.17) and by Gronwall's inequality one easily derives the second estimate of 2". The proof is complete.

-.

12.6 Some Lemmas

Lemma 358 E s u ~ ~Ixt , ~- xt ~ , x:" ~ ~- x:"l Proof. Set y; = x; - xt - x:'" - x21" t

Then

2

=o

385

(2) .

-

+ + + + ++ + +c(s, X , + xi1€+ x'$€,u;) - a(s,x,, u;) + c ( s ,x,, u;) - c ( s ,x,, us))dws + j," jZ (c(s,x;, 2 ) - x, + xitE + x p , z ) +c(s,5 , + xi" + x i 1 €Z,) x S ,z ) ) S k ( d sdz) , t 1 = So So bX(s,xs + x$' + x:lE + XyQE, u;)dXy:ds + J: c X ( sx,, + xilE+ xzrE+ Xy;, u;)dXy~dw, + Sz J: c X ( sx,, + x:lE + xz1"+ Xy;, z ) d ~ ~ , ~ @dz) ~(ds,

YE

= j;(b(s, x;, uz) - b(s,xs zitE x:*€,u;) x:lE,uz) - b(s,x,, u;) b(s,x,, uz) - b(s,x,, u,))ds - x ; , ~- x 2 , ~ J;(c(s, xz, uz) - a ( s ,x, xilE xi>€, u;)

+b(s,X S +

C(S,

C(S,

+sotJ z [ ~ c X x ( s , ~ , , x:~ ) (+~x ~, ,x,~ 1

E

2

E

2,E

2,"

+So1 (1 - X)(cXx(s,X , + X(xirE+ x : ~ ~Z)) -, cXx(s,xsz))dX .(Xft"+ X ~ ~ " ) ( X ; > + " x:?')]fik(ds,dz)

+

+

j," R € ( ~ ) ( s ) ~jots R ~ ( C ) ( S ) ~ ~ , , where RE(b)(s)= (bx(s,2 3 , u;) - bx(s,x S ,U , ) ) X : > ~ f L b X x ( sx,, , u s ) . (2xi3Ex:2E+ x ' $ ~ x ~ ~ ~ ) j? +T(bx,(s, xs, u;) - b,,(s, xs, us))(x;lE X ; + ) ( X : > ~ + x:vE) 1 + So (1 - A)(bxx(s,xs X ( X : ' ~ x : ' € ) ,u:) -bxx ( s ,~ , u E ) ) d X ( x i >x:>€) ~ (x:lE+ x:lE), R E ( c ) ( s )is similarly defined. Now by applying Ito's formula, G r ~ n ~ a l l ' ~ inequality and Lemma 356, one obtains the stated conclusion.

+ +

+

+

Lemma 359

Proof. Since (xt,ut) is optimal, we have

E[S,T l(t,xZ, uZ)dt + h(x?)]- E [ J l(t, ~ x t , ut)dt + h ( x T ) ] = ~ [ ~ ? ( l xt ( t+ , x:lE + xTvE, - l ( t ,x t , ut))dt + x;') - h(xT)]+ o ( E ) + E [ ~ ( x+ T = E [ / c ( l ( t ,xt + x:" + x:lE,u t ) - l(t,xt, ut))dt

0

UZ)

x$€

386

12. Maximum Principle for Stochastic Systems with Jumps

1,e 1,e

+$E ~ : ( 1 ~ ~xt, ( tu:) , - lxx(t,xt, ut))xt xt dt ' l e l e +E[hX( X T ) ( x $ € x : ~ ) ] E[hxx( X T ) X + a+ ] o ( E )

+

+

+

Thus the conclusion now follows from Lemma 356 rn

12.7 Proof of Theorem 354 To derive the Maximum Principle from Lemma 359 we need to introduce an adjoint process pt such that it makes E l x ( t lx t , ut)xil'dt E h x ( ~ ( T ) ) ( x = $ ~E) f ( t ,utr vt, xt, pt)dt. For this it is necessary to notice that by Ito's formula (or say, using integration by parts) -E c ( d p t ) x i l E = ptdx:'E J: dptdxi" d(ptx~'E)]. So if we introduce (Pt,Qt, Rt) as the solution of (12.4),then by (12.16) we have e E l x ( t ,X ~ , U ~ ) X E~~ S, ( x~ (~T )~) ( x $ "= ) -E J,T ( xlt > )*dpt

SOT

+

E[C

Jr

+

Now let

. . . x;:(x;;) ............. 1E X ; ~ ~ ( X : : ) . . . xis ( d s .::(x;:)

X; = x ; ' E ( x ; ' ~ ) *

+

=

I

12.7 Proof of Theorem 354

Then we have

387

Appendix A A Short Review on Basic Probability Theory

A . l Probability Space, Random Variable and Mathematical Expectation In real analysis a real-valued function f (x) defined on R1 is called a Borel - measurable function, if for any c E R1 {x E R1 : f (x) 5 c) is a Borelmeasurable set. If we denote the totality of Borel - measurable sets in R1 by %(R1), and call (R1,%(R1)) a measurable space, then we may write {x E R1 : f (x) 5 c) E %(R1),Vc E R1, for a Borel - measurable function f , and say that f is a measurable function defined on the measurable space (R1, !?3(R1)). The definition for a Lebesgue - measurable function is given in similar fashion.. F'urthermore, in real analysis, the Lebesgue measure m is also introduced and used to construct a measure space (R1, %(R1),m), so that one can define the Lebesgue - integral for a Lebesgue - measurable function under very general conditions and obtain its many important properties and applications. Such concepts and ideas can be applied to probability theory. For this we first need to introduce a probability space (a,5,P ) . Definition 360 (SZ,5, P ) is called a probability space, if SZ is a set, 5 is a family of subsets from 0 , which is a a-field, and P is a probability measure. Definition 361 5 is called a a-field (c-algebra), i f it satisfies the following conditions: 1) 0 E 5, 2)AE5 +AC=a-AE5,

Appendix A. A Short Review on Basic Probability Theory

390

It is easily seen that a a-field is closed under the operations of U, n, in a countable number of times.

Definition 362 A set function P defined on a a-field ability measure, or simply a probability, i f 1) P(A) L 0,VA E 5, 2) P(+) = 0, where 9 is the empty set, 3)Ai E 5 , A i n A j = + , V i # j ; i , j = 1,2,... +P( U g l Ai) = Cz1P(Ai), 4 ) P ( 0 ) = 1.

5 is called a prob-

Usually, a set function P with properties 1) - 3) is called a measure. Now we can introduce the concept of a random variable.

Definition 363 A real - valued function J defined on a probability space ( Q 5 ,P ) is called a random variable, if for any c E R1, { W E0 : J ( w ) 5 C ) E 5. So, actually, a random variable is a measurable function defined on a probability space. Recall that in real analysis if two Lebesgue - measurable functions f and g are equal except at a set with a zero Lebesgue measure, then we will consider them to be the same function and show this by writing f =g b ) , ax. Such a concept is also introduced into probability theory.

(XI

Definition 364 If J and r] are two random variables on the same probability space (Q, 5,P ) , such that they equal except at a set with a zero probability, then we will consider them to be the same random variable, and show this by writing [ ( w ) = r](w),P - a s . Or, simply, J = 77, a.s. Here "a.s." means "almost sure". Similar to real analysis we may also define the integral of a random variable as follows: (Recall that a random variable is a measurable function defined on a probability measure space). If J 0 is a random variable defined on a probability space (a,5,P ) , we will write EE = S , J(w)P(dw)= S, J(w)dP(w)= S, JdP = limn-+mCY:; $ P ( { w : [(w)E [$, For a general random variable J = J f - J - we will write

>

&)I).

EJ

= EJ+ - EJ-,

if the right side makes sense.

Definition 365 EJ defined above is called the mathematical expectation (or, simply, the expectation) of J and D ( J ) = E lJ - EJ~' is called the variance of (.

A.1 Probability Space, Random Variable and Mathematical Expectation

391

By definition E J may not exist for a general random variable J. However, if EJ+ < cw or EJ- < cw, then EJ exists. By definition one also sees that E J is simply the mean value of the random variable J. So to call EJ the expectation of J is quite natural. (Sometimes we also call Ef the mean of J). Obviously, D(J) describes the mean square error or difference between J and its expectation EJ. In real analysis the Fourier transform is a powerful tool when simplifying and treating many kinds of mathematical problem. Such idea can also be introduced into Probability Theory, leading to the so-called a characteristic function of a random variable J. Definition 366 If J is a random variable, p(t), t E R1 defined by p(t) = ~ e j ~where t , i is the imaginary unit number, is called the characteristic function of J. The charateristic function is a powerful tool when simplifying and treating many probability problems. Here, we can use it to distinguish different specific useful random variables.

Example 367 J is called a Normal (distributed) random variable, or a Guassian random variable, if and only if its characteristic function p(t) = eimt-+02t . In this case E J = rn, E IJ - E E =~a 2~. That is, m and a2 are the expectation and variance of J , respectively. One can show that if J is such a random variable that Vx E R1, e-wdx, P([ 5 x) = 1 &u J-cO then p(t) = Eeitt = eimt-+oZt. So J is a Normal random variable. Moreover, b y the definition of expectation and by the definition of derivative one can show that - m x f (x)dx, and E f = Ja J d P = Jm $YE I x) = f (x), e - w . Naturally, we call f (x) = k e - w the where f (x) = lJi;;u probability density function of the Normal random variable J. So this (probability) density function also completely describes a Normal random variable. Furthermore, we also call the function denoted by F(x) = P ( J 5 x),Vx E R1, the (probability) distribution function of a random variable J. Obviously, any (probability) distribution function F(x) has the following properties: 1) F(-co) = lim,,-, F(x) = 0, F(W) = lim,,, F(x) = 1; 2) F(x) is an increasing and right continuous function on x E R1. Conversely, one can show[40]~[78] that if F(x) is such a function satisfying the above two properties, then there exists a probability space (a,Z,P) and a random variable J defined on it such that F(x) = P(f 5 x),Vx E R1. That is,

392

Appendix A. A Short Review on Basic Probability Theory

F ( x ) must be a distribution function of some random variable J. So the distribution function completely describes a random variable. Now let us discuss the n-dimensional case. For a random vector J = (J1,. . . ,Jn), where each component ti,i = 1,. . . ,n is a real-valued random variable, we may also introduce a n-variable characteristic function cp(t) as follows: Definition 368 cp(t) = cp(tl,. . . ,t,) = ~ e ~ ( ~=l Eei c ) xE=l the characteristic function of [.

is called

Now, we can use the characteristic function to define the independence of two random variables. Definition 369 We say that two real random variables J and q- are independent, if they satisfy that cp&) = cpt,&, t2) = cp&)cp,(t2), vt = (tl ,t2) E R2, ) , cpE(tl) l c ~= Eeitlc, etc. where cpc,,(tl, t2) = ~ e ~ t(a q ~ or, equivalently, VA, B E B(R1), P(J E A, 7) E B) = P ( J E A)P(q- E B). In similar fashion we can define the independence of the three random variables J , and q-. If n random variables J,, . . . ,[, are independent, then we will say that {Jk)i=l is an independent system. The intuitive meaning of independence for two random variables is that they can take values independently in probability. One easily sees that if {Jk);=, is an independent system, then each pair of random variables is independent, i.e. all (ti, Jj) form an independent system, i # j ; i, j = 1 , . . . ,n. One should pay attention to the fact that, the inverse statement is not necessary true.[126] However, if {Jk);=l is a Gaussian system, then it will certainly hold true. See the next section. This is why the Gaussian system is quite useful and easy to treat in practical cases, because it is much easier to check whether each pair of random variables forms an independent system, than to check the whole n-dimensional system. The amount of calculation work will be considerably saved. (Calculations on involving 2 x 2- matrices are much easier than calculation on involving an n x n-matrix).

A.2 Gaussian Vectors and Poisson Random Variables Naturally, we generalize the concept of a real normal random variable to an n-dimensional normal random variable (also called a Gaussian vector) as follows: Definition 370 An n-dimensional random variable J* = (El, . . . ,J,) is called a Gaunssian vector if it its characteristic function is

A.2 Gaussian Vectors and Poisson Random Variables

393

Cn t, cp(tl,... ,t,) = it.<] = eim't-it* =~, where t* = ( t l , . .. ,t,),m* = (ml, ... ,m,), En= [ ~ i j ] r ~and mi = EE,, aij = E[(& - mi) . ( t j - mj)]. (Here t* = tT means the transpose oft, etc.).

En

Usually, m and are called the mean vector and covariance matrix of the Gaussian vector J, respectively. As in the 1-dimensional normal random variable case one can show that if E* = (&, . . . ,t,) is such a random vector that V(xl ,...,x,) E Rn,

then cp(t) = ~ e ~ ( ~=l eim't-it* t) En t . So [ is a normal random vector. Moreover, by the definition of the expectation one can also show that Vi = I,... ,n, 03 03 Eti = Ja JidP = JPm ... J-coxif (21,. .. ,xn)dxn. .. dxl = mil 1 e-t ("-m)* (~,)-'("-m) is the sowhere f (xl, ... ,x,) = (6)nIdetC n y called (n-dimensional) probability density function of the normal random vector J. So this (probability) density function also completely describes a normal random vector. Now as in the 1-dimensional case let us call the function denoted by F ( x l , . . . ,x,) = P ( t l 5 x l , . . . , t n 5 x,),Vx= ( x l , . . . ,x,) E Rnr the n-dimensional (probability) distribution function of a random vector J. Obviously, any n-dimensional (probability) distribution function F(xl , ... , x,) has the following properties: 1) F(-m, 22,. .. ,x,) = limxl,-, F(x1,. .. ,x,) = 0, . . . . . . . . . ,F ( x l , x 2 , . . . , -m) = limx,--t-03 F(x1,... ,x,) = 0; F(cm,. . . ,m ) = limxl,m ,... F(x1,. .. ,x,) = 1; 2) F ( x l , ... ,x,) is increasing and right continuous in each xi E R1, Vi = I , . . . ,n; 3) forV(xl,... ,x,),( Y I , . . . ,Y,) E Rn 21 I ?/I,... , X n I Yn XI,... ,Xn;Yl,... r ~ n = ) = F ( y l , . . . j ~ n ) - C y = ~ F ( ~ l ,,Yi-l,xi,Yi,... ... ,yn) +Cyl,iz=l,il
*

+

394

Appendix A. A Short Review on Basic Probability Theory

Proof. Obviously, the characteristic function o f q is cp(tl1.. . ,t k )= E[etT]= ei~'t-3t*Ekt1 where t* = ( t l , .. . ,tk),E*= (m1, .. . , E k ) , = [&j]f,j=lland mi = Eqi = j=niw1+lc j r n j i fs .-. aij = E[(qi - E i ) . ( q j - E j ) ]= En" 3=ni-i+l C-"j j=nj-l+l c3.e 3 23 T h e conclusion is now follows.

zk

x'?

Proposition 372 If J* = ( J 1 , . . . ,J n ) is a Gaussian vector, then the following statements are equivalent: 1) {Ji)y=l is an independent system; 2) each two random variables Ji and t j (i # j ) are independent, i # j ; i l j = l , ... I n; 3) each two random variables Ji and J j (i # j ) are uncorrelated, that is, E[(&-rni).(Jj-rnj)]=O,i#j;i,j=lI...,n.

-

(el, ,en)

Proof. Since J* = .. . is a Gaunssian vector, { J i ) L 1 is an independent system cp(tl,.. . ,t k )= eim.t-+t* Ent = ny=l pi(&) = ei m*. t*. - ~2 u . . t* ? ++ a i j = O , i # j ; i , j = l , . . . , n .

nLl

Proposition 373 If J is a 1-dimensional Gaunssian variable with the mean rn = E J , and the variance a 2 = E[lJ - rn12],then

I =o,

.

Proof. Notice that the characteristic function o f J is cp(t)= Eeit[ = eimt-+u2t2 1

and t h e roba ability density function o f J is f ( x ) = -&-e-*. nu w - 1 e-*dz E I I C- m12k+1~ = Lm 15 - ml and from the integration by parts 1

m

Hence

= 0,

From the above discussion one sees how powerful and useful the characteristic function technique can be, when proving some probabilistic statements! This also happens as in real analysis by using the Fourier transform t o treat some problems. Now let us look at the Poisson random variables.

A.3 Conditional Mathematical Expectation and its Properties

395

Definition 374 A 1-dimensional random variable J only taking values on the set {0,1,2,3, . . . ) is called a Poisson variable i f its characteristic function is 'p(t)= ~ [ ~ i t=EeA(eit-l) ] where X > 0 is its intensity such that E J = X. One easily sees that for the above Poisson random variable its probability distribution is P ( t = n) = e - A n! x ' and E J = A, D ( J ) = E(IJ - Et12) = A.

A.3 Conditional Mathematical Expectation and its Properties Up to now we have reviewed the elementary theory of Probability needed in this book. Now we will enter a higher level of Probability theory. The first important concept is the conditional mathematical expectation of 6 under a given a-algebra Q , where Q ~ 5 .

Definition 375 Suppose that Q C$ is a a-algebra, J is a random variable. If there exzsts a Q-measurable function denoted by E [ J J Qsuch ] that V A E Q JA E[JIBldP = JA EdP, then E[JIQ]is called the conditional expectation o f t given Q .

m,

If one takes Q = (0,A , A", 41, where A E 0, then one easily sees that EKlQ](w)=

5 is any fixed set with P ( A ) >

as w E A; and EKIQ](w)= w , as w E AC. (To sees notice that E[JIQ]is G-measurable. So one should have that E[JIQ](w)= cl IA(w) c z I ~(= w ) , where cl and cl are constants only depending on the given [. On the other hand, by cl P ( A ) = JA E[JIQ]dP= JA EdP one sees that cl = Similarly, one has that cz = w . From which the conclusion follows). This tells us that if one takes the conditional expectation of J given the condition that A has happened (that is, w E A ) , then one should consider ( A ,A n S , P ( . ) / P ( A ) )as a new probability space, and take the expectation of t in this new probability space, that equals SA J d P / P ( A ) = The intuitive meaning of the conditional expectation E[JIQ]is the expectation of t taken under the given information Q. Notice that by definition the conditional expectation is usually a random variable, unlike the expectation which is a deterministic value. This also can be explained by intuition. For example, a person at the present time s has wealth x , for a game, and at the future time t he will have the wealth xt for this game. In this game the expected money for this person at the future time t is naturally expressed as E[xt15,],where 5, means the information up to time s,

+

m.

m.

Appendix A. A Short Review on Basic Probability Theory

396

I ~a ~ random ] variable which is known by the gambler. Naturally, E [ X ~ is for any t > s. The important thing is that the conditional expectation has many useful properties which we will use a lot in this book, so we must discuss it. Existence and Properties of the Conditional Expectation Now let us use the Radon-Nikodyn theorem to show the existence of the conditional expectation. Assume that J is a random variable such that E < m , and 9 CZ is a a-algebra. Write p(A) = JA N= 0), then EJn EE. However, under the same condition it is not true[lg8] that E[J,14] -+ E[@7],P - a.s.! As examples of what technique to use when proving the properties of conditional expectations let us show 6) and 7). For 6) we have that VA E 91,

*

+

-+

{J,)2.1

+

A.4 Random Processes and the Kolmogorov Theorem

397

JA E[JlG~ldp= JA EdP = JA E[ElGIdP = J A E(E[JIGIIGl)dP. So EIJIGl] = E(EIJIQ]IG1),P - a.s. Simlarly, one also sees that E[JJGl]= E(EIJIG1]IQ),P - a . ~The . property 6) holds true. Now for 7) we have that VA E G, JA E[JIGIdP = J A EP'lmn-+wJn(G]dP= JA limn-w JndP = limn,, JA JndP = limn,, JA EIJnIG]dP = JA limn-+, E[JnlGIdP, where we have applied the Lebesgue's dominated convergence theorem to the expectation, because that lJnl I g,Eg < m , and lEIJnlG]l I E[gIG],EE[glG] < m . Thus E[JIG]dP = limn+, EIJnIG],P - a.s. The property 7) holds true. The other properties can be proved in completely the same way. Let us give one more property of the conditional expectation which is used a lot in the discussion of the filtering problem. 9) (Integral property). If at(w) = a ( t , w) is jointly measurable in ([0, T]x a,B([o, 2] x 5 ) , with E lat!dt < m and G c5 is a a-algebra, then E(/: a,ds IG) = Sot E ( a s 19) ds, P - a s . for all t E [O,T]. This conclusion is easily derived from the definition of the conditional expectation and Fubini's theorem. Indeed, VA E G E[IAE(J," asds [G)] = E(IA a&) = JotE ( I ~ a ~ ) d s = E ( I A E ( ~IG))ds , = E[IA E ( a s IG))dsl. So the conclusion follows. 10) (Jensen's ineguality).[25]~[951~uppose that G CSis a a-algebra, J is ' is a convex function a random variable such that E (Jl < m . If f (x), x E R such that E 1f (<)I < m , then f (E(JIG)) 5E(f

Jot

Jot

A.4 Random Processes and the Kolmogorov Theorem In nature we meet a lot of "random processes" xt(w): like the temporature in some area, the wind speed over the sea, the population of fish in some large pond, etc. All of them have much the same features: at each time to, xto(w) is a random variable; after each observation, that is for each wo, xt(wo) can be recorded as a function or a trajectory of changing time t. These are exactly the concepts of random processes.

Definition 376 We call a family of random variables { X ~ ( W )a )ran~~~ dom process, i.e. for each t 0, xt is a random variable. If the xt(w) are ) )called ~~~ jointly measurable with respect to B([O,m ) ) x 5, then { X ~ ( W is a measurable random process.

>

As in the case of Gaussian vector, we can have a Gaussian process as follows:

Appendix A. A Short Review on Basic Probability Theory

398

Definition 377 A random process { x ~ ) ~ . is , ~called a Gaussian process, if for each 0 5 tl < . . . < t,, ( x t l , .. . ,xtn)% a Gaussian vector. W e can also have a Poisson process as follows: Definition 378 A random process {N(t))t.,o is called a Poisson process, if for each t , N ( t ) is a Poisson random vasable (r.v.) with intensity At, and it has independent increments; that is, for any 0 tl < t2 < . . . < t , { N ( t z )- N ( t l ) ,. . . ,N(t,) - N(t,-l)) is an independent system.

<

Obviously, the trajectory of a Poisson r.v. N ( t ,w ) , as w is fixed, is RCLL (right continuous with left limit), and it is a step function with the jump equal to 1 at each jump time. Similar to the random vector we want to introduce some distribution functions t o describe a random process {xt),>,,. Naturally, for arbitrary ( t l ,t 2 , .. . ,t,) E RT, where Rn+ is the set of alrpoints in Rn with all nonnegative components, we will have a finite - dimensional (probability)distribution function Ftl,... ,tn ( x l ,. . . ,x,) from the random vector (xtl, . . . ,xt,). Thus we will have a family of all finite - dimensional distribution functions {Ftl,...,tn( X 1 , . . . ,x,), V(t1,t2, . . . ,t,) E ~ n +~, n from ) a random process { ~ t ) ~ Obviously, ,~. this family has the following properties: 1) [Symmetric property). Exchanging any positions of the indexes, the values of the distribution functions do not change. For example, if we exchange the positions of the indexes i and Ic, then Ftl,... ,ti-l,ti,ti+l,...,tk-l,tkrtk+12,...tn(x1,. . . 7 Xi-l,xi, x i t l , . . . , xk-1, xk,xk+lr ' ' ' ,x ~ Z ) = Ftl,...,ti-l,tk,ti+l,'.' ,tk-l,tirtk+12r"'tn(x1~ ' ' ' > xi-1, xk, xi+l, ' ' ' ,xk-1, xi,xk+l, ' ' ' ,xn). 2) (Consistent property). Restricting the higher dimensional distribution function to a lower dimensional space will cause it to coincide with the lower dimensional distribution function. For example, Ftl,...,tk,... ,tn(xlr' ' ' ,xk, 00,' ' ' ,00) = Ftl,... ,tk(xl,' ' ' ,xk). Conversely,one can show that the following Kolmogorov's theorem holds. Theorem 379 [40J74suppose that {Ftl,...,t,,(~i,...r ~ n ) , v ( t l r t 2 , . .,tn) . E Rn+,Vn) is a family of finite - dimensional distribution functions. If it satisfies the Symmetric property and Consistent property stated above, then there exists a probability sapce (R,5,P) and a random process { x ~ ) , defined ,~ on it such that {Ftl,... ,t, ( X I , ... ,x,),V(tl, t2, . . . ,t,) E ~ n + , V ni )s just the family of finite - dimensional distribution functions from { x ~ )-~ >. , , So the Symmetric and Consistent family of finite - dimensional distribution functions completely describes a random process. An important special kind of random processes is useful for us in this book.

A.4 Random Processes and the Kolmogorov Theorem

399

Definition 380 A real random process {x~),,,, is called a stationary Marlcov a Bore1 set in R1, Vs t process, if for any A E %(R1), that is A

is

6'0 5 7)

<

P(xt E A l x u , ~ s)= P(xt-s E Airs), where P ( ~ J X ~5, U s) = EIIxlu(xu, u probability.

<

< s)] is the so-called conditional

~ folThe physical meaning of a stationary Markov process { x ~ ) ~is ,as lows: 1) It is stationary; that is, starting from any time, its probability distributions remain the same; or say, its probability distributions do not depend on the starting time. More precisely, Vs 0, tl, , . . . ,t, 2 O,VAl,A,, ..- ,An E %(R1), P(xt1 E A1,. . . ,~ t E , An) = P(xtl+s E AI,. . . xtn+s E An). 2) If its "presentttis known, then its "future" behavior does not depend on its "past" behavior. A stationary strong Markov process can be similarly defined, if we substitute the Vs t by Vu T , where u and T are two 3:-stopping times. Here 3: = o(xu,u t). (For the definition of stopping times see Definition

>

<

5).

<

<

Appendix B Space D and Skorohod's Metric

Write D = D([O,w ) , Rd), where D([O, ca), Rd) is the space of RCLL functions X = (Xt)tlo,with values in R ~ Let . us introduce a metric in this space. We will first define a metric d in space D([O, 11) = D([O, I]),R). Recall that for space C([O, 11) = C([O, I]), R)-the totality of continuous functions X = (Xt)tE[o,ll,with values in R- it is well known that the metric between two element X, Y E C([O, 1)) is defined by maxtE~o,l~ lxt , under this metric the space C([O,11) becomes a complete metric space, and the convergence in this space is uniform convergence. However, for the space D([O, 11)such metric is not suitable, because now elements have discontinuous points. In particular if we want to discuss how X n E D([O, 11) converges to X E D([O, I]), where the discontinuous points of X n also approximate the discontinuous points of X, then such convergence usually cannot be uniform. For example, Xn(t) = I[;- n (t) j, I13,,l(t)= X(t), for each t E [0, 11. But SuPt€[o,l]IXn(t) - X(t)l = 1 + 0. This is because of the discontinuous point t of Xn(t) also moves. (Notice that by Dini's theorem, if X n , X E C([O, I]), and Xn(t) j, X(t), V t E [0, I], then up,,[^,^^ IXn(t) - X(t)l -+ 0). So we need to introduce a metric that allows t to change a little between two elements from D([O, 11). Skorohod introduce the following metric: for X, Y E D([O, 11)

402

Appendix B. Space D and Skorohod's Metric

where

1

X = X(t) : X is strictly increasing, continuous on t E [O, 11, such that X(0) = 0, X(1) = 1 . One can verify that d(X, Y) is a metric in D([O, 11). To show this let us first

A={

establish a lemma which indicates that, similar the continuous function case, the elements in D([O, 11) can also be approximated by step functions uniformly. Such a result is very useful. Moreover, the proof of this lemma also gives us with a nice property of the RCLL functions.

Lemma 381 ('3 For each X E D([O, 11) there exist step functions X n E D such that 0, as n m. SUP,€[0,1] IXS - Xsnl -+

+

> 0 such Proof. In fact, since X is RCLL, there exists a point ~ [ -~ Xtl , ~ < ~ )E. Write that sups,[o T ) IX, - Xol < &/2. SO S U ~ ~ , ~ IX, ' .L tl sup^^^^ { t l ) . Then suP,,t~[o,t~) 1x8 - Xtl I E. Again starting from tl there exists a maximum point tz > tl such that SUPs,t~[tl,t2) IX, - Xtl E. By induction there exists a maximum point t, > t,-l such that SUPs,tE[t,-l,t,) IX, - Xtl I E. NOWsuch a procedure should stop at finitely many steps. Otherwise, let T = sup,t,. As XTIX, - Xtl E. exists, one easily finds a t; > t, such that This is a contradiction with the definition of t,. Obviously, we should have T = 1. So we have proved that there exist finite many points 0 = to < t l < . . < t,, = 1 such that

<

<

xz=l

+

Now let XE(t) = I[t,-l,t,)(t)Xt,-l I{l)Xl. Then X E E D([O, 11) is a step function and S U ~ , ~ [ ~IX, , ~I Xi1 5 E. By (B.2) we immediately obtain the following corollary.

Corollary 382 If X E D([O, I]), then VE > 0 there exist finitely many points 0 = to < t l < . . . < t,, = 1 such that SUPs,t~[t,-~,t,)1x3 - Xtl 5 E,n = 17%.. . 1%. By means of this corollary one immediately finds the following facts: If X E D([O, I]), then 1) SUPt€[O,l]lXtl < 00; 2) VE > 0, there exist only finitely many points 0 = to < tl < . . . < t,, = 1 such that AX,, > E, i = 1,2,. . . ,n,. Thus, X only has at most countably many discontinuous points. In fact, by Corollary 382 taking E = 1, one sees that SUP~E[O,I, IXtl 5 C;tl(lxtn-l + IXtn-l - X t l ) I t ~ [ t ~ - l , t ~ ) < n l + C:L1 Ixtn-,I< 03.

1

Appendix B. Space D and Skorohod's Metric

403

F'urthermore, by Corollary 382 for any given E > 0 it only can be true that a t t h e p o i n t s o = to < t l < . . . < t,, = I , lAXt,l > ~ , i = 1 , 2 , . . . , n ., Finally, {laxtl> o) = {t E [o, 11: lnxtl > 0) = u=:, {Inxtl> So X only has at most countably many discontinuous points. Thus, facts 1) and 2) hold true. These two facts mean that any RCLL function defined on [0,1] is bounded, and for any given E > 0 it has only finitely many jump points where its jumps can exceed E, and so it can only have at most countably many discontinuous points. Now let us show the following lemma.

i).

Lemma 383 d(X, Y) defined by (B.l) is a metric in D([O, 11).

Proof. First, take X(t) = t,Vt E [O,l]. then 0 5 d(X, Y) 5 suPt,[o,l IXt - Ytl < m , because X - Y E D([O, l]), and any function in D([0,1]) is bounded. Secondly, let us show that d(X, Y) satisfies the three properties for metric. 1) Notice that X E A x-' E A. For each so E [O,1] write to = X(so). Then SO - X(SO)= X-l(t0) - t o , and X,, - Yx(so)= X,-I(,,) - Go. So SUPo 0 there exists a sequence of {t,) such that all {t,) are continuous points of Y and I&, - Yt-1 5 E, t, t, because there are only finitely many points s such that lAKl > E , and Y can only have countably many discontinuous points. So IXt,-l'-l<~,tnf t,andIXt--&-lS~. This shows that Xt- = I$- = Xt. By symmetry one also has that Yt- = Xt- = K. So t is a continuous point of Y, and we complete the proof that d(X,Y) = 0 ==+ Xt = Y,,W E [O, 11. 3) Notice that if XI, X2 E A, then their composition XI o X2 E A. By this one sees that for any XI, X2 E A; x , Y, E W O , 11)l SUPtE[o,i~ IX1 h ( t ) - tl 5 SUPtE[o,l] h ( t ) - X2(t)l IXl(t) , l ] - tl + S U P t ~ [ ~ 1X2(t) ,l] + S u p t ~ [ ~1X2(t) , l ] - tl = S U P t ~ [ ~ and

X-Wl

A.

r

r

404

Appendix B. Space D and Skorohod's Metric

1

SUPt€[0,1]IXx1ox,(t) - XI 5 suPt,[o,l] IXx,ox2(t) - zx,(t) + suPt,[o,l] I-%,(t! - XI = suPt,[o.l] I ~ * l ( t )- ~t l+~u~t,[O.l]I-%,(t) - Xl . So, if we take the infimum among A1 E A first, then take the infimum among A2 E A secondly, we arrive at the result d(X, Y) i d(X, 2 ) d(Z, Y). Thus d(X, Y) is a metric. Now one sees that under metric d Xn = -+ I[+,ll = X, in D([0,1]). In fact, one easily finds that d ( X n , X ) = $ -+ 0. However, one should notice that this convergence is not equivalent to pointwise convergence; that is, for X n , X E D([O, I]), d(Xn,X) -+ 0 e Xn(t) -+ X(t), V t E [O,l]. For example, write Xn(t) = tn,W E [O, 11; and X(t) = 0, as t E [O, 1); X(t) = 1, as t = 1. Then Xn(t) -+ X(t), V t E [O,l]. However, d ( X n , X ) = 1 -t, 0. So Xn(t) -+ X(t), v t E [O, 11 ;tj d(Xn, X ) -+ 0. Such a situation happens here because X n E C([0, 11) is continuous for each n , but its limit X E D([O, 11) has a discontinuous point. Another example is X n = I[++;,,],X = I[+,,]. Then X n , X E D([O, I]), and d ( X n , X ) = 0 -+ 0. However, x n ( $ ) = 0 .tt x(;) = 1. So d ( X n , X ) -+O+Xn(t) -+ X(t), V t E [O,l]. The situation happens here because the jump points move, and at the limit of jump points we change the value of the limit function to fit that d(Xn, X ) = 0. Anyway, we have the following lemma, which indicates that d(Xn, X ) -+ 0 is equivalent to some kind of uniformly convergence.

+

Lemma 384 For X n , X E D([O, I]), d(Xn,X ) -+ 0 is equivalent to there existing a sequence An E A such that as n -+ m , SUPt€[0,1]IAn(t) - tl + suPt,[o,l] IXn(An(t)) - X(t)l 0. -+

Proof. In fact, if d(Xn,X) -+ 0, then for each n there exists a An E A such that IXn(An(t)) - X(t)l < SuPt6[o,l]IXn(t) - tl + SUPt€[o,l] Conversely, if as n -+ m , SuPt~[o,l] IXn(t) - tl + suPt,[o,l] IXn(An(t))- X(t)l 0, then VE > 0 there exists a N such that as n 2 N 4xn,X ) 5 SUPt,[0,1 IAn(t) - tl + SUPt€[0,1]IXn(An(t))- X(t)l < E . SO as n -+ m , ~ ( x " , x \ -+0. Furthermore, we have that for C([O, 11)the convergence in d here is equivalent to uniform convergence. Shown in the following lemma.

5

-+

Lemma 385 If X n , X E C([O,I]), then d(Xn, X ) SUPt,[0,l] IXYt) - X(t)I 0.

-+

0 is equivalent to that

+

Proof. For X n , X E D([O, I]), taking X(t) = t, t E [O,l], one sees that

Appendix B. Space D and Skorohod's Metric

4xn,X ) I

-

405

SUPt,(O,l] l X n ( t ) - X ( t ) l ,

so,

SUPt,[O,l] IXn(t) - X(t)l 0 d ( X n , X ) 0. Conversely, notice that IXnW - X ( t ) l I IXn(t) - x((w1(t)>l +IWW1(t)> -X(t)l , where An E A are taken from Lemma 384, so i f X n , X E C([O,I ] ) , and d ( X n ,X ) -+ 0 , then SUPte[0,1]IXn(t) - X(t)l I + SUPt,[O,l (X((Xn)-l(t))'- X ( t ) l . As X ( t ) is uniformly continuous on t E 10, l ] and limn,, (A")-' ( t )- t = 0 , +

+

;

I

SO

1

1

limn-+, su~t,[o,l]Ix((Xn)-l ( t ) )- ~ ( t=)0. ,l - X ( t ) J-,0. Hence as n -+ m, s ~ p , , ~ ~ ,IXn(t) So, one sees that the Skorohod's metric is a nice generalization o f the uniform metric from the space C([O,11) t o the space D([O,11). However, this Skorohod's metric is not a complete metric, that is, the space D([O,11) is not a complete metric space under this metric. So it is not so convenient, and we need t o introduce an equivalent complete metric

L e m m a 386 13 ' The space (D([O,I ] ) ,do) is a complete separable metric space. The metric do is equivalent to the above Skorohod's metric.

Remark 387 We say that two metrices do and d are equivalent, i f the topologies generated by do and d, respectively, are equivalent. In other words, write S d o ( x , ~=) {y E D([O,11) : do(x,y) < E ) , (that is, the ball with center x and radius E under the metric do), we have that for each given S d 0 ( x , & ) , there exists a Sd(x,E ' ) such that Sd(x,E') c Sdo( x ,E ) . Vice versa. Actually, for the Skorohod's metric one can show that ~ ( x , YI ) 2do(x,Y ) . However, the inverse is not true. So the completeness i n metric do cannot imply the completenes i n metric d.

. Now let us introduce a metric t o the space D = D([O,m),R ~ )Let DO([O,11) = { X E D([O,11) : X 1 = X I - ) . Obviously, (Do( [ O , I ] ) ,do) is a closed subspace o f (D([O,I ] ) ,do). Similarly, define Do = { X E D : limt,, Xt exists and finite). Let P0(X,Y ) = do(X', Y ' ) , where XI =X,,-l(,), =Y+-l(,), $ ( t ) = - log(1 - t ) , 0 I t < 1; and $(t)= m, as t = 1. Then obviously, ( D o ,pO) is a complete separable metric space. Now let X,k = Xt . gk ( t ) , where

406

Appendix B. Space D and Skorohod's Metric

gk(t)= I [ o , ~ ]f( ~(k) -k 1 - t ) I ( k , k + l ~ ( ~ ) . Set

k=l Then we have the following

Lemma 388 [741(Lindvall, 1973). (D, p) is a complete separable metric space. Moreover, the convergence i n the metric p is equivalent to the existence of functions An E A,,n 2 1, such that for each L > 0

x?&)I

sup,
+

Appendix C Monotone Class Theorems. Convergence of Random Processes

C.1 Monotone Class Theorems In this Appendix we will present some useful tools used quite frequently in this book. The first one is the so-called: monotone class theorems.

Definition 389 Let 6 be a set, C be a family of subsets of 6. 1) C is called a T-system, if it is closed under finite intersection; 2) C is called a A-system, if (i) 6 E C , (~~)A,BEC,ACB==+B\AEC (iii) A, E C , A, f A ===+ A E C. 3) C is called a monotone system, if it is closed under all monotone limits. Obviously, a A-system is a monotone system. If a A-system is also a T-system, then it is a a-field. If a field is also a monotone system, then it is a a-field. The following monotone class theorem on a family of sets is useful.

Theorem 390 1) If C is a T-system, denote by A(C) the smallest A-system containing C, then A(C) = a(C). 2) If C is a field, denote by m(C) the smallest monotone system containing C, then m(C) = a(C). Proof. 1): Let us show this by using a method we will call the technique of "collecting right setstt. First, we note that the intersection of all Xsystems containing C is A(C). Now set

408

Appendix C. Monotone Class Theorems. Convergence of Random Processes

~;={BEA(C):VAEC,B~AEA(C)). Then 3 1 > C, and F1is a A-system. So 3; = A(C). Set 3 2 = { B E A(C) : V A E F ~ , B ~EA A(C)). Again F2> C, and F2is a A-system. So F2= A(C). This means that A(C) is a also T-system. So it is a a-field, i.e. A(C) > a(C). However, a a-field is also a A-system. Therefore A(C) c a(C). 2): Notice that the intersection of all monotone systems containing C is M(C). Similarly as in 1) we find that it is a T-system. Now set 3 3 = {A E M(C) : ACE M(C)) . Since C is a field, then C c 33. Moreover, 3 3 is obviously a monotone system. So F3 = M(C). This means that M(C) is also a field, since it is a T-system, and F3is closed under the complement operation. So it is a a-field, i.e. iM(C) > a(C). However, a a-field is also a monotone system. Therefore M(C) c a(C). rn The following theorem is the monotone class theorem on a function family.

Theorem 391 Assume that 3-1 is a linear space of some real functions defined on fi, and C is a n-system of some subsets i n 0. If (i) 1 €3-1, (ii) fn E 'H, 0 5 f, T f , f is finite (resp. bounded)* f E 'H; (iii) A € C I A E 3-1; then 3-1 contains all a(C)-measurable real (resp. bounded) functions defined on 5.

*

1

Proof. Let 3 = A C fi : I A E 3-1 . Then by assumption one sees that 3 is a A-system, and 3 > C. Hence a(C) = A(C) c 3. Now for any a(C)-measurable real (resp. bounded) function [ let

{

n2n i

= Ci=oI;;'{~[+,+)}. Then E 3-1, since { E E E a(C) c 3,and 3-1 is a linear space. Obviously, 0 5 En T [+ = J V 0. Hence [+ E 3-1. Similarly, we argue that 6- E 3-1. Therefore = [ + - [ - E 'H. rn En

En

e))

[&,

Now we formulate the following frequently used a monotone class theorem:

Theorem 392 If 3-1 is a linear space of real finite (bounded) measurable processes satisfying (i) 0 L, fn E 3-1, f , T f , f is finite (resp. bounded) ===+ f E 3-1; (ii) 3-1 contains all St- left continuous processes; then 3-1 contains all real (resp. bounded) gt- predictable processes. (For the definition of a predictable process, etc. see section 2 of Chapter 1 in this book).

Remark 393 If i n the condition (ii) we substitute the left - continuous processes by the right - continuous processes, then i n the conclusion the

C.2 Convergence of Random Variables

409

predictable processes will also be substituted by the optional processes. The proof for this new case is the same. Now let us prove Theorem 392. Proof. Notice that f is St- predictable, means that, f E P , where P = a(C), and C ={&fZ:l {(ai, bi)) c [O, co) x R : Vk, Vai < bi, 'dfi - Zt - left continuous process) Obviously, C is a n-system. By Theorem 391 one only needs to show that VA E C ==+ IAE 7-t. In fact, there exists a sequence of bounded continuous functions 0 I cp; defined on R such that cp;(x) f I(ai,bi) (x), as n f co. Hence as n f co, q:p..(fi(t, w)) T I(.,,~,,(fi(t, w)) = I , : = ~ ~ ; ~ { ( ~ , , ~ ~(t,) )w ) . Since the left hand side is St-left continuous, so it belongs to 7-t. Now ) The proof for predictable by assumption (i) Ink f L 1 ~ ( a i , b i ) ) ( tE, ~7-t. r=l processes is complete.

nL

nt,

C.2 Convergence of Random Variables Now let us discuss the convergence of random variables. Recall that in real analysis we have several different concepts for the convergence of measurable functions: a.e. convergence (almost everywhere convergence); convergence in measure; and weak convergence. Since random variavles are measurable functions defined on a probaility space, all such convergent concepts can be introduced into probability theory. Definition 394 A sequence of real random variables tn defined on a probability space is said to be convergent to a real random variable J almost surely (a.~.), and we will write:

if P(w : limn,, IJ,(w) - J(w)l = 0) = 1; and En is said to be convergent to J in probability PI and we will write:

if for every E > 0 limn,, P(w : [tn(w)- t(w)I > E) = 0; and tn is said to be weakly convergent to t , and we will write:

410

Appendix C. Monotone Class Theorems. Convergence of Random Processes

if for arbitrary f E Cb(R), where Cb(R) is the totality of all real bounded continuous functions defined on R, S f (Jn(w))dP-,S f (J,(w))dP, as n -,00. Naturally, we will have the following relations between these types of convergence.

Proposition 395 Assume that J n , n = 1,2, . . . , and J are real random ((7.3). variables. Then (C.l) ==+ ((7.2) Proof. C . l ) ==+ C.2): Let 1 A = nz=1 u g = , nn>N {IJn - 61 < ,). If C.1) holds, then P ( A C )= 0 , where AC is the complement of A. So limN-too P(Un>N {Itn- 2 limN-oo P({IEN - El 2 = P ( n g = , U n > { ~I J n -[I > I P ( A C )= 0 , for each m. C.2) ==+ C.3): For any f E C b ( R ) ,and f is uniformly continuous, ISf (J,(w))dP -,S f (J(w))dpI 5 E l f (Jn) - f (J)l I E I f ( e n ) - f(E)I I1<,(w)-<(w)l
A))<

A))

A))

i

{

::

= lim

I

n-+w

< <

4(mp(Jn(w),Fi))dP 2 &,,,P(J,

E Fi).

(C.4)

Now write A i = { x : q 5 f(~)<~)=F~-~-F~,,i=1.2..~.,k. Then k-1 1 XiZo rP(E E Fi) - = ~ t Y = p ( J~E Ai) S f ( J ( w ) ) ~ P < ~ t ~ =P ( EJ~Ai) = ~ P ( EJ Fi). The same inequality holds for 6,. Therefore, by this and by (C.4) k-1 1 limn,, J f (Jn(w))dPI limn,, Xi=o r P ( J n E Fi) < $ p ( J E Fi) 5 S f (J(w))dP. Letting k -+ CO, one obtains that limn,, f (J,(w))dP 5 S f (J(w))dP. Write 7 = 1 - f. One easily sees that limn,, J?(J,(w))~P 2 f f ( J ( w ) ) d ~ . Thus C.3) is proved.

i

<

~fzt

~ f ~ i i+

S

C.3 Convergence of Random Processes and Stochastic Integrals

411

Using real analysis one easily construct a counter example that C.2) +C.l). Let us give an example to show that C.3) + C.l). Example 396 Write [ ( t ) = w(t),[,(t) = -w(t),Vn = 1,2,... ; where w ( t ) , t 2 0, is a BM. Then, obviously, 3) holds, and 2) fails. The convergence in probability has a nice physical meaning. Since En E, in P is equivalent to the case that YE,6 > 0,3N such that as n 2 N P(w : IE,(w) - E(w)I > E ) < 6. So this obviously means that with any given probability 1 - 6 (or say possibility (1 - 6)%)the error between J , and can be less than or equal to any given positive number E , when n is large enough (that is, n 2 N ) . In this book we also see that convergence in probability is quite helpful when discussing whether approximation solutions from the approximation SDE converge to the true solution o f the true SDE, because such convergence needs weaker conditions. Moreover, it has a nice property that simplifies the discussions. See the following remark. -+

Remark 397 Suppose that a sequence of random process xZ); x:, . . . ,x?, . . . , t E [O,T] is given, which is uniformly bounded in probability, that is, as ... P(suptE[o,T1 I x ? ~ > N ) 0. Then N + W , SUP,,,,,, T ) - Fo(xZ),tI > E) P(IFn(@lt I 5 ~ U P ~ = O ,P(SUP~E[O,T~ I,... 1x211 > N ) + P b ~ t E [ o ,IxZ) ~j >N) +P(IF71 - F ~ l > E l S U P t ~Ix?1[ ~ I, ~ N~ , s u ~ t E [ ~ IxZ)I , ~ ] I N). This obviously means that proving P ( / F , ( x ? , ~I T ) - ~,(xZ),tI > E ) -+ o is equivalent to proving that P ( ] F , ( x : , ~I T ) - F ~ ( X Z ) , ~ IT)^ > E ) + O under the condition that all {x?,t E [0,TI):=, are uniformly bounded, that is, Ixy1 I ko,W E [O,T],Vn=0,1,2,... . Finally, let us point out that E - 1x?12 I kT < oo, Vn = 0,1,2,. . . imples that as N -+ oo, sup,,,,, ,... P ( s ~ p , , [ , ,1x71 ~ ~ > N ) 0. (In fact, supn=o,1,... P(~~P,,[,,T] lx?l > N ) I +). -)

T)I

I

T)I

-)

C.3 Convergence of Random Processes and Stochastic Integrals Finally, let us discuss the convergence o f random processes and stochastic integrals. First we will quote a Skorohod theorem, which is very useful in the discussion of the convergence of random processes.

412

Appendix C. Monotone Class Theorems. Convergenceof Random Processes

Theorem 398 (Skorohod, 1961)['~41.Let {[r)t>o,n = 1,2,. - . , be a sequence of d-dimensional RCLL random satisfying the following two conditions: for each T 2 0 , E > 0 limnr-, supn S U P t q P {ICl > N ) = 0 , l i m h l o s u ~ ~ S U P t ~ , t ~ s ~ , lPt ~{It: -t~l -~C2 h > E ) = 0. Then there exists a subsequence { n k ) of {n) , a probability space (Q5,P ) , (actually, 6 = [O, 1],%= 23([0,I ] ) ) , and d-dimensional RCLL random

I

processes

{ck}

tto

---

{ E } , defined ~ ~ on it such that distributions of T k { )t,O coincide with the finite-

, k = 1 , 2 , . . . ,and

1) all finite-dimensional

dimensional distributions of {Jyk)t,O, - k = 1,2,. . . ;

2)

Kk -tt,i n probability p, as k --+

+ m,

V t 2 0.

Here we give the following lemma, and quote two more lemmas that will be needed for the discussion in the book. T o save space we will not give the proofs o f the last two lemmas. (For their proofs see [ I g 4 ] ) . Lemma 399 Let {x;, w;, c y ) t y o ,n = 1,2,. . . , be a sequence of d x r x d-dimensional RCLL random processes satisfying the two conditions stated i n Theorem 398. are BMs, n = 1,2,. . . , then the {G~k)t,O 1) If, i n addition, obtained from Theorem 398 also are BMs on ( R ,5,P ) , k = 1,2, . . and so also is the limit {Gt)t,O. 2) If, in addition, g y ) t Z o ,n = 1,2, . . . , are random processes with in-

-

---

nk} dependent increments, then { - Ct

tzo

obtained from Theorem 398 also are

---

random processes with independent increments on ( R ,5,P ) , k = 1,2,. . . and so also is the limit . Write tzo pn(dt, d z ) = C s E d t I A C : Z O I A C : E ~ ~ ~ Pk(dt,d z ) = C s E d t I n ~ : k o I A p E d Z , F(dt, dz) = C s E d t I A T , + ~ I A ~ , E ~ ~ If, besides, all pn(dt,dz) are Poisson random measures with the same compensator A such that V n = 1,2,. . .

{L}

I Z I8 d + l 8

qn(dt,d z ) = pn(dt, d z ) - &, . . where qn(dt,d z ) ,n = 1,2, . . . , are Poisson martingale measures, then P k ( d t , d z ) , k= 1,2, ... , also are Poisson random measures with the same compensator on

&

z,

(6, F) such that Vk = 1,2,. .d.z p*(dt,d z ) = jTk (dt,d z ) - p, where pk(dt, d z ) are Poisson martingale

I-,

-- -

measures on (a,5,P ) ; and the same conclusions hold true for the limits F(dt, d z ) and ij'(dt,dz).

C.3 Convergence of Random Processes and Stochastic Integrals

413

Proof. 1). Write 3: = a ( x ~ , w : , < ~ L: ; st ) , = a ( Z : , G ~ , ~ L: ~ ;t )s, and gt = ~ ( L , G , ?s ,5; t ) . Since by assumption for each n, { w : , ~ ) ~ ~ , is a B M , so WE - w c , t L: t l < t2, is independent of g, and w? - w:, 0 s < t , is Guassian N(0,t - s). However, {GFk)t,o has the same finite-

<

eK, e'}

independent of that is, {GFk,

. So Gt"," - G T , t

5 tl < t2,is also and Gyk -G:k, 0 L: s < t , is also Guassian N(0, t - s);

distributions as that of

wz>o

-L,o

also is a BM. Notice that G;* -+ u?,,in probability t>o P, as Ic -+ ca. Hence the same conclusion holds true for

2) is proved in a similar fashion. w Lemma 400 Assume that pn(dt,dz) n = 1,2, ...,are the Poisson counting measures with the same compensator ~ ( d z ) d=t A d t such that lzldfl , , qn(dt,dz) = pn(dt, dz) - ~ ( d z ) d n t , = 1,2, .... Set zpn(ds,dz),n = 172, ..-. C: = J; &lrl zqn(ds,dz)+ J; If as n CQ 5; <:, in probability, for all t > 0, then 1) for any finite Bore1 measurable function f ( z ) and for any E > 0 as n--+cs f (z)p0(ds,d . 4 , in probabJity, J; &I>, f (z)pn(ds,dz) --+ J; for all t 2 0, where --+

--+

where qO(dt,dz) = pO(dt,dz) - *dt.; 3) zf 0 t 5 t + h, and xy --+ xt, in probability, as n --+ CQ, such that z y , xt are finite, then for any bounded jointly continuous function f (s,x , z ) as n --+ CQ, J:+h JlzI>, f ( t ,x;, z)pn(ds,d.4 ~ z I , , f ( t ,xt, zlpO(ds,dz), in probability.

<

+

f

Lemma 401 [1841 Assume that fn(t) satisfies the following conditions: for anyOIT N ) =0 1' l i m ~ -sup, + ~ P~( S U P ~Ifn(t)l F' for any E > 0 lim6+0 P(lfn(t)- fn(s)I > E ) = 0. Moreover, if as n -+ ca fn(t) --+ f ( t ) ,in probability,

414

Appendix C. Monotone Class Theorems. Convergence of Random Processes

where the stochastic integrals fn(t)dwn(t) and f (t)dw(t) exist, w,(t) and w ( t ) are r-dimensional BM1s, 0 5 T < oo is arbitrary given, and as n 4 co, wn(t) -+ w ( t ) , i n probability, then as n co S,T fn(t)dwn(t) -+ f (t)dw(t),in probability.

References

[I] Adams, R. A. (1975). Sobolev spaces. Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975. [2] Arnold, L. (1974). Stochastic Differential Equations: Theory and Applications. Wiley, New York. [3] Anulova, S.V. (1978). On process with Levy generating operator in a half-space. Izv. AH. SSSR ser. Math. 47:4, 708-750. (In Russian). [4] Anulova, S.V. (1982). On stochastic differential equations with boundary conditions in a half-space. Math. USSR-Izv. 18, 423-437. [5] Anulova, S.S. and Pragarauskas, H. (1977). On strong Markov weak solutions of stochastic equations. Lit. Math. Sb. XVII:2, 5-26. (In Russian). [6] Anulova, A., Veretennikov, A., Krylov, N.. Liptser, R. and Shiryaev, A. (1998). Stochastic Calculus. Encyclopedia of Mathemetical Sciences. Vol 45, Springer-Verlag. [7] Aase K.K. (1988). Contingent claims valuation when the security price is a combination of an Ito process and a random point process. Stochastic Process. Appl., 28: 185-220. [8] Bardhan I. and Chao X. (1993). Pricing options on securities with discontinuous returns. Stochastic Process. Appl., 48: 123-137.

416

References

[9] Barles G.. Buckdahn R , and Pardoux E. (1997). Backward stochastic differential eqnations and integral-partial differential equations. Stochastzcs and Stochastzcs Reports, 60: 67-83 [lo] Bensoussan A. (1982). Lectures on stochastic control. In: Mittler S.K. & Moro A. eds. Nonlinear Filtering and Stochastic Control. LNM, 972: 1-62. [ll]Bensoussan A. and Lions J.L. (1984). Impulse Control and QuasiVariational Inequalities. Gautheir-Villars.

[12] Bismut J. M. (1973). Theorie probabiiste du controle des diffusions. Mem. Amer. Math. Soc., 176: 1-30. [13] Billingsleys, P. (1968). Convergence Wiley and Sons? New York.

0.f

Probability Measwcs. John

[14] Black, F. and Scholes, M. (1971). The pricing of options and corporate liabilities. J. Polit. Econom. 8, 637-869. [l5] Chen S. and Yu X. (1999). Linear quatlratic optimal control: froln det,erministic to stochast,ic cases. In: Chen S., Li X., Yong J., Zhou X.Y. eds. Control o,f Distributed Parameter and Stochastic Sgstems. Kluwer Acad. Pub., Boston, 181-188. [16] Chen, Z. J. and Peng, S. G. (2000). A general downcrossing i~icyuality for g-marbingales. Statistics & Prob. Letters, 46, 169-175. [17] Chen, Z.J. and Wang, B. (2000). Infinite time interval BSDEs and the convergence of g-martingales, J. Austral. Math. Soc. (Ser A ) 69, 187-211. [18] Chen Z. J. (1997). Existence of solution to backward stochastic difFerential equation with stopping times. Chin. Bulletin, 42(2): 2379-2382. [19] Chow G. C. (1997). Dynamic Economics: Optimization by The Lagrange Method, NewYork: Oxford Univ. Press. [20] Cox J . C.. Ingersoll Jr. J . E.. and Ross S. A. (1985). An International general eq~~ililrirl~n model of asset priccs. Econ.ometricta, 53(2): 363384. [21] Clrrtain R . F. and Pritchard A. J . (1977). Rcnctional Anal~sisIN Modern Applied Mathematics. New York: Academic Press. [22] Cvitanic J. and Karatzas I. (1993). Hedging contingent claims with constrained portfolios. Annals 0.f Applied Probab.! 3: 652-681. [23] Cvitanic J , and Karatzas I. (1996). Backward SDE's with reflection and Dynkin galnes. Ann. Prohab. 24(4): 2021-2056.

References

417

[24] Darling R. & Pardoux E. (1997). Backward SDE with random time and applications to semi-linear elliptic PDE. Annal. Prob., 25: 11351159. [25] Dellacherie, C. and Meyer,, P.A. (1980). Probabilitts et Potentiel. Thtorie des Martingales. Hermann, Paris. [26] Doob, J.L. (1953). Stochastic Processes. John Willey & Sons, Inc. [27] Doob, J.L. (1983). Classical Potential Theory and Its Probabilistic Counterpart. Springer-Verlag. [28] Dunford, N. and Schwartz, J. T. (1958). Linear Operators - Part I: General Theory. Interscience Publishers, INC., New York. [29] Durrett, R. (1996). Stochastic Calculus : A Practical Introduction. CRC Press. [30] El Karoui N., Kapoudjian C., Pardoux E., Peng S. and Quenez M.C. (1997). Reflected solutions of backward SDE's and related obstacle problems for PDE's. The Annals of Probab., 25(2): 702-737. [31] El Karoui N, and Mazliak L. (eds.) (1997). Backward Stochastic Differential Equations. Addison Wesley Longman Inc. USA. [32] El Karoui N., Pardoux E., and Quenez M.C. (1997). Reflected backward SDEs and American options. In: Rogers L.C.G. & Talay D. e d ~ . Numerical Methods in Finance. Pub. Newton Inst., Cambridge Univ. Press, 215-231. [33] El Karoui N., Peng S., and Quenez M.C. (1997). Backward stochastic differential equations in finance. Math. Finance, 7, 1-71. [34] Elliott, R.J. (1982). Stochastic Calculus and Application. SpringerVerlag. N.Y. [35] Friedman, A. (1975). Stochastic Differential Equations and Applications. I. Academic Press. [36] Friedman, A. (1976). Stochastic Differential Equations and Applications. II. Academic Press. [37] Elliott, R.J. (1982). Stochastic Calculus and Application. SpringerVerlag. N.Y. [38] Gihman, 1.1. and Skorohod, A.V. (1979). The Theory of Stochastic Processes 111, Springer-Verlag, Berlin. [39] Gikhman, I. I .; Skorokhod, A. V. (1982). Stochastic differential equations and their applications. "Naukova Dumka", Kiev, 612 pp. (In Russian).

418

References

[40] Halmos, P.R. (1950). Measure Theory. Van Nostrand, New York. [41] Hamadene S. (1998). Reflected BSDEs and mixed game Problem. Stoch. Proc. Appl., 85(2): 177-188. [42] Hamadhe S. and Lepeltier J.P. (2000). Backward - forward SDE's and stochastic differential games. Stoch. Proc. Appl., 77: 1-15. [43] He S.W., Wang J.G. and Yan J.A. (1992). Semimartingale Theory and Stochastic Calculus. Science Press & CRC Press Inc. [44] Hida, T. (1980). Brownian motion. Springer-Verlag. [45] Hu, Y. and Peng, S. (1990). Maximum principle for semilinear stochastic evolution control systems. Stochastics and Stochastic Reports, 33: 159-180. [46] Hu, Y. and Peng, S. (1991). Adapted solution of a backward semilinear stochastic evolution equation. Stochastic Analysis and Applications, 9(4): 445-459. [47] Hu Y. and Peng S. (1995). Solution of forward-backward stochastic differential equations. Probab. Theory Relat. Fields, 103: 273-283. [48] Hu Y. and Yong J. (2000). Forward-backward stochastic differential equations with nonsmooth coefficients. Stoch. Proc. Appl. 87: 93-106. [49] Ikeda N. and Watanabe S. (1989). Stochastic Differential Equations and Diffusion Processes. North-Holland. [50] Ito, K. (1942). Differential equations determining Markov processes. Zenkoku Shijo Sugalco Danwakai, 244:1077,1352-1400. (In Japanese). [51] Ito, K. (1944). Stochastic integral. Proc. Imp. Acad. Tokyo, 20, 519524. [52] Ito, K. (1951). On Stochastic differential Equations. Mem. Amer. Math. Soc., 4. [53] Ito, K. (1951). On a formula concerning stochastic differentials. Nagoya Math. J. 3, 55-65. [54] Ito, K. (1972). Poisson point process attached to Markov processes. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 111, 225-239. Univ. California Press, Berkeley. [55] Jacobson D.H., Martin D.H., Pachter M. and Geveci T. (1980). Extensions of Linear-Quadratic Control Theory. LNCIS 27, SpringerVerlag.

References

419

[56] Jacod, J. (1979). Calcul Stochastique et Problemes de Martingales. L. N. Math. 714. 1571 Kallianpur, G. (1980). Stochastic Filtering Theory. Springer-Verlag. [58] Karatzas I., Lehoczky J. P., Shreve S. E. and Xu G. L. (1991). Martingale and duality methods for utility maximization in incomplete market. SIAM J. Control Optim., 29(3): 702-730. 1591 Karatzas I. and Shreve S.E. (1987). Brownian Motion and Stochastic Calculus. Springer-Verlag. 1601 Karatzas I. and Shreve S.E. (1998). Methods of Mathematical Finance. Springer-Verlag. 1611 Klebaner, F. C. (1998). Introduction to stochastic calculus with applications. Imperial College Press, London. [62] Kohlmann M. (1999). Reflected forward backward stochastic differential equations and contingent claims. In Chen S., Li X., Yong J., Zhou X.Y. eds. Control of Distributed Parameter and Stochastic Systems. Kluwer Acad. Pub., Boston, 223-230. 1631 Krylov, N. V. (1974). Some estimates on the distribution densities of stochastic integrals. Izv. A.N. USSR, Math. Ser., %:I, 228-248. (In Russian). 1641 Krylov N.V. (1980). Controlled D i f i i o n Processes. Springer-Verlag. [65] Krylov N. V. and Paragarauskas, H. (1980). On the Bellman equations for uniformly non-degenerate general stochastic processes. Liet. Matem. Rink., XX: 85-98. (In Russian). 1661 Kunita, H. and Watanabe, S. (1969). On square integrable martingales. Nagoya Math. J. 30, 209-245. 1671 Kunita H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Univ. Press, World Publ. Corp. 1681 Ladyzenskaja 0 . A., Solonnikov V. A. and Uralceva N. N. (1968). Linear and Quasilinear Equations of Parabolic Qpe. Translation of Monographs 23, AMS Providence, Rode Island. 1691 Lamberton, D. and Lapeyre, B. (1996). Introduction to stochastic calculus applied to finance. Chapman & Hall, New York. 1701 Le Gall, J.F. (1983). Applications du temps local aux equations differentielles stochastiques unidimensionnelles. Lect. Notes Math. 986, Springer-Verlag, 15-31.

420

References

[71] Le Gall, J.F. (1984). One- dimensional stochastic differential equations involving the local time of the unknown processes. Lect. Notes Math. 1095, Springer-Verlag. 1721 Lepeltier J.P. and San Martin J . (1997). Backward stochastic differential equations with continuous coefficient. Statistics & Probab. Letters, 32: 425-430. [73] Li, C. W., Dong, Z. and Situ, R. (Situ Rong). (2002). Almost sure stability of linear stochastic differential equations with jumps. Probab. Theory Related Fields 123(1), 121-155. 1741 Lindvall, T . (1973). Weak convergence of probability measures and random functions in the function space D[O, 00). AppLProbab., 10, 109-121. [75] Liptser, R.S. and Shiryaev. (1989). Theory of Martingales. Kluwer Academic Publishers. Boston. 1761 Liptser, R.S. and Shiryaev. (2001). Statistics of Random Processes, I, II. 2nd ed. Springer-Verlag. N.Y. [77] Li Xiaojun. (1993). Maximum principle of stochastic evolution e q u a tion. Chin. Ann. Math. 14A(6): 647-653. [78] Loeve, M. (1977). Probability Theory. Springer-Verlag, New York. [79] Ma J., Protter P. and Yong J . (1994). Solving forward-backward stochastic differential equations explicitly - a four step scheme. Probab. Theory Relat. Fields, 98: 339-359. [80] Ma J . and Yong J. (1995). Solvability of forward-backward SDEs and the nodal set of Hamilton-Jacobi-Bellman equations. Chinese Ann. of Math. 16B(3): 279-298. [81] Ma J. and Yong J. (1997). Adapted solution of a degenerate backward SPDE with applications. Stoch. Proc. Appl., 70: 59-84. [82] Ma J . and Yong J. (1999). On linear degenerate backward stochastic partial differential equations. Prob. Th. 0 Relat. Fields, 113(2): 135170. 1831 Ma J. and Yong J. (1999). Forward - Backward Stochastic DiSferential Equations and their Applications. LNM 1702, Springer-Verlag. 1841 Malliavin, P. and Thalmaier, A. (2003). Stochastic calculus of uariations i n mathematical finance. Springer- Verlag.

References

421

[85] Mao X. (1995). Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients. Stoch. Proc. Appl., 58: 281292. [86] Marhno S.'(1976). The first boundary problem for integral-differential equations. Theory of Stochastic Processes 4, Kiev, 66-72. (In Russian). [87] Mckean, H. (1969). Stochastic Integrals. Academic, New York. [88] McShane, E. J. (1974). Stochastic calculus and stochastic models. Academic Press, New York. [89] Menaldi, J.L. and Robins, M. (1985). Reflected diffusion processes with jumps. The Annals of Probab. 13:2, 319-342. [90] Merton R. (1971). Optimal consumption and portfolio rules in a continuous time model, Journal of Economic Theory, 3: 373-413. [91] Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous. J. Finan. Econom. 3, 125-144. [92] MBtivier, M. and Pellaumail, J. (1980). Stochastic Integration. Academic, New York. [93] Meyer, M. (2001). Continuous stochastic calculus with applications to finance. Chapman & Hall/CRC. [94] Meyer P.A. (1966). ProbabilitCs et Potentiel, Hermann, Paris. [95] Meyer P.A. (1976). Un Cours sur les integrales stochasticques. Lect. Notes Math., 511, Springer. [96] Meyer P.A. (1989). A Short presentation of stochastic calculus. In: Stochastic Calculus i n Manifolds, M. Emery (ed). Springer-Verlag. [97] Modeste N'zi. (1997). Multivalued backward stochastic differential equations with local Lipschitz drift. Stochastics and Stochastics Reports, 60: 205-218. [98] Musiela, M. and Rutkowski. (1997). Martingale Methods i n Financial Modelling. Springer-Verlag. [99] Novikov A.A. (1973). On moment inequalities and identities for stochastic integrals. Proc. Second Japan-USSR Symp. Prob. Theor., Lect. Notes in Math., 330, Springer-Verlag, 333-339.

[loo] Bksendal, B. (2003). Stochastic Differential Equations: A n Introduction with Applications. 6th ed. Springer-Verlag, N.Y.

422

References

[loll Pardoux E. (1979). Stochastic partial differential equations and filtering of diffusion processes. Stochastic, 9: 127-167.

I1021 Pardoux E. and Peng S. (1990). Adapted solution of a backward stochastic differential equation, Systems €4 Contr. Lett., 14: 55-61. [I031 Pardoux E. and Peng S. (1992). Backward stochastic differential equations and quasi-linear parabolic partial differential equations. Lect. Notes i n CIS, 176, Springer, 200-21 7. [I041 Pardoux E. and Rbcanu A. (1988). Backward stochastic differential equations with subdifferential operator and related variational inequalities. Stoch. Proc. Appl., 76(2): 191-215. [I051 Pardoux E. and Veretennikov A. Yu. (1997). Averaging of backward stochastic differential equations, with application to semi-linear PDE's. Stochastics and Stochastics Reports, 60: 255-270. [I061 Pardoux E. and Zhang S. (1998). Generalized BSDE's and nonlinear Neumann boundary value problems. Probab. Th. Relat. Fields, 110: 535-558. I1071 Peng S. (1990). A general stochastic maximum principle for optimal control problems. SIAM J. Control and Optimization, 28(4): 966979. [108] Peng S. (1991). Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics and Stochastics Reports, 32: 61-74. [log] Peng S. (1992). A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation. Stochastics and Stochastics Reports, 38: 119-134. [I101 Peng S. (1992). Stochastic Hamilton-Jacobi-Bellman Equations. SIAM J. Control 63' Optim., 30(2): 284-304. [ I l l ] Peng S. (1993). Backward stochastic differential equations and applications to optimal control. Appl. Math. Optim., 27: 125-144. 11121 Peng S. (1994). Backward stochastic differential equations. CIMPA School: Lect. Notes on Stochastic Calculus and Applications to Mathematical Finance, Beijing. [I131 Peng S. (1994). Backward stochastic differential equations and exact controllability of stochastic control systems. Progress i n Natuml Science 4 (9): 274-284. [I141 Peng S. (1997). BSDE and related g-expectation. Pitman Research Notes i n Mathematics Series 364: 141-159.

References

423

[115] Peng S. (1999). Open problems on backward stochastic differential equations. In Chen S., Li X., Yong J., Zhou X.Y. eds. Control of Distributed Parameter and Stochastic Systems. Kluwer Acad. Pub., Boston, 265-273. [I161 Peng S. (1999). Monotonic limit theorem of BSDE and its application to Doob-Meyer decomposition theorem. Probab. Theory Relat. Fields, 113: 473-499. [I171 Peng S. (2000). Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions. Stoch.Proc. Appl. 88(2): 259-290. [118] Peng S. and Shi Y. 2000. Infinite horizon forward-backward stochastic differential equations. Stoch.Proc. Appl. 85(1): 75-92. [I191 Peng S. and Wu Z. (1999). Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J. Control. Optim., 37(3): 825-843. [120] Protter, P. (1990). Stochastic Integral and Differential Equations - A New Approach. Springer-Verlag. [I211 Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer-Verlag. [122] Mikosch, T. (1998). Elementary stochastic calculus with finance i n view. World Scientific Publ. Singapore. [I231 Rogers, L.C. and Williams, D. (1987). Diffusions, Markov Processes and Martingales. V.2: Ito Calculus. John Wiley & Sons, New York. [124] Roman, S. (2004). Introduction to the Mathematics of Finance. Springer-Verlag. [125] Saisho, Y.S. (1987). Stochastic differential equation for multidimensional domain with reflecting boundary. Probability Th.& Rel. Fields 74, 455-477. [I261 Shiryayev, A. N. (1996). Probabilty. 2nd edition. Springer-Verlag. New York. [127] Shiryayev, A. N. (1999). Essentials of Stochastic Finance. World Scientific, Singapore. 11281 Shreve, S. E. (2004). Stochastic Calculus for Finance 11: ContinuopusTime Models. Springer-Verlag. [129] Situ Rong. (1983). Non-convex stochastic optimal control and maximum principle. IFAC 3rd Symposium, Control of Distributed Parameter System 1982, Eds. J.P. Babary and L.L. Letty, Pergamon Press, 401-407.

424

References

[130] Situ Rong. (1984). An application of local time to stochastic differential equations in m-dimensional space. ACTA Scientiarum Natur. Univ. Sunyatseni, 3, 1-12. [131] Situ Rong. (1984). On strong solutions and stochastic pathwise BangBang control for a non-linear system. IFAC 9th Triennial Word Congress, Hungary, 1984, Proceedings uol. 3, Large-Scale Systems., Decision Making, Mathematics of Control, Ed. A. Titili, etc.14331438.(1985). [132] Situ Rong. (1985). On strong solution, uniqueness, stability and comparison theorem for a stochastic system with Poisson jumps. In: Kappel F., Kunisch K., & Schappacher W. eds. Distributed Parameter System. Lect. Notes i n Contr. Inform. Scienc. 75, Springer, 352-381. [I331 Situ Rong. (1985). Strong solutions and pathwise Bang-Bang control for stochastic system with Poisson jumps in 1-dimensional space. Proceedings of 5th National Conference on Control Theory and its Applications, 319-321. [I341 Situ Rong. (1986). Strong solutions and pathwise Bang-Bang control for multidimensional non-linear stochastic system with jumps. 4th IFAC Symposium of Control of Distributed Parameter System, LA. Calif. USA, June, 1986. [135] Situ Rong. (1987). On weak, strong solutions and pathwise BangBang controls for non-linear degenerate stochastic systems. In: Shinha N.K. and Telksnys L. eds. IFAC Stochastic Control, USSR, 1986. Proceedings Series (1987), No.2, Pergamon Press, 145-150. [136] Situ Rong. (1987). A non-linear filtering problem and its application. Chin. Ann. Math. 8B(3), 296-310. [I371 Situ Rong. (1987). Strong solutions, pathwise control for stochastic nonlinear system with Poisson jumps in n-dimensional space and its application to parameter estimation. Proc. of Sino-American Statistical Meeting, Beijing, 389-392. [138] Situ Rong. (1987). Local time, Krylov estimation, Tanaka formula, strong comparison theorem and limit theorem for 1-dimensionalsemimartingales. Acta Sci. Naturali Univ. Sunyatseni 10, 14-26. [139] Situ Rong. (1987). On non-linear Voterra Ito stochastic system and pathwise control. Proc. of National Annual Conf. on Control Theory and its Applications, China, 341-344. [140] Situ Rong (1990). On existence and stability of solutions for reflecting stochastic differential equations with rectangular boundary. Acta Scientiarum Naturalium Universitatis Sunyatseni 29:3, 1-11.

References

425

[I411 Situ Rong (1990). Stochastic differential equations with jump reflection on half space and pathwise optimal control. Proc. of National Annual Conference on Control Theory and its Applications, 679-682. [142] Situ Rong (1990). Strong solutions and pathwise control for nonlinear stochastic system with Poisson jumps in n-dimensional space. Chinese Ann. Math., Ser. B , 11:4, 513-524. [I431 Situ Rong. (Rong Situ). (1991). Theory and Application of stochastic differential equations in China. Contemporary Math. 118, Amer. Math. Soci., 263-280. [I441 Situ Rong (1991). Strong solutions and optimal control for stochastic differential equations in duals of nuclear spaces. Lect. Notes in Control and Inform. Sci. 159, Control Theory of Distributed Parameter Systems and Applications, Springer-Verlag, 144-153. [I451 Situ Rong (1991). Reflecting stochastic differential equations with jumps and stochastic population control. In Control Theory, Stochastic Analysis and Applications, eds. S. Chen and J . Yong, World Scientific, Singapore, 193-202. [146] Situ Rong (1991). Stochastic population control problem on white noise measure space. International Conf. on White Noise Analysis and Applications , May Nagoya, Japan. [I471 Situ Rong (1993). Non-linear filtering for reflecting stochastic differential systems. Proc. of National Annual Conf. on Control Theory and its Appl., Ocean Press, 210-213. [I481 Situ Rong (1994). Anticipating stochastic differential equations and optimal control. Proc. of CSIAM Conf. on Systems and Control, Shangdong, China, 64-69. [I491 Situ Rong (1995). On existence of strong solutions and optimal controls for anticipating stochastic differential equations with nonLipschitzian coefficients. Proc. of 1995 Chinese Control Conference, Chinese Sci.& Tech. Press, 537-540. [I501 Situ Rong. (1995). On backward stochastic differential equations with jumps and with non-Lipschitzian coefficients and applications. The 23rd Conference on Stochastic Processes and their Applications, 1923 June, Singapore. [151] Situ Rong (1996). On strong solutions and pathwise optimal control for stochastic differential equations in Hilbert space. Proc. of 1996 Chinese Control Conference, Shangdong, 396-399.

426

References

[I521 Situ Rong (1996). On comparison theorem of solutions to backward stochastic differential equations with jumps and its applications. Proc. of the 3rd CSIAM Conf. on Systems and Control, LiaoNing, China, 46-50. [I531 Situ Rong. (1997). On solutions of backward stochastic differential equations with jumps and applications. Stochastic Process. Appl., 66: 209 -236. [I541 Situ Rong (1997). One kind of solution for optimal consumption to financial market with jumps by Lagrange multipliers in stochastic control. Proc. of 1997 Chinese Control Conference, Wuhan Press, 1205-1209. [I551 Situ Rong (1997). Optimization for financial market with jumps by Lagrange's method. The 2nd International Symposium of Econometrics on Financial Topics, Guangzhou. [156] Situ Rong (1998). FBSDE and financial market with applications to stochastic control. Proceedings of 1998 Chinese Control Conference, National Defense University Press, Beijing, 301-305. [I571 Situ Rong (1998). On solutions of reflecting backward stochastic differential equations with jumps. Conference on Mathematical Finance, Oct. Shanghai, China. [I581 Situ Rong (1998). Comparison theorem of solutions to BSDE with jumps and viscosity solution to a generalized Hamilton-JacobiBellman equation. Conference on Control of Distributed Parameter and Stochastic Systems, June 19-22, Hangzhou, China. [159] SITU Rong. (R. Situ) (1999). Comparison theorem of solutions to BSDE with jumps and viscosity solution to a generalized HamiltonJacobi-Bellman equation. In Chen S., Li X., Yong J., Zhou X.Y. eds. Control of Distributed Parameter and Stochastic Systems. Kluwer Acad. Pub., Boston, 275-282. [I601 Situ Rong (1999). On solutions of FBSDE with jumps and with nonLipschitzian coefficients. Workshop on Infinite Dimensional Analysis and Dirichlet Forms, March 28 - April 1, WuHan, China. [I611 Situ Rong. (1999). On comparison theorems and existence of solutions to backward stochastic differential equations with jumps and with discontinuous coefficients. The 26th Conference on Stochastic Processes and their Applications, 14-18, June, Beijing. [I621 Situ Rong. (1999). Optimization for financial market with jumps by Lagrange's method. Pacific Economic Review, 4(3): 261-275.

References

427

[163] Situ Rong. (1999). On solutions of backward stochastic differential equations with jumps and stochastic control. Intent. Workshop on Markov Processes & Controlled Markov Chains, Changsha, Hunan, China, 22-28 Aug. [164] Situ Rong. (1999). Option Pricing in Mathematical Financial Market with Jumps and Related Problems. Colloquium held in Math. Institute of Vietnam Academy, Dec. 1999, Hanoi, Vietnam. [165] Situ Rong. (2000). Reflecting Stochastic Differential Equations with Jumps and Applications. Chapman & Hall/CRC Research Notes in Mathematics 408, Chapman & Hall/CRC. [166] Situ Rong. (2000). Backward Stochastic Differential Equations with Jumps and Applications. Guangadong Science and Technology Press, Guangzhou. [I671 Situ Rong. (2002). Option Pricing in Mathematical Financial Market with Jumps and Related Problems. Vietnam Journal of Mathematics 30(2), 103-112. [I681 Situ Rong. (2002). On solutions of backward stochastic differential equations with jumps and with non-Lipschitzian coefficients in Hilbert spaces and stochastic control. Statistics and Probability Letters 60(3), 279-288. [I691 Situ Rong. (2002). On solutions of backward stochastic differential equations with jumps and stochastic control. In Hou Z.T. and Jerzy Filar et a1 eds. Markov Processes and Controlled Markov Chains. 331340, Kluwer Acad. Pub., Boston. [I701 Situ Rong (2002). On solutions of reflecting backward stochastic differential equations with jumps in finite and infinite horizon. The 3rd Conference on "Backward Stochastic Differential Equations and Applications" - Satellite Conference of International Congress Mathematics 2002, Aug 29 - Sept. 2, 2002, Weihai, China. [171] Situ Rong. (2004). Hedging contingent claims in financial market with jump perturbation and Backward Stochastic Differential Equations. Proceedings of the International Conference "Abstract and Applied Analysis", eds. N.M. Chuong, L. Nirenberg, and W. Tutschke, Hanoi, Vietnam, 13-17 August 2002, Word Scientific, pp 515-532. [172] Situ Rong & W.L.Chen. (1992). Existence of solutions and optimal control for reflecting stochastic differential equations with applications to population control theory. Stochastic Analysis and Applications 10:1, 45-106.

428

References

[173] Situ Rong and Hwang Min. (2001). On solutions of backward stochastic differential equations with jumps in Hilbert space (I). Acta Sci. Naturali Univ. Sunyatseni. 40(3), 1-5. [174] Situ Rong and Hwang Min. (2001). On solutions of backward stochastic differential equations with jumps in Hilbert space (11). Acta Sci. Naturali Univ. Sunyatseni. 4O(4), 20-23. [175] SITU Rong and HUANG Wei. (2004). On solutions of BSDEs with jumps and with coefficients having a quadractic growth (I). Acta Sci. Naturali Univ. Sunyatseni. 43(6), 48-51. [176] SITU Rong and HUANG Wei. (2005). On solutions of BSDEs with jumps and with coefficients having a quadractic growth (11). Acta Sci. Naturali Univ. Sunyatseni. 44(1), 1-5. 11771 SITU Rong and WANG Yueping. (2000). On solutions of BSDE with jumps, with unbounded stopping times as terminals and with nonLipschitzian coefficients, and probabilistic interpretation of solutions to quasi-linear elliptic integro-differential equations. Applied Math. f4 Mechanics, 21 (6): 659-672. [I781 Situ Rong and Xu Huanyao. (2001). Adapted solutions of backward stochastic evolution equations with jumps on Hilbert space (I). Acta Sci. Naturali Univ. Sunyatseni. 40(1), 1-5. [179] Situ Rong and Xu Huanyao. (2001). Adapted solutions of backward stochastic evolution equations with jumps on Hilbert space (11). Acta Sci. Naturali Univ. Sunyatseni. 40(2), 1-5. [NO] Situ Rong and Yang Yan. (2003). Upcrossing inequality and Strong g-supermartingales (I). Acta Sci. Naturali Univ. Sunyatseni 42(5), 1-5. [I811 Situ Rong and Yang Yan. (2003). Upcrossing inequality and Strong g-supermartingales (11). Acta Sci. Naturali Univ. Sunyatseni. 42(6), 1-3. [182] Situ Rong and Zeng A.T. (1999). On solutions of BSDE with jumps and with unbounded stopping terminals I. Acta Sci. Naturali Univ. Sunyatseni, 38(2), 1-6. [I831 Situ Rong and Zeng, A.T. (1999). On solutions of BSDE with jumps and with unbounded stopping terminals 11. Acta Sci. Naturali Univ. Sunyatseni. 38(4), 1-5. [I841 Skorohod, A.V. (1965). Studies i n the Theory of Random Processes. Addison-Wesley, Reading, Massachusetts.

References

429

[I851 Skorohod, A. V. (1991). Random processes with independent increments. Translated from the second Russian edition by P. V. Malyshev. Mathematics and its Applications (Soviet Series), 47. Kluwer Academic Publishers Group, Dordrecht, 1991. [186] Steele, J. M. (2001). Stochastic calculus and financial applications. Springer-Verlag. 11871 Stroock, D.W. and Varadhan, S.R.S. (1979). Multidimensional Diffusion Processes. Springer-Verlag. N.Y. [I881 Tang S. and Li X. (1994). Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control and 0 p timization, 32(5): 1447-1475. [189] Teo, K. L. and Wu, Z. S. (1984). Computational methods for optimizing distributed systems. Mathematics in Science and Engineering, 173. Academic Press, Inc., Orlando, FL, 1984. [I901 Veretennikov, A.Ju. (1981). On the strong and weak solutions of onedimensional stochastic equations with boundary conditions. Theor. Prob. Appl., 26:4, 685-700. (In Russian). [I911 Watanabe, S. and Nair, M. G. (1984). Lectures on Stochastic dzfferential Equations and Malliavin Calculus. Springer-Verlag. [I921 Yeh, J. (1973). Stochastic Processes and Wiener Integral. Marcel Dekker Inc. N.Y. [I931 Yin Juliang and Situ Rong. (2003). On solutions of forward-backward stochastic differential equations with Poission jumps. Stochastic Analysis and Applications 21:6, 1419-1448. [194] Yin Juliang and Situ Rong. (2004). Existence of solutions for forwardbackward stochastic differential equations with jumps and nonLipschitzian coefficients. J. Math. Res. Exposition 24:4, 577-588. [195] Yong J. (1997). Finding adapted solution of forward-backward stochastic differential equations - method of continuation. Prob. Th. & Relat. Fields, 107: 537-572. [196] Yong J . and Zhou X.Y. (1999). Stochastic Controls. Hamiltonian Systems and HJB Equations. Appl. of Math. 43, Springer-Verlag. [197] Yu, J.Y., Guo, B.Z. and Zhu, G.T. (1987). The control of the semidiscrete population evolution system. J. Syst. & Math. Scien. 7:3, 214-219. (In Chinese).

430

References

I1981 Zheng, W. (1980). A note on the convergence of sequence of conditional expectations of random variables. 2. Wahrsch. Verw. Gebiete, 53:3, 291-2

Index

a s . , 390 arbitrage-free market, 253 backward stochastic differential equation (BSDE), 206 solution, 212 Black-Scholes formula for continuous market, 225 for market with jumps, 233 Brownian Motion nowhere differentiable, 42 stationary strong Markov property, 68 Brownian Motion (BM) definition, 39 standard, 42 comparison theorem for solutions of BSDE, 243 for solutions of population SDE in components, 366 for solutions of RSDE in components, 367 for solutions of SDE in 1- dimensional space, 292

in components, 293 contingent claim, 206 hedging, 222 convergence almost surely (as.), 409 in probability P, 409 weakly, 409 Doob's stopping theorem, 8, 19 Doob-Meyer decomposition class (D), 20 class (DL), 20 theorem, 22 Fatou's lemma for conditional expectation, 396 for expectation, 396 Feynman-Kac formula by solution of BSDE, 273 by solution of SDE, 274 field U-, 389 filtering 1'inear continuous, 194

432

Index

more general continuous, 200 optimal, 193 non-linear, 184 filtration a-field, 4 Girsanov type theorem, 89,96,97 for SDE with functional coefficients, 173 Gronwall's inequality, 82, 217 stochastic, 115 Hamilton-Jacobi-Bellman equation, 278 independence of BM and Poisson point process, 64 independent random variables, 392 independent system, 392 integration by parts, 63 Ito's formula for continuous martingale, 57 for continuous semi-martingale, 58 for semi-martingale with jumps, 58 d-dimensional case, 62 Ito's integral w.r.t Brownian Motion, 43 Jensen's inequality, 397 Kalman-Bucy filtering equation in 1-dimensional case, 194 in multi-dimensional case, 196 Kolmogolov's theorem, 398 Krylov type estimate in 1-dimensional space, 129 in d-dimensional space, 131 Kunita-Watanabe's inequality, 52 Lagrange functional, 282 method, 279 multiplier, 280, 282

Lebesgue's dominated convergence theorem for conditional expectation, 396 for expectation, 396 Levi's theorem, 15 local time, 124 occupation density formula, 126 martingale, 4 inequality, 8, 17 limit theorem, 11, 17 locally square integral, 35 representation theorem, 69 for SDE with functional coefficients, 177 square integrable, 35 with continuous times, 17 with discrete time, 8 maximum principle adjoint equations, 380 for SDE, 379 measure characteristic, 29 counting, 28 compensator (of), 33 random, 26 Poisson, 27, 28 Weiner, 41 monotone class thoerems, 407 Novikov's theorem, 174 optimal consumption, 262, 279 control, 264 option, 207 pricing, 208 by partial differential equation approach, 227 by probability approach, 225 P-a.s., 390 Pathwise Bang-Bang control in 1-dimensional space

Index

non-degenerate case, 312 partial- degenerate case, 316 in d-dimensional space non-degenerate case, 319 partial-degenerate case, 325 population SDE optimal control, 369 solution calculation, 372 convergence, 353 stability, 360 strong, 352 portfolio, 206 self-financed, 206 price equation bond, 205 stock, 206 probability measure, 389 space, 389 process St-adapted, 4 characteristic cross predictable, 49 predictable, 49 innovation, 181 integrable and increasing, 21 measurable, 6 progressive, 6 natural, 21 optional, 7 point, 28 St-adapted, 32 a-finite, 32 class (QL), 32 Poisson, 28 St-Poisson, 33 stationary Poisson, 29 Poisson, 29, 398 with intensity, 29 Poisson point stationary strong Markov property, 68 predictable, 7, 34 stationary Markov, 399

433

stationary strong Markov, 399 variational, 49 predictable quadratic, 49, 184 Weiner, 39 Radon-Nikodym derivative, 184, 187, 188 random process, 397 finite-dimensional distribution functions, 398 Gaussian, 397 random variable, 390 characteristic function, 391 distribution function, 391 Gaussian, 391 mathematical expectation, 390 conditional, 395 Normal, 391 Poisson, 395 variance, 390 random vector, 392 characteristic function, 392 distribution function, 393 Gaussian, 392 reflecting stochastic differential equation (RSDE), 332 solution pathwise uniqueness, 334 strong, 334 uniqueness in law, 335 weak, 334 right continuous with left limit (RCLL), 17 semi-martingale continuous, 58 with jumps, 58 Skorohod metric, 401 problem, 335 theorem, 411 topology, 341 weak convergence technique, 145

434

Index

result of SDE from, 145 standard measurable space, 89 stochastic differential equation (SDE), 75 linear solution, 86 solution, 76 minimal, 306 pathwise uniqueness, 76 strong, 76 weak, 93 weak uniqueness, 94 with functional coefficient, 169 strong solution, 171 stochastic Fubini theorem, 127 stochastic integral w.r.t. Brownian Motion, 46, 48 w.r.t. counting measure, 32, 34 w.r.t. martingale, 49 w.r.t. martingale measure, 34 stochastic population control, 330 stopping time, 5 submartingale, 4 regular, 25 right-closed, 14 uniformly integrable, 14 with continuous time, 17 with discrete time, 8 supermartingale, 4 with continuous times, 17 with discrete time, 8 Tanaka type formula for RSDE in components, 364 for SDE d-dimensional case, 110 in 1-dimensionalspace, 116 in components, 121 uniform inner normal vector positive projection condition, 332, 335

uniformly integrable, 12 variational equation first order, 383 second order, 383 wealth process, 206 more general, 234 Yamada-Watanabe type theorem for RSDE, 348 for SDE, 104 Zakai's equation, 202