Taylor Approximations for Stochastic Partial Differential Equations
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Arnulf Jentzen Princeton University Princeton, New Jersey
Peter E. Kloeden Goethe University Frankfurt am Main, Germany
Taylor Approximations for Stochastic Partial Differential Equations
Society for Industrial and Applied Mathematics Philadelphia
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Copyright © 2011 by the Society for Industrial and Applied Mathematics. 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA. Trademarked names may be used in this book without the inclusion of a trademark symbol. These names are used in an editorial context only; no infringement of trademark is intended. Maple is a trademark of Waterloo Maple, Inc. Mathematica is a registered trademark of Wolfram Research, Inc. MATLAB is a registered trademark of The MathWorks, Inc. For MATLAB product information, please contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 USA, 508-647-7000, Fax: 508-647-7001,
[email protected], www.mathworks.com. Figure 3.1 and 6.1 used with permission from Springer. Figure 4.1 used with permission from Cambridge University Press. Figures 8.1-8.10 used with permission from the American Institute of Mathematical Sciences. Library of Congress Cataloging-in-Publication Data Jentzen, Arnulf. Taylor approximations for stochastic partial differential equations / Arnulf Jentzen, Peter E. Kloeden. p. cm. -- (CBMS-NSF regional conference series in applied mathematics ; 83) Includes bibliographical references and index. ISBN 978-1-611972-00-9 1. Stochastic partial differential equations. 2. Approximation theory. I. Kloeden, Peter E. II. Title. QA274.25.J46 2011 515'.353--dc23 2011029546
is a registered trademark.
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Contents Preface
xi
List of Figures 1
I 2
3
xiii
Introduction 1.1 Taylor expansions for ODEs . . . . . . . . . . . . . . . . . . . 1.1.1 Taylor schemes for ODEs . . . . . . . . . . . . . 1.1.2 Integral representation of Taylor expansions . . .
Random and Stochastic Ordinary Differential Equations Taylor Expansions and Numerical Schemes for RODEs 2.1 RODEs . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Equivalence of RODEs and SODEs . . 2.1.2 Simple numerical schemes for RODEs 2.2 Taylor-like expansions for RODEs . . . . . . . . . 2.2.1 Multi-index notation . . . . . . . . . . 2.2.2 Taylor expansions of the vector field . 2.3 RODE–Taylor schemes . . . . . . . . . . . . . . . 2.3.1 Discretization error . . . . . . . . . . 2.3.2 Examples of RODE–Taylor schemes .
1 1 3 4
7 . . . . . . . . .
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SODEs 3.1 Itô SODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Existence and uniqueness of strong solutions of SODEs . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Simple numerical schemes for SODEs . . . . . . 3.2 Itô–Taylor expansions . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Iterated application of the Itô formula . . . . . . . 3.2.2 General stochastic Taylor expansions . . . . . . . 3.3 Itô–Taylor numerical schemes for SODEs . . . . . . . . . . .
9 9 10 12 13 14 14 15 17 22 25 25 26 27 29 29 31 32
vii
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viii
Contents 3.4 3.5
4
II 5
6
7
Pathwise convergence . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Numerical schemes for RODEs applied to SODEs Restrictiveness of the standard assumptions . . . . . . . . . . . 3.5.1 Counterexamples for the Euler–Maruyama scheme
Numerical Methods for SODEs with Nonstandard Assumptions 4.1 SODEs without uniformly bounded coefficients . . . . . . 4.2 SODEs on restricted regions . . . . . . . . . . . . . . . . . 4.2.1 Examples . . . . . . . . . . . . . . . . . . . . 4.3 Another type of weak convergence . . . . . . . . . . . . .
. . . .
. . . .
Stochastic Partial Differential Equations
35 36 37 37 43 43 44 46 47
51
Stochastic Partial Differential Equations 5.1 Random and stochastic PDEs . . . . . . . . . . . . . . . . 5.1.1 Mild solutions of SPDEs . . . . . . . . . . . . 5.2 Functional analytical preliminaries . . . . . . . . . . . . . 5.2.1 Hilbert–Schmidt and trace-class operators . . 5.2.2 Hilbert space valued random variables . . . . 5.2.3 Hilbert space valued stochastic processes . . . 5.2.4 Infinite dimensional Wiener processes . . . . . 5.3 Setting and assumptions . . . . . . . . . . . . . . . . . . . 5.4 Existence, uniqueness, and regularity of solutions of SPDEs 5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Finite dimensional SODEs . . . . . . . . . . . 5.5.2 Second order SPDEs . . . . . . . . . . . . . . 5.5.3 Fourth order SPDEs . . . . . . . . . . . . . . 5.5.4 SPDEs with time-dependent coefficients . . .
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
53 53 54 55 55 56 56 56 57 58 60 60 60 65 67
Numerical Methods for SPDEs 6.1 An early result . . . . . . . . . . . . . . . . . . 6.2 Other results . . . . . . . . . . . . . . . . . . . 6.3 The exponential Euler scheme . . . . . . . . . . 6.3.1 Convergence . . . . . . . . . . . . 6.3.2 Numerical results . . . . . . . . . 6.3.3 Restrictiveness of the assumptions
. . . . . .
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69 70 71 73 74 75 76
Taylor Approximations for SPDEs with Additive Noise 7.1 Assumptions . . . . . . . . . . . . . . . . . . . . . 7.1.1 Properties of the solutions . . . . . . . 7.2 Autonomization . . . . . . . . . . . . . . . . . . . 7.3 Examples . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Semigroup generated by the Laplacian
. . . . .
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. . . . .
79 80 83 88 91 91
. . . . . .
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Contents
7.4 7.5 7.6
7.7
8
ix 7.3.2 The drift as a Nemytskii operator . . . . . . . . 7.3.3 Stochastic process as stochastic convolution . . 7.3.4 Concrete examples of SPDEs with additive noise Taylor expansions . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Integral operators . . . . . . . . . . . . . . . . Abstract examples of Taylor expansions . . . . . . . . . . . Examples of Taylor approximations . . . . . . . . . . . . . . 7.6.1 Space–time white noise . . . . . . . . . . . . . 7.6.2 Taylor approximations for a nonlinear SPDE . . 7.6.3 Smoother noise . . . . . . . . . . . . . . . . . Numerical schemes from Taylor expansions . . . . . . . . . 7.7.1 The exponential Euler scheme . . . . . . . . . . 7.7.2 A Runge–Kutta scheme for SPDEs . . . . . . .
Taylor Approximations for SPDEs with Multiplicative Noise 8.1 Heuristic derivation of Taylor expansions . . . . . . . . . . 8.2 Setting and assumptions . . . . . . . . . . . . . . . . . . . 8.2.1 Stochastic heat equation . . . . . . . . . . . . 8.3 Taylor expansions for SPDEs . . . . . . . . . . . . . . . . 8.3.1 Integral operators . . . . . . . . . . . . . . . 8.3.2 Derivation of simple Taylor expansions . . . . 8.3.3 Higher order Taylor expansions . . . . . . . . 8.4 Stochastic trees and woods . . . . . . . . . . . . . . . . . 8.4.1 Stochastic trees and woods . . . . . . . . . . 8.4.2 Construction of stochastic trees and woods . . 8.4.3 Subtrees . . . . . . . . . . . . . . . . . . . . 8.4.4 Order of stochastic trees and woods . . . . . . 8.4.5 Stochastic woods and Taylor expansions . . . 8.5 Examples of Taylor approximations . . . . . . . . . . . . . 8.5.1 Abstract examples of Taylor approximations . 8.5.2 Application to the stochastic heat equation . . 8.5.3 Finite dimensional SODEs . . . . . . . . . . . 8.6 Numerical schemes for SPDEs . . . . . . . . . . . . . . . 8.6.1 The exponential Euler scheme . . . . . . . . . 8.6.2 An infinite dimensional analogue of Milstein’s scheme . . . . . . . . . . . . . . . . . . . . . 8.6.3 Linear-implicit Euler and Crank–Nicolson schemes . . . . . . . . . . . . . . . . . . . . 8.6.4 Global and local convergence orders . . . . . 8.6.5 Numerical simulations . . . . . . . . . . . . . 8.7 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 Proof of Lemma 8.3 . . . . . . . . . . . . . . 8.7.2 Proof of Lemma 8.5 . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . .
92 93 94 96 98 104 114 114 116 121 121 123 124
. . . . . . . . . . . . . . . . . . .
127 127 130 133 135 136 138 139 140 140 142 145 146 147 149 149 151 152 154 155
. . 156 . . . . . .
. . . . . .
157 157 158 160 160 163
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x
Contents 8.7.3 8.7.4
A
Some additional lemmas . . . . . . . . . . . . . . 164 Proof of Theorem 8.4 . . . . . . . . . . . . . . . 171
Regularity Estimates for SPDEs A.1 Some useful inequalities . . . . . . . . . . . . . . . . . . A.1.1 Minkowski’s integral inequality . . . . . . . A.1.2 Burkholder–Davis–Gundy-type inequalities A.2 Semigroup term . . . . . . . . . . . . . . . . . . . . . . A.3 Drift term . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Diffusion term . . . . . . . . . . . . . . . . . . . . . . . A.5 Existence and uniqueness . . . . . . . . . . . . . . . . . A.6 Regularity . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
173 174 174 175 177 182 193 204 206
Bibliography
209
Index
219
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Preface The numerical approximation of stochastic partial differential equations (SPDEs), specifically, stochastic evolution equations of the parabolic or hyperbolic type, encounters all of the difficulties that arise in the numerical solution of both deterministic PDEs and finite dimensional stochastic ordinary differential equations (SODEs) as well as many more due to the infinite dimensional nature of the driving noise processes. The state of development of numerical schemes for SPDEs compares with that for SODEs in the early 1970s. Most of the numerical schemes that have been proposed to date have a low order of convergence, especially in terms of an overall computational effort, and only recently has it been shown how to construct higher order schemes. The breakthrough for SODEs started with the Milstein scheme and continued with the systematic derivation of stochastic Taylor expansions and the numerical schemes based on them. These stochastic Taylor schemes are based on an iterated application of the Itô formula. The crucial point is that the multiple stochastic integrals which they contain provide more information about the noise processes within discretization subintervals, and this allows an approximation of higher order to be obtained. This theory is presented in detail in the monographs Kloeden & Platen [82] and Milstein [95]. There is, however, no such Itô formula for the solutions of stochastic PDEs in Hilbert spaces or Banach spaces (see Chapter 7 for more details). Nevertheless, it has recently been shown that Taylor expansions for the solutions of such equations can be constructed by taking advantage of the mild form representation of the solutions. Moreover, such expansions are robust with respect to the noise in the additive noise case, i.e., hold for other types of stochastic processes with Hölder continuous paths such as fractional Brownian motion. This book is based on recent work of the coauthors. Its style, contents, and structure follow the series of lectures given by the second author, Peter Kloeden, in August 2010 at the Illinois Institute of Technology in Chicago. The main difference from the lectures is the existence and uniqueness theorem in Chapter 5. Most of that chapter and the entire appendix were written by the first coauthor, Arnulf Jentzen.
xi
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xii
Preface
The book also includes new developments on numerical methods for random ordinary differential equations and SODEs, since these are relevant for solving spatially discretized SPDEs as well as in their own right. The focus is on pathwise and strong convergences. In finance mathematics, weak convergence is of primary interest, but strong convergence is nevertheless important, too, as an essential component of the multilevel Monte Carlo method introduced recently in Giles [35] (see also Heinrich [54]). Much of this book was written during an extended stay by the second author at the Isaac Newton Institute for Mathematical Sciences at the University of Cambridge during a special half year on SPDEs in the first half of 2010. The financial support and congenial working atmosphere are gratefully acknowledged. We also thank SIAM and the National Science Foundation for sponsoring and funding these CBMS lectures, as well as Jinqiao Duan, Igor Cialenco, and Fred J. Hicknell of the Illinois Institute of Technology for hosting the lectures and for their local organization. In particular, we thank Michael Tretyakov for carefully reading parts of the manuscript and Sebastian Becker for his assistance with the computations, figures, and LATEX. Arnulf Jentzen, Princeton
Peter Kloeden, Frankfurt am Main
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List of Figures 3.1
4.1
6.1
Stochastic Taylor trees for the stochastic Euler scheme (left) and the Milstein scheme (right). The multi-indices are formed by concatenating indices along a branch from the right back toward the root ∅ of the tree. Dashed line segments correspond to remainder terms. . . . . . . . . . (0.5)
34
(1.0)
Empirical distributions of K0.001 and K0.001 (sample size: N = 104 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
Mean-square error vs. computational effort as log-log plot with the function f (u) = 12 u. . . . . . . . . . . . . . . . . . . . . . .
77
7.1
Pathwise approximation error of the Taylor approximations (7.54)– (7.59) vs. t = t − 21 for different t ∈ [ 12 , 1] and one random realization of (7.18). . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.2
Mean square approximation error of the Taylor approximations (7.54)–(7.59) vs. t = t − 12 for different t ∈ [ 12 , 1] and (7.18). . . 120
8.1
Two examples of stochastic trees. . . . . . . . . . . . . . . . . . 141
8.2
The stochastic wood w0 in SW. . . . . . . . . . . . . . . . . . . 142
8.3
The stochastic wood w1 in SW. . . . . . . . . . . . . . . . . . . 143
8.4
The stochastic wood w2 in SW. . . . . . . . . . . . . . . . . . . 143
8.5
The stochastic wood w3 in SW. . . . . . . . . . . . . . . . . . . 144
8.6
The stochastic wood w4 in SW. . . . . . . . . . . . . . . . . . . 144
8.7
The stochastic wood w5 in SW. . . . . . . . . . . . . . . . . . . 145
8.8
Subtrees of the right tree in Figure 8.1. . . . . . . . . . . . . . . 146 xiii
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xiv
List of Figures 8.9
Root-mean-square approximation error of (8.42), (8.43), and (8.46) vs. time steps t = t − t0 = t − 12 of the size t ∈ { 214 , 215 , . . . , 218 }. . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.10
Root-mean-square discretization error (8.57) of the exponential Euler approximation Yk128,M,128 , k ∈ {0, 1, . . . , M}, given by (8.54), and of the linear-implicit Euler approximation Yk128,M,128 , k ∈ {0, 1, . . . , M}, given by (8.55), vs. M ∈ {2, 4, 8, 16} and M ∈ {4, 16, 64, 256}, respectively. . . . . . . . . . . . . . . . . . . . . 160
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Chapter 1
Introduction
Taylor expansions are a very basic tool in numerical analysis and other areas of mathematics which require approximations. In particular, they enable the derivation of one-step numerical schemes for ordinary differential equations (ODEs) of arbitrary high order, although in practice such Taylor schemes are rarely implemented but are used instead as a theoretical comparison for determining the convergence orders of other schemes that have been derived by more heuristic methods. A similar situation holds for random ordinary differential equations (RODEs), which are pathwise ODEs, where the vector field is continuous but not differentiable in time due to the nature of the driving noise processes.
1.1 Taylor expansions for ODEs The Taylor expansion of a p + 1 times continuously differentiable function x : R → R is given by x(t) = x(t0 ) + x (t0 ) h + · · · +
1 (p) 1 x (t0 ) hp + x (p+1) (θ ) hp+1 p! (p + 1)!
(1.1)
with h = t − t0 and the remainder term evaluated at some intermediate value θ ∈ [t0 , t], which is usually unknown. Now let x(t) = x(t; t0 , x0 ) be the solution of a scalar ODE dx = f (t, x) dt
(1.2)
1
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2
Chapter 1. Introduction
with the initial value x(t0 ) = x0 and define the differential operator L by ∂g ∂g (t, x) + f (t, x) (t, x), ∂t ∂x i.e., Lg(t, x(t)) is the total derivative of g(t, x(t)) with respect to a solution x(t) of the ODE (1.2), since Lg(t, x) :=
d ∂g ∂g g(t, x(t)) = (t, x(t)) + (t, x(t)) x (t) = Lg(t, x(t)) dt ∂t ∂x by the chain rule. In particular, for any such solution x (t) = f (t, x(t)), x (t) =
d d x (t) = f (t, x(t)) = Lf (t, x(t)), dt dt
x (t) =
d d x (t) = Lf (t, x(t)) = L2 f (t, x(t)) , dt dt
and, in general, x (j ) (t) = Lj −1 f (t, x(t)), j = 1, 2, . . . , provided f is smooth enough. For notational convenience, define L0 f (t, x) ≡ f (t, x). If f is p times continuously differentiable, then the solution x(t) of the ODE (1.2) is p + 1 times continuously differentiable and has a Taylor expansion (1.1), which can be rewritten as p 1 j −1 x(t) = x(t0 ) + L f (t0 , x(t0 )) (t − t0 )j (1.3) j! j =1
+
1 Lp f (θ, x(θ)) (t − t0 )p+1 . (p + 1)!
On a subinterval [tn , tn+1 ] with h = tn+1 − tn > 0, the Taylor expansion is x(tn+1 ) = x(tn ) +
p 1 j −1 1 L f (tn , x(tn )) hj + Lp f (θn , x(θn )) hp+1 j! (p + 1)! j =1
(1.4) for some θn ∈ [tn , tn+1 ], which is usually unknown. Nevertheless, the error term can be estimated and is of order 0(hp+1 ), since hp+1 p hp+1 L f (θn , x(θn ; tn , xn )) ≤ max Lp f (t, x) ≤ Cp,T ,D hp+1 , (p + 1)! (p + 1)! t0 ≤t≤T x∈D
where D is some sufficiently large compact subset of R containing the solution over a bounded time interval [t0 , T ], which contains the subintervals under consideration.
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1.1. Taylor expansions for ODEs
3
The maximum can be used here since Lp f is continuous on [t0 , T ] × D. Note that Lp f contains the partial derivatives of f of up to order p.
1.1.1 Taylor schemes for ODEs The Taylor scheme of order p for the ODE (1.2), p hj j −1 L f (tn , xn ), xn+1 = xn + j!
(1.5)
j =1
is obtained by discarding the remainder term in the Taylor expansion (1.4) and replacing x(tn ) by xn . The Taylor scheme (1.5) is an example of a one-step explicit scheme of the general form xn+1 = xn + hF (h, tn , xn )
(1.6)
with an increment function F defined by p 1 j −1 L f (t, x) hj −1 . F (h, t, x) := j! j =1
Such a scheme is said to have order p if its global discretization error Gn (h) := |x(tn , t0 , x0 ) − xn | ,
n = 0, 1, . . . , Nh :=
T − t0 , h
converges with order p, i.e., if max Gn (h) ≤ Cp,T ,D hp .
0≤n≤Nh
A basic result in numerical analysis says that a one-step scheme converges with order p if its local discretization error converges with order p + 1. This is defined by Ln+1 (h) := |x(tn+1 ) − x(tn ) − hF (h, tn , x(tn ))| , i.e., the error on each subinterval takes one iteration of the scheme starting at the exact value of the solution x(tn ) at time tn . Thus, the Taylor scheme of order p is indeed a pth order scheme. The simplest Taylor scheme is the Euler scheme, xn+1 = xn + hf (tn , xn ),
(1.7)
with p = 1.
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4
Chapter 1. Introduction
The higher coefficients Lj −1 f (t, x) of a Taylor scheme of order p > 1 are, however, very complicated. For example, ∂ ∂ [Lf ] + f [Lf ] ∂t ∂x ∂f ∂ ∂f ∂f ∂ ∂f +f +f +f = ∂x ∂x ∂t ∂x ∂t ∂t
L2 f = L[Lf ] =
=
∂ 2f ∂ 2f ∂ 2 f ∂f ∂f + + f + f +f ∂t ∂x ∂t∂x ∂x∂t ∂t 2
∂f ∂x
2 +f2
∂ 2f . ∂x 2
Taylor schemes are thus rarely used in practice,1 but they are very useful for theoretical purposes, e.g., for determining by comparison the local discretization order of other numerical schemes derived by heuristic means, such as the Heun scheme, 1 xn+1 = xn + h [f (tn , xn ) + f (tn+1 , xn + hf (tn , xn ))] , 2 which is a Runge–Kutta scheme of order 2.
1.1.2
(1.8)
Integral representation of Taylor expansions
Taylor expansions of a solution x(t) = x(t, t0 , x0 ) of an ODE (1.2) also have an integral derivation and representation. These are based on the integral equation representation of the initial value problem of the ODE, t f (s, x(s)) ds, (1.9) x(t) = x0 + t0
and, by the Fundamental Theorem of Calculus, the integral form of the total derivative, t g(t, x(t)) = g(t0 , x0 ) + Lg(s, x(s)) ds. (1.10) t0
Note that (1.10) reduces to the integral equation (1.9) with g(t, x) = x, since Lg = f in this case. Applying (1.10) to g = f over the interval [t0 , s] to the integrand of the integral equation (1.9) gives t s f (t0 , x0 ) + Lf (τ , x(τ )) dτ ds x(t) = x0 + t0
t0
= x0 + f (t0 , x0 )
t
t0
ds +
t
s
Lf (τ , x(τ )) dτ ds, t0
t0
1 Symbolic manipulators now greatly facilitate their use. Indeed, Coomes, Koçak & Palmer [20] applied a Taylor scheme of order 31 to the three-dimensional Lorenz equations.
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1.1. Taylor expansions for ODEs
5
which is the first order Taylor expansion. Then applying (1.10) to g = Lf over the interval [t0 , τ ] to the integrand Lf in the double integral remainder term leads to t t s x(t) = x0 + f (t0 , x0 ) ds + Lf (t0 , x0 ) dτ ds t0
+
t s t0
t0
t0 τ
t0
L2 f (ρ, x(ρ)) dρ dτ ds.
t0
In this way one obtains the Taylor expansion in integral form: x(t) = x(t0 ) +
p
L
j −1
t f (t0 , x0 ) t0
j =1
+
t
s1
sj
... t0
s1
t0
t0
t0
···
sj −1
dsj · · · ds1
(1.11)
t0
Lp f (sj +1 , x(sj +1 )) dsj +1 · · · ds1 .
(For j = 1, there is just a single integral over t0 ≤ s1 ≤ t.) This is equivalent to the differential form of the Taylor expansion (1.3) by the Intermediate Value Theorem for Integrals and the fact that t s1 sj −1 1 ··· dsj · · · ds1 = (t − t0 )j , j = 1, 2, . . . . j! t0 t0 t0
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Chapter 2
Taylor Expansions and Numerical Schemes for RODEs
Random ordinary differential equations (RODEs) are pathwise ODEs that contain a stochastic process in their vector field functions. They have been used for many years in a wide range of applications but have been very much overshadowed by stochastic ordinary differential equations (SODEs). They are intrinsically nonautonomous ODEs due to the noise. Typically, however, the driving stochastic process has at most Hölder continuous sample paths, so the sample paths of the solutions are certainly continuously differentiable, but the derivatives of the sample paths are at most Hölder continuous in time. Thus, after insertion of the driving stochastic process, the resulting vector field is at most Hölder continuous in time, no matter how smooth the vector field is in its original variables. Consequently, although classical numerical schemes for ODEs can be used pathwise for RODEs, they rarely attain their traditional order and new forms of higher order schemes are required.
2.1
RODEs
Let (, A, P) be a probability space and let ζ : [0, T ] × → Rm be an Rm -valued stochastic process with continuous sample paths. In addition, let f : Rm × Rd → Rd be a continuous function. A RODE in Rd , dx x ∈ Rd , (2.1) = f (ζt (ω), x), dt is a nonautonomous ODE dx = Fω (t, x) := f (ζt (ω), x) (2.2) dt for almost every realization ω ∈ . 9
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10
Chapter 2. Taylor Expansions and Numerical Schemes for RODEs A simple example of a scalar RODE is dx = −x + sin(Wt (ω)) , dt
(2.3)
where Wt is a scalar Wiener process. Here f (z, x) = −x + sin z and d = m = 1. RODEs with other kinds of noise, such as fractional Brownian motion, have been used, e.g., in Garrido-Atienza, Kloeden & Neuenkirch [34]. For convenience, it will be assumed that the RODE (2.2) holds for all ω ∈ , by restricting to a subset of full probability if necessary, and that f is infinitely often continuously differentiable in its variables, although k-times continuously differentiable with k sufficiently large would suffice. In particular, f is then locally Lipschitz in x, so the initial value problem d xt (ω) = f (ζt (ω), xt (ω)), dt
x0 (ω) = X0 (ω),
(2.4)
where the initial value X0 is an Rd -valued random variable, has a unique pathwise solution xt (ω) for every ω ∈ , which will be assumed to exist on the finite time interval [0, T ] under consideration. Sufficient conditions that guarantee the existence and uniqueness of such solutions can be found in Arnold [2] and Bunke [12]. They are similar to those for ODEs. The solution of the RODE (2.4) is a stochastic process (xt )t∈[0,T ] . Its sample paths t → xt (ω) are continuously differentiable but need not be further differentiable since the vector field Fω (t, x) of the nonautonomous ODE (2.2) is usually only continuous, but not differentiable in t, no matter how smooth the function f is in its variables.
2.1.1
Equivalence of RODEs and SODEs
RODEs occur in many applications; see, for example, [2, 12, 113, 114] and the papers cited therein. Moreover, RODEs with Wiener processes can be rewritten as SODEs, so results for one can be applied to the other. For example, the scalar RODE (2.3) can be rewritten as the two-dimensional SODE Xt −Xt + sin(Yt ) 0 d = dt + dWt . Yt 0 1 On the other hand, any finite dimensional SODE can be transformed to an RODE. In the case of commutative noise this is the famous Doss–Sussmann result [31, 116], which was generalized to all SODE in recent years by Imkeller & Schmalfuss [60]. It is easily illustrated for a scalar SODE with additive noise: The equation dXt = f (Xt ) dt + dWt
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2.1. RODEs
11
is equivalent to the RODE dz = f (z + Ot ) + Ot , dt
(2.5)
where z(t) := Xt −Ot and Ot is the stochastic stationary Ornstein–Uhlenbeck process satisfying the linear SODE dOt = −Ot dt + dWt .
(2.6)
To see this, subtract integral versions of both SODE and substitute to obtain z(t) = z(0) +
t
[f (z(s) + Os ) + Os ] ds.
0
It then follows by continuity and the Fundamental Theorem of Calculus that z is pathwise differentiable. In particular, deterministic calculus can be used pathwise for RODEs. This greatly facilitates the investigation of dynamical behavior and other qualitative properties of RODEs. For example, suppose that f in the RODE (2.5) satisfies a one-sided dissipative Lipschitz condition, x − y, f (x) − f (y) ≤ −L|x − y|2 for all x, y ∈ Rd and some L > 0. Then, for any two solutions z1 (t) and z2 (t) of the RODE (2.5),
d dz1 dz2 |z1 (t) − z2 (t)|2 = 2 z1 (t) − z2 (t), − dt dt dt = 2 z1 (t) − z2 (t), f (z1 (t) + Ot ) − f (z2 (t) + Ot ) = 2 (z1 (t) + Ot ) − (z2 (t) + Ot ) , f (z1 (t) + Ot ) − f (z2 (t) + Ot ) ≤ −2L |z1 (t) − z2 (t)|2 , from which it follows that |z1 (t) − z2 (t)|2 ≤ e−2Lt |z1 (0) − z2 (0)|2 → 0
as t → ∞.
From the theory of random dynamical systems [2] there thus exists a pathwise asymptotically stable stochastic stationary solution. Transforming back to the SODE, one concludes that the SODE also has a pathwise asymptotically stable stochastic stationary solution.
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12
2.1.2
Chapter 2. Taylor Expansions and Numerical Schemes for RODEs
Simple numerical schemes for RODEs
The rules of deterministic calculus apply pathwise to RODEs, but the vector field function in Fω (t, x) in (2.2) is not smooth in t. It is at most Hölder continuous in time like the driving stochastic process ζt and thus lacks the smoothness needed to justify the Taylor expansions and the error analysis of traditional numerical methods for ODEs in Chapter 1. Such methods can be used but will attain at best a low convergence order, so new higher order numerical schemes must be derived for RODEs. For example, let ζt be pathwise Hölder continuous of order 12 . Then the Euler scheme with step size n Yn+1 = (1 − n ) Yn + ζtn n for the RODE dx = −x + ζt dt attains the pathwise order 12 . One can do better, however, by using the pathwise averaged Euler scheme tn+1 ζt dt, Yn+1 = (1 − n ) Yn + tn
which was proposed by Grüne & Kloeden [40] (see also Talay [119, 120]). It attains the pathwise order 1 provided the integral is approximated with Riemann sums Jn tn+1 ζt dt ≈ ζtn +j δ δ tn
j =1
≈ n and δ · Jn = n . In fact, this was done with the step size δ satisfying more generally in [40] for RODEs with an affine structure, i.e., of the form δ 1/2
dx = g(x) + G(x)ζt , dt
(2.7)
where g : Rd → Rd and G : Rd → Rd × Rm . The explicit averaged Euler scheme then reads Yn+1 = Yn + [g (Yn ) + G (Yn ) In ] n ,
(2.8)
tn+1 1 ζs ds. (2.9) n tn For the general RODE (2.1) this suggests that one should pathwise average the vector field, i.e., tn+1 1 f (ζs , Yn ) ds, n tn where
In :=
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2.2. Taylor-like expansions for RODEs
13
which is computationally expensive even for low-dimensional systems. An alternative is to use the averaged noise within the vector field, which leads to the explicit averaged Euler scheme Yn+1 = Yn + f (In , Yn ) n .
(2.10)
A systematic derivation of higher order numerical schemes for RODEs based on this idea using Taylor-like expansions from [61] and [79] will be outlined below. The next two subsections come from [79]. See also Carbonell et al. [15] for the local linearization method for RODEs.
2.2 Taylor-like expansions for RODEs For simplicity, our attention will be restricted here to the one-dimensional case d = m = 1; readers are referred to [66] for vector valued RODEs with multidimensional noise processes. To emphasize the role of the sample paths of the driving stochastic process ζt , a canonical sample space = C([0, ∞), R) of continuous functions ω : [0, ∞) → R will be used, so ζt (ω) = ω(t) for t ∈ [0, ∞) and, henceforth, the RODE (2.1) will be written dx = f (ω(t), x). dt Since f is assumed to be infinitely often continuously differentiable in its variables, the initial value problem dx (2.11) = f (ω(t), x), x(t0 ) = x0 , dt has a unique solution, which will be assumed to exist on a finite time interval [t0 , T ] under consideration. Hence, by the continuity of the solution x(t) = x(t; t0 , x0 , ω) on [t0 , T ], there exists an R = R(ω, T ) > 0 such that |x(t)| ≤ R for all t ∈ [t0 , T ]. In addition, it will be assumed that P almost all sample paths of ζt are locally Hölder continuous with the same Hölder exponent, i.e., there is a γ ∈ (0, 1] such that for P almost all ω ∈ and each T > 0 there exists a Cω,T > 0 such that |ω(t) − ω(s)| ≤ Cω,T · |t − s|γ
for all
|t|, |s| ≤ T .
(2.12)
Let θ be the supremum of all γ with this property. Two cases will be distinguished later: Case A, in which (2.12) also holds for θ itself, and Case B, when it does not. The Wiener process with θ = 12 is an example of Case B. For such sample paths ω define ω ∞ := sup |ω(t)|, t∈[t0 ,T ]
so
ω γ := ω ∞ + sup
|ω(t) − ω(s)| ≤ ω γ · |t − s|γ
s =t∈[t0 ,T ] |t−s|≤1
for all
|ω(t) − ω(s)| , |t − s|γ
s, t ∈ [t0 , T ], |s − t| ≤ 1.
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14
2.2.1
Chapter 2. Taylor Expansions and Numerical Schemes for RODEs
Multi-index notation
Let N0 denote the nonnegative integers. For a multi-index α = (α1 , α2 ) ∈ N20 define |α| := α1 + α2 ,
α! := α1 ! α2 ! .
In addition, for a given γ ∈ (0, 1], define the weighted magnitude of a multi-index α by |α|γ := γ α1 + α2 , ∞ and for each K ∈ R+ with K ≥ |α|γ let |α|K γ := K − |α|γ . For f ∈ C (R × R, R ), denote ∂ α f := (∂1 )α1 (∂2 )α2 f
with ∂ (0,0) f = f and (0, 0)! = 1. Finally, for brevity, write fα := ∂ α f . Let ω and R > 0 be fixed, the latter being chosen, as above, as an upper bound on the solution of the initial value problem (2.11) corresponding to the sample path ω on a fixed interval [t0 , T ]. Define f k := max
sup
|α|≤k |y|≤ ω ∞ |z|≤R
|fα (y, z)|
for all k ∈ {0, 1, 2, . . . } and, for brevity, write f := f 0 . Note that the solution of the initial value problem (2.11) is Lipschitz continuous with |x(t) − x(s)| ≤ f |t − s| for all t, s ∈ [t0 , T ] .
2.2.2 Taylor expansions of the vector field The solution x(t) of the initial value problem (2.11) is only once differentiable, so the usual Taylor expansion cannot be continued beyond the linear term. Nevertheless, the special structure of a RODE and smoothness of f in both of its variables enables one to derive implicit Taylor-like expansions of arbitrary order for the solution. Fix k ∈ Z+ and ω ∈ and write ωs := ω(s) − ω, ˆ where
ωˆ := ω(tˆ),
xs := x(s) − x, ˆ xˆ := x(tˆ),
for an arbitrary tˆ ∈ [t0 , T ). Then the usual Taylor expansion for f in both variables gives f (ω(s), x(s)) =
1 ˆ x) ˆ (ωs )α1 (xs )α2 + Rk+1 (s) ∂ α f (ω, α!
(2.13)
|α|≤k
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2.3. RODE–Taylor schemes
15
with remainder term Rk+1 (s) =
|α|=k+1
1 α ∂ f (ωˆ + ξs ωs , xˆ + ξs xs ) (ωs )α1 (xs )α2 α!
(2.14)
for some ξs ∈ [0, 1]. Substituting this into the integral equation representation of the solution of (2.11), t x(t) = xˆ + f (ω(s), x(s)) ds, (2.15) tˆ
gives xt =
t t 1 (ωs )α1 (xs )α2 ds + ∂ α f (ωˆ , x) ˆ Rk+1 (s) ds , α! tˆ |α|≤k tˆ
remainder Taylor-like approximation
or, more compactly, as xt =
Tα (t; tˆ) +
|α|≤k
tˆ
t
Rk+1 (s) ds,
(2.16)
where Tα (t; tˆ) :=
1 ˆ x) ˆ fα (ω, α!
tˆ
t
(ωs )α1 (xs )α2 ds.
The expression (2.16) is implicit in xs and so is not a standard Taylor expansion. Nevertheless, it can be used as the basis for constructing higher order numerical schemes for the RODE (2.1).
2.3
RODE–Taylor schemes
The RODE–Taylor schemes are a family of explicit one-step schemes for RODEs (2.1) on subintervals [tn , tn+1 ] of [t0 , T ] with step size hn = tn+1 − tn , which are derived from the Taylor-like expansion (2.16). The simplest case is for k = 0. Then (2.16) reduces to t t 1 x(t) = xˆ + ˆ (ωs )0 (xs )0 ds + R1 (s) ds ∂ (0,0) f (ωˆ , x) (0, 0)! tˆ tˆ t = xˆ + f (ωˆ , x) ˆ t + R1 (s) ds, tˆ
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16
Chapter 2. Taylor Expansions and Numerical Schemes for RODEs
which leads to the well-known Euler scheme yn+1 = yn + hn f (ω(tn ), yn ). In order to derive higher order schemes, the xs terms inside the integrals must also be approximated. This can be done with a numerical scheme of a lower order than that of the scheme to be derived. Higher order schemes can thus be built up recursively for sets of multi-indices of the form AK := α = (α1 , α2 ) ∈ N20 : |α|θ = θα1 + α2 < K , where K ∈ R+ and θ ∈ (0, 1] is specified by the noise process in the RODE, i.e., the supremum of the Hölder coefficients of its sample paths. For K ∈ [0, ∞) consider the first step (K)
y1K,h (tˆ, y) ˆ = yˆ + yh (tˆ, y) ˆ
(2.17)
of a numerical approximation at the time instant tˆ + h for a step size h ∈ (0, 1] and (K) initial value (tˆ, y) ˆ ∈ [t0 , T ] × R, where the increments yh are defined recursively. (0) Specifically, let yh := 0 and define
(K)
ˆ := yh (tˆ, y)
Nα(K) (tˆ + h, tˆ, y), ˆ
(2.18)
|α|θ
where Nα(K) (tˆ + h, tˆ, y) ˆ :=
1 ˆ y) ˆ fα (ω, α!
tˆ+h
tˆ
α2 (|α|K ) (ωs )α1 ys θ (tˆ, y) ˆ ds
(2.19)
(K) with s = s − tˆ and |α|K are θ = K − θ · α1 − α2 ; i.e., in nontrivial cases, the Nα (L) evaluated in terms of previously determined ys with L := K − θ · α1 − α2 < K.
This procedure is repeated for each time step to give the AK -RODE–Taylor scheme K,h := ynK,h + Nα(K) tn+1 , tn , ynK,h (2.20) yn+1 α∈AK
on discretization subintervals [tn , tn+1 ] with step sizes hn = tn+1 − tn > 0 for n = 0, 1, . . . , NT − 1. This will often be called the K-RODE–Taylor scheme.
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2.3. RODE–Taylor schemes
2.3.1
17
Discretization error
The increment function of a K-RODE–Taylor scheme α2 tˆ+h 1 1 (|α|K (K) α1 θ ) ˆ ˆ fα (ω, ys (t , y) ˆ := ˆ y) ˆ (ωs ) ˆ ds F (h, t , y) h α! tˆ |α|θ
is continuous in its variables as well as continuously differentiable, hence locally Lipschitz, in the yˆ variable with ˆ = f (ω, ˆ y). ˆ lim F (K) (h, tˆ, y)
h→0+
The RODE–Taylor schemes are thus consistent and hence convergent for each K > 0 (see, e.g., Deuflhard & Bornemann [30]). Moreover, the classical theorem for ODEs on the loss of a power from the local to global discretization errors also holds for the RODE–Taylor schemes. Consequently, it suffices to estimate the local discretization error, which is defined here as (K) ˆ , ˆ := x(tˆ + h; tˆ, y) ˆ − y1K,h (tˆ, y) Lh (tˆ, y) where x(tˆ + h; tˆ, y) ˆ is the value of the solution of the RODE at time tˆ + h with initial ˆ value (t , y) ˆ and y1K,h (tˆ, y) ˆ is the first step of the numerical scheme with step size h for the same initial value. Define R˜ 0 := 0 and for K > 0 define R˜ K := sup
max
0
In addition, let
k = kK :=
and define
(h,t,x)∈ [0,1]×[t0 ,T ]×[−R,R]
K θ
|F (L) (h, t, x)|.
( ·
integer part)
RK := max R˜ K , f k+1 .
(2.21)
Two cases will be considered: Case A, in which the Hölder estimate (2.12) also holds for the supremum θ of the admissible exponents itself, and Case B, when it does not. Theorem 2.1. The local discretization error for an RODE–Taylor scheme in Case A satisfies (K) ˆ ˆ ≤ CK hK+1 Lh (t , x) for each 0 ≤ h ≤ 1, where K+1 . CK := e ω θ +2RK
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18
Chapter 2. Taylor Expansions and Numerical Schemes for RODEs
In Case B it satisfies
(K) ˆ ε ˆ ≤ CK · hK+1−ε Lh (t , x)
for ε > 0 arbitrarily small, where K+1 ε := e ω γε +2RK , CK
γε := θ −
ε . (k + 1)2
Case A will be proved in detail with some remarks provided on the essential differences which arise in Case B. Proof. The terms Tα in the Taylor-like expansion of (2.16) of the RODE solution are easily estimated to give Tα (tˆ + t; tˆ) ≤ f |α| · 1 ω α1 f α2 (t)|α|γ +1 , γ α!
(2.22)
where t := t − tˆ ∈ (0, 1] and t ∈ (tˆ, T ]. Then the estimate t 1 ω αγ 1 f α2 (t)|α|γ +1 ˆ Rk+1 (s) ds ≤ f k+1 α! t
(2.23)
|α|=k+1
of the remainder term follows from 1 α α1 α2 |Rk+1 (s)| = ∂ f (ωˆ + ξs ωs , xˆ + ξs xs ) (ωs ) (xs ) α! |α|=k+1
≤
|α|=k+1
1 α ∂ f (ωˆ + ξs ωs , xˆ + ξs xs ) ωs |α1 |xs |α2 α!
≤ f k+1
|α|=k+1
≤ f k+1
|α|=k+1
α 1 ω γ (s)γ 1 ( f s )α2 α! 1 ω αγ 1 f α2 (s)|α|γ . α!
Case A is proved by mathematical induction on K ≥ 0. Here estimates such as (2.22) and (2.23) hold for all γ ≤ θ, including θ itself, which will be used below. For K = 0 the assertion is trivial, because (0) ˆ := x(tˆ + h; tˆ, x) ˆ − y10,h (tˆ, x) ˆ = x(tˆ + h) − x(tˆ) ≤ f h. Lh (tˆ, x)
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2.3. RODE–Taylor schemes
19
For K ∈ (0, θ ), the index k = kK = 0 and the multi-index set AK = {(0, 0)}. Hence (K) ˆ ˆ := x(tˆ + h; tˆ, x) ˆ − y1K,h (tˆ, x) Lh (tˆ, x) := xt − y1K,h (tˆ, x) ˆ tˆ+h (K) = Tα (tˆ + h; tˆ) + R1 (s) ds − Nα (tˆ + h; tˆ, x) ˆ tˆ |α|≤0 AK tˆ+h (K) R1 (s) ds − N(0,0) (tˆ + h; tˆ, x) ˆ = T(0,0) (tˆ + h; tˆ) + tˆ
tˆ+h θ+1 R1 (s) ds ≤ f 1 ( ω θ + f ) h . =
tˆ ≤ CK
≤ hK+1
This completes the first step of the induction argument. Suppose now that the assertion is true for K < L for a fixed L = l θ, where l ∈ N. The aim is to show that the assertion is then true for any K ∈ [L, L + θ). Then k = kK = l and (K) ˆ ˆ := x(tˆ + h; tˆ, x) ˆ − y1K,h (tˆ, x) Lh (tˆ, x) tˆ+h (K) ≤ Rk+1 (s) ds − Nα (tˆ + h; tˆ, x) ˆ Tα (tˆ + h; tˆ) + tˆ |α|≤k α∈AK ˆ Tα (tˆ + h; tˆ) + ≤ ˆ Tα (t + h; tˆ) − Nα(K) (tˆ + h; tˆ, x) |α|≤k,α∈AcK
1. error
α2 >0,α∈AK
tˆ+h + Rk+1 (s) ds . ˆ t
2. error
3. error
Estimates of the first and third errors were given, respectively, in (2.22) and (2.23). The second error can be estimated through the following lemma. Lemma 2.2. For α ∈ AK with α2 > 0 α2 (K) ˆ ω αθ 1 (RK )α2 −1 hK+1 (2.24) ˆ − Tα (tˆ + h; tˆ) ≤ f k CK−1 Nα (t + h; tˆ, x) α! with the convention that Cr := 1 for r < 0.
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20
Chapter 2. Taylor Expansions and Numerical Schemes for RODEs (|α|K θ )
Proof. For brevity, omit the variables tˆ and xˆ in ys
and write
ˆ − Tα (tˆ + h; tˆ) . E2 := Nα(K) (tˆ + h; tˆ, x) Then α2 tˆ+h 1 (|α|K α1 α2 θ ) ys ds ˆ x) ˆ (ωs ) − (xs ) E2 ≤ fα (ω, α! tˆ α2 tˆ+h 1 (|α|K ) − (xs )α2 ds . ω αθ 1 (s)θα1 y s θ f |α| ≤ α! tˆ By the inequality |y p − x p | ≤ p · |y − x| · (max(|x|, |y|) )p−1 ,
x, y ∈ R,
p ∈ N,
the difference in the integrand can be estimated by α2 K (|α|Kθ ) α2 −1 (|α|Kθ ) α2 y (|α|θ ) | |x , y ≤ α y max − (x ) − x s 2 s s s s s (2.25) where, by the definition of RK , (|α|Kθ ) (2.26) max |xs | , ys ≤ RK s. Hence α2 ω αθ 1 E2 ≤ f k α!
tˆ
tˆ+h
(|α|K )
Ls θ (tˆ, x) ˆ (s)θα1 (RK s)α2 −1 ds
α2 ≤ f k ω αθ 1 (RK )α2 −1 α!
tˆ+h
tˆ
(|α|K )
Ls θ (tˆ, x) ˆ (s)|α|θ −1 ds.
(|α|K )
ˆ ≤ C|α|K (s)|α|θ By the induction assumption, Ls θ (tˆ, x)
K +1
θ
, so
tˆ+h α2 K α1 α2 −1 E2 ≤ f k C|α|K · (s)|α|θ +|α|θ ds ω θ (RK ) θ α! tˆ α2 α1 α2 −1 ω θ (RK ) · hK+1 , ≤ f k CK−1 · α! where Cr := 1 for r < 0. This completes the proof of Lemma 2.2.
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2.3. RODE–Taylor schemes
21
Returning to the proof of Theorem 2.1, the three error estimates (2.22), (2.23), and (2.24) combine to give (K)
Lh (tˆ, x) ˆ ≤ f k+1 CK−1 hK+1
|α|≤k+1
1 ω αθ 1 (RK )α2 . α!
Finally, using the double exponential expansion ex+y =
α2 1 x α1 y α2 −1 x α1 y α2 = α! α! 2
α∈ N0
one obtains
α2 >0
(K) Lh (tˆ, x) ˆ ≤ f k+1 e ω θ +RK CK−1 hK+1 ≤ CK hK+1 .
This completes the proof of Case A of Theorem 2.1. The main difference in the proof of Case B of Theorem 2.1 is in the estimate of the second error there given in Lemma 2.2. The required estimate is given by the following lemma, where the convention that Cr := 1 for r < 0 is also used. Lemma 2.3. For α ∈ AK with α2 > 0, h ∈ (0, 1], and ε > 0 arbitrarily small, α2 (K) ˆ ε∗ ω αγε1 (RK )α2 −1 hK+1−ε , ˆ − Tα (tˆ + h; tˆ) ≤ f k CK−1 Nα (t + h; tˆ, x) α! (2.27) where k ε ε∗ := ε , γε := θ − . k+1 (k + 1)2 Proof. As in Lemma 2.2, write E2 for the expression to be estimated. Now (ωs )α1 ≤ ω αγε1 (s)α1 γε ≤ ω αγε1 (s)
α1 θ−α1 ε
1 (k+1)2
1
≤ ω αγε1 (s)θα1 −ε k+1 ,
since s ∈ (0, 1] and α1 θ − ε
1 1 ≤ α1 θ − α1 ε k+1 (k + 1)2
for α1 ≤ K/θ ≤ kK + 1 = k + 1. Thus α2 tˆ+h 1 (|α|K α1 α2 θ ) E2 ≤ fα (ω, ys ds ˆ x) ˆ (ωs ) − (xs ) α! tˆ α2 tˆ+h 1 1 (|α|K ) f |α| ω αγε1 (s)θα1 −ε k+1 y s θ − (xs )α2 ≤ α! tˆ
ds,
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22
Chapter 2. Taylor Expansions and Numerical Schemes for RODEs
so, using the inequalities (2.25) and (2.26) to estimate the difference in the integrand, tˆ+h 1 α2 (|α|K ) E2 ≤ f k ω αγε1 Ls θ (tˆ, x) ˆ (s)(θα1 −ε k+1 ) (RK s)α2 −1 ds α! ˆt tˆ+h 1 α2 (|α|K ) ≤ f k Ls θ (tˆ, x) ˆ (s)|α|θ −1−ε k+1 ds, ω αγε1 (RK )α2 −1 α! tˆ (|α|K )
K +1−ε ∗
∗
|α|θ ε ˆ ≤ C|α| where Ls θ (tˆ, x) K (s)
by the induction assumption. Hence
θ
α2 ε∗ E2 ≤ f k C|α| ω αγε1 (RK )α2 −1 K · α! θ
tˆ+h
tˆ
(s)|α|θ
K +|α| −ε θ
ds
∗
ε ≤ CK−1 ∗
ε ≤ f k CK−1
α2 ω αγε1 (RK )α2 −1 hK+1−ε , α!
since |α|K θ + |α|θ − ε = K − ε. This completes the proof of Lemma 2.3.
2.3.2
Examples of RODE–Taylor schemes
Some explicit examples of RODE–Taylor schemes will be presented here for scalar RODEs, all but one with scalar noise processes. Two representative noise processes will be considered: Brownian motion (Wiener process) and fractional Brownian motion with Hurst coefficient H = 34 . MATLAB software for RODE–Taylor schemes can be found in [6]. RODEs with Brownian motion The following examples have the Brownian motion or Wiener process as the driving process, which falls into Case B with θ = 12 . The AK -RODE–Taylor schemes, which have pathwise global convergence order K − ε, are given for K = 0, 0.5, 1.0, 1.5, and 2.0. Since θ = 12 AK := α : |α| 1 < K = { α : α1 + 2α2 ≤ 2K − 1 } 2
for these K. Example 2.4. The 0-RODE–Taylor scheme corresponding to A0 = ∅ is yn ≡ y0 , which is an inconsistent scheme. Example 2.5. The 0.5-RODE–Taylor scheme corresponding to A0.5 = { (0, 0) } is the classical Euler scheme yn+1 = yn + hf (ω(tn ), yn ),
(2.28)
which has order 0.5 − ε.
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2.3. RODE–Taylor schemes
23
Example 2.6. The 1.0-RODE–Taylor scheme corresponding to A1.0 = {(0, 0), (1, 0)} is the “improved” Euler scheme, tn+1 yn+1 = yn + hf (ω(tn ), yn ) + f(1,0) (ω(tn ), yn ) ωs ds. tn
Its order 1 − ε is comparable to that of the Euler scheme for smooth ODEs. In the following schemes the coefficient functions on the right side are evaluated at (ω(tn ), yn ). Example 2.7. The 1.5-RODE–Taylor scheme corresponding to A1.5 = {(0, 0), (1, 0), (2, 0), (0, 1)} is tn+1 f(2,0) tn+1 h2 (ωs )2 ds + f(0,1) f . ωs ds + yn+1 = yn + hf + f(1,0) 2 2 tn tn tn+1 (0.5) Here |(0, 1)|1.5 (ys ) ds 1 = 1.5−1 = 0.5 and the last term is obtained from f(0,1) tn 2
(0.5)
with ys = (s − tn )f coming from the Euler scheme (2.28). Example 2.8. In the next case A2.0 = {(0, 0), (1, 0), (2, 0), (3, 0), (0, 1), (1, 1)} and (0.5) (1.0) |(1, 1)|21 = 0.5, |(0, 1)|21 = 1, so the terms ys , ys corresponding to the 0.52
2
and 1.0-RODE–Taylor schemes are required in the right-hand side of the new scheme. The resulting 2.0-RODE–Taylor scheme is then tn+1 f(2,0) tn+1 yn+1 = yn + hf + f(1,0) ωs ds + (ωs )2 ds 2 tn tn f(3,0) tn+1 h2 + (ωs )3 ds + f(0,1) f 6 2 tn tn+1 s tn+1 + f(0,1) f(1,0) ωv dv ds + f(1,1) f ωs s ds. tn
tn
tn
Example 2.9. Finally, consider a two-dimensional Brownian motion ω(t) = (ω1 (t), ω2 (t)). The 1.5-RODE–Taylor scheme here corresponds to the indicial set A1.5 = {(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0), (2, 0, 0), (0, 2, 0), (0, 0, 1)} and is given by tn+1 tn+1 1 yn+1 = yn + hf + f(1,0,0) ωs ds + f(0,1,0) ωs2 ds + f(1,1,0)
tn
tn+1 tn
f(0,2,0) + 2
tn+1 tn
ωs1 ωs2 ds +
f(2,0,0) 2
(ωs2 )2 ds + f(0,0,1) f
tn
tn+1 tn
(ωs1 )2 ds
h2 . 2
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24
Chapter 2. Taylor Expansions and Numerical Schemes for RODEs
RODEs with fractional Brownian motion Fractional Brownian motion with Hurst coefficient H = 34 also falls into Case B with θ = 34 . The RODE–Taylor schemes generally contain fewer terms than the schemes of the same order with Brownian motion or attain a higher order when they contain the same terms. Example 2.10. For AK = { (0, 0) } with K ∈ (0, 34 ] the RODE–Taylor scheme is the classical Euler scheme in Example 2, but now the order is 34 − ε. Example 2.11. AK = {(0, 0), (1, 0)} for K ∈ ( 34 , 1] : the RODE–Taylor scheme is the same as that in Example 3 and also has order 1 − ε. Example 2.12. For AK = {(0, 0), (1, 0), (0, 1)} with K ∈ (1, 32 ] the RODE–Taylor scheme, tn+1 h2 yn+1 = yn + hf + f(1,0) ωs + f(0,1) f , 2 tn which has order 1.5 − ε, omits one of the terms in the RODE–Taylor scheme of the same order for Brownian motion given in Example 2.7. Example 2.13. For AK = {(0, 0), (1, 0), (0, 1), (2, 0)} with K ∈ ( 32 , 74 ] the Taylor scheme is the same as that in the Brownian motion case in Example 2.7 but now has order 74 − ε instead of order order 32 − ε. Remark 1. The K-RODE-Taylor schemes are not necessarily optimal in the sense of involving the minimum number of terms for the given order (see also [66]).
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Chapter 3
SODEs
Deterministic calculus is much more robust to approximation than stochastic calculus because the integrand function in a Riemann sum approximating a Riemann integral can be evaluated at an arbitrary point of the discretization subinterval, whereas for an Itô stochastic integral the integrand function must always be evaluated at the left-hand endpoint. Consequently, considerable care is needed in deriving numerical schemes for stochastic ordinary differential equations (SODEs) to ensure that they are consistent with Itô stochastic calculus. In particular, stochastic Taylor schemes are the essential starting point for the derivation of consistent higher order numerics schemes for SODEs. Other types of schemes for SODEs, such as derivativefree schemes, can then be obtained by modifying the corresponding stochastic Taylor schemes. The theory of stochastic Taylor schemes is briefly outlined here in order to highlight the methods and assumptions used and their shortcomings. Most of this chapter comes from [78] and [58, 71, 81].
3.1
Itô SODEs
Consider a scalar Itô SODE dXt = a(t, Xt ) dt + b(t, Xt ) dWt ,
(3.1)
where (Wt )t∈R+ is a standard Wiener process, i.e., with W0 = 0, w.p.1, and increments Wt − Ws ∼ N (0; t − s) for t ≥ s ≥ 0, which are independent on nonoverlapping subintervals. The SODE (3.1) is, in fact, only a symbolic representation for the stochastic integral equation t t Xt = Xt0 + a(s, Xs ) ds + b(s, Xs ) dWs , (3.2) t0
t0
25
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26
Chapter 3. SODEs
where the first integral is pathwise a deterministic Riemann integral and the second an Itô stochastic integral, which looks as if it could be defined pathwise as a deterministic Riemann–Stieltjes integral, but this is not possible because the sample paths of the Wiener process, though continuous, are not differentiable or even of bounded variation on any finite subinterval. The Itô stochastic integral of a nonanticipative mean-square integrable integrand g is defined in terms of the mean-square limit, namely,
T
g(s) dWs := ms − lim
N T −1
→0
t0
g(tn , ω) Wtn+1 (ω) − Wtn (ω) ,
n=0
taken over partitions of [t0 , T ] of maximum step size := maxn n , where n = tn+1 −tn and tNT = T . Nonanticipativeness here means, in particular, that the random variables g(tn ) and Wtn+1 − Wtn are independent, from which follow two simple but very useful properties of Itô integrals: E
T
g(s) dWs = 0,
t0
3.1.1
E
2
T
g(s) dWs t0
=
T
Eg(s)2 ds.
t0
Existence and uniqueness of strong solutions of SODEs
The following theorem is a standard existence and uniqueness theorem for SODEs. The vector valued case is analogous. Proofs can be found, for example, in G¯ı hman & Skorohod [36], Kloeden & Platen [82], and Mao [91]. Theorem 3.1. Suppose that a, b : [t0 , T ] × R → R are continuous in (t, x) and satisfy the global Lipschitz condition |a(t, x) − a(t, y)| + |b(t, x) − b(t, y)| ≤ L|x − y| uniformly in t ∈ [t0 , T ] and that the random variable X0 is nonanticipative with respect to the Wiener process Wt with E(X02 ) < ∞. Then the Itô SODE dXt = a(t, Xt ) dt + b(t, Xt ) dWt has a unique strong solution on [t0 , T ] with initial value Xt0 = X0 . Alternatively, one could assume just a local Lipschitz condition. Then the additional linear growth condition |a(t, x)| + |b(t, x)| ≤ K(1 + |x|)
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3.1. Itô SODEs
27
or the Hasminskii condition xa(t, x) + |b(t, x)|2 ≤ K(1 + |x|2 ) ensures the existence of the solution on the entire time interval [0, T ], i.e., prevents explosions of solutions in finite time. Cherny & Engelbert [17] give a systematic investigation of existence theorems (both with and without uniqueness) for SODEs under different assumptions on the coefficients. For SODEs with a discontinuous but monotone increasing drift, such as a Heaviside function, see Halidias & Kloeden [47].
3.1.2
Simple numerical schemes for SODEs
The stochastic counterpart of the Euler scheme, usually called the Euler–Maruyama scheme, for the Itô SODE (3.1) is given by Yn+1 = Yn + a(tn , Yn ) n + b(tn , Yn ) Wn with time and noise increments tn+1 ds, n = tn+1 − tn = tn
(3.3)
Wn = Wtn+1 − Wtn =
tn+1
dWs . tn
It seems to be consistent (and, indeed, is) with the Itô stochastic calculus because the noise term in (3.3) approximates the Itô stochastic integral in (3.2) over a discretization subinterval [tn , tn+1 ] by evaluating its integrand at the left-hand endpoint of this interval: tn+1 tn+1 tn+1 b(s, Xs ) dWs ≈ b(tn , Xtn ) dWs = b(tn , Xtn ) dWs . tn
tn
tn
Convergence for numerical schemes can be defined in a number of different useful ways. It is usual to distinguish between strong and weak convergence, depending on whether the realizations or only their probability distributions are required to be close, respectively. Consider a fixed interval [t0 , T ] and let be the maximum step size of any partition of [t0 , T ]. To emphasize the dependence on the step size, the () numerical iterates will be written Yn . Then a numerical scheme is said to converge with strong order γ if, for each sufficiently small , () E XT − YNT ≤ KT γ (3.4) and with weak order β if () E (g(XT )) − E g(YNT ) ≤ Kg,T β
(3.5)
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28
Chapter 3. SODEs
for each polynomial g. These are global discretization errors, and the largest possible values of γ and β give the corresponding strong and weak orders, respectively, of the scheme. The stochastic Euler scheme (3.3) has strong order γ = 12 and weak order β = 1, which are quite low, particularly in view of the fact that a large number of realizations need to be generated for most practical applications. To obtain a higher order, one should avoid heuristic adaptations of well-known deterministic numerical schemes because they are usually inconsistent with Itô calculus or, when they are consistent, then they do not improve the order of convergence. Example 3.2. The deterministic Heun scheme, a second order Runge–Kutta scheme, adapted to the Itô SODE (3.1) has the form Yn+1 = Yn +
1 [a(tn , Yn ) + a(tn+1 , Yn + a(tn , Yn )n + b(tn , Yn )Wn )] n 2
1 + [b(tn , Yn ) + b(tn+1 , Yn + a(tn , Yn )n + b(tn , Yn )Wn )] Wn . 2 In particular, for the Itô SODE dXt = Xt dWt it simplifies to 1 Yn+1 = Yn + Yn (2 + Wn ) Wn , 2 so Yn+1 − Yn = Yn (1 + 12 Wn ) Wn . The conditional expectation Yn+1 − Yn x 1 1 x 1 2 E 0 + n = x Yn = x = E Wn + (Wn ) = 2 n 2 2 n n
should approximate the drift term a(t, x) ≡ 0 of the SODE. The adapted Heun scheme is thus not consistent with Itô calculus and does not converge in either the weak or strong sense. A higher order of convergence cannot be obtained with a deterministic numerical scheme adapted to SODEs, when it happens to be consistent, because such a scheme involves only the simple increments of time n and noise Wn , the latter being a poor approximation for the highly irregular Wiener process within the discretization subinterval [tn , tn+1 ]. To obtain a higher order of convergence one needs to provide more information about the Wiener process within the discretization subinterval. Such information is provided by multiple integrals of the Wiener process, which arise in stochastic Taylor expansions of the solution of an SODE. Consistent numerical schemes of arbitrarily desired higher order can be derived by truncating appropriate stochastic Taylor expansions.
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3.2. Itô–Taylor expansions
3.2
29
Itô–Taylor expansions
Itô–Taylor expansions or stochastic Taylor expansions of solutions of Itô SODEs are derived through an iterated application of the stochastic chain rule, the Itô formula. The nondifferentiability of the solutions in time is circumvented by using the integral form of the Itô formula. The Itô formula for scalar valued function f (t, Xt ) of the solution Xt of the scalar Itô SODE (3.1) is t t f (t, Xt ) = f (t0 , Xt0 ) + (3.6) L0 f (s, Xs ) ds + L1 f (s, Xs ) dWs , t0
t0
where the operators L0 and L1 are defined by ∂ 1 ∂2 ∂ +a + b2 2 , ∂t ∂x 2 ∂x
L0 =
L1 = b
∂ . ∂x
(3.7)
This differs from the deterministic chain rule by the additional third term in the L0 operator, which is due to the fact that E |W |2 = t. When f (t, x) ≡ x, the Itô formula (3.6) is just the integral version (3.2) of the SODE (3.1) in its integral form, i.e., t t Xt = Xt0 + a(s, Xs ) ds + b(s, Xs ) dWs . (3.8) t0
3.2.1
t0
Iterated application of the Itô formula
Applying the Itô formula to the integrand functions f (t, x) = a(t, x) and f (t, x) = b(t, x) in (3.8) gives t s s Xt = Xt0 + a(t0 , Xt0 ) + L0 a(u, Xu ) du + L1 a(u, Xu ) dWu ds t0
+
t0
t
b(t0 , Xt0 ) +
t0
= Xt0 + a(t0 , Xt0 )
t
t0
s
t0
ds + b(t0 , Xt0 )
t0
L0 b(u, Xu ) du +
s
L1 b(u, Xu ) dWu dWs
t0 t
dWs + R1 (t, t0 )
(3.9)
t0
with the remainder s s s s 0 R1 (t, t0 ) = L a(u, Xu ) du ds + L1 a(u, Xu ) dWu ds t0
+
t0
t t0
t0 s
t0
t0
L0 b(u, Xu ) du dWs +
t t0
s
L1 b(u, Xu ) dWu dWs .
t0
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30
Chapter 3. SODEs
Discarding the remainder gives the simplest nontrivial stochastic Taylor approximation t
Xt ≈ Xt0 + a(t0 , Xt0 )
t0
t
ds + b(t0 , Xt0 )
dWs ,
(3.10)
t0
which has strong order γ = 0.5 and weak order β = 1. Higher order stochastic Taylor expansions are obtained by successively applying the Itô formula to the integrand functions in the remainder, there being an increasing number of different alternatives. For example, applying the Itô formula to the integrand L1 b in the fourth double integral of the remainder R1 (t, t0 ) gives the stochastic Taylor expansion t t Xt = Xt0 + a(t0 , Xt0 ) ds + b(t0 , Xt0 ) dWs t0
+ L1 b(t0 , Xt0 )
t t0
t0
s
dWu dWs + R2 (t, t0 )
(3.11)
t0
with the remainder s s s s 0 L a(u, Xu ) du ds + a(u, Xu ) dWu ds R2 (t, t0 ) = t0
+
t0
t t0
t0 s
L0 b(u, Xu ) du dWs +
t0
+
t s t0
t0
t0
t s t0
u
t0
u
L0 L1 b(v, Xv ) dv dWu dWs
t0
L1 L1 b(v, Xv ) dWv dWu dWs .
t0
Discarding the remainder gives the stochastic Taylor approximation t t Xt ≈ Xt0 + a(t0 , Xt0 ) ds + b(t0 , Xt0 ) dWs t0
+ b(t0 , Xt0 )
∂b (t0 , Xt0 ) ∂x
t
t0
s
dWu dWs , t0
(3.12)
t0
∂b , which has strong order γ = 1 (and also weak order β = 1). since L1 b = b ∂x
To obtain even higher order expansions one continues the above procedure of expanding integrand functions in appropriate remainder terms with the help of the Itô formula. There are thus many different possible stochastic Taylor expansions. Remark 2. The two examples above already indicate the general pattern of the stochastic Taylor schemes: (i) They achieve their higher order through the inclusion of multiple stochastic integral terms.
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3.2. Itô–Taylor expansions
31
(ii) An expansion may have different strong and weak orders of convergence. (iii) The possible orders for strong schemes increase by a fraction 21 , taking values 1 3 2 , 1, 2 , 2, . . . , whereas the possible orders for weak schemes are whole numbers 1, 2, 3, . . . .
3.2.2
General stochastic Taylor expansions
Multi-indices provide a succinct means to describe the terms that should be included or expanded to obtain a stochastic Taylor expansion of a particular order as well as for representing the iterated differential operators and stochastic integrals that appear in the terms of such expansions. Consider now a scalar SODE dXt = a(t, Xt ) dt +
m
j
bj (t, Xt )dWt
(3.13)
j =1
. . . , Wtm and differential operators with m independent scalar Wiener processes Wt1 , j 2 0 2 L now equal to that in (3.7) with b replaced by m j =1 (b ) and Lj = bj
∂ , ∂x
j = 1, . . . , m.
Writing b0 (t, x) for a(t, x) and dW 0 (t) for dt, the Itô formula (3.6) for a solution of the SODE (3.13) takes the compact form f (t, Xt ) = f (t0 , Xt0 ) +
m
t
j
Lj f (s, Xs ) dWs
j =0 t0
and reduces to the SODE (3.13) when f (t, x) ≡ x, i.e., the identity function id(x) ≡ x, since Lj id ≡ bj for j = 0, 1, . . . , m. In general, a multi-index α of length l(α) = l is an l-dimensional row vector α = (j1 , j2 , . . . , jl ) with components ji ∈ {0, 1, . . . , m} for i ∈ {1, 2, . . . , l}. Let Mm be the set of all multi-indices of length greater than or equal to zero, where a multiindex ∅ of length zero is introduced for convenience. Given a multi-index α ∈ Mm with l(α) ≥ 1, write −α and α− for the multi-index in Mm obtained by deleting the first and the last component, respectively, of α. For such a multi-index α = (j1 , j2 , . . . , jl ) with l ≥ 1, the multiple Itô integral Iα [g(·)]t0 ,t of a nonanticipative mean–square integrable function g is defined recursively by t j Iα− [g(·)]t0 ,s dWs l , I∅ [g(·)]t0 ,t := g(t). Iα [g(·)]t0 ,t := t0
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32
Chapter 3. SODEs
Similarly, the Itô coefficient function fα for a deterministic function f is defined recursively by fα := Lj1 f−α , f∅ = f . The multiple stochastic integrals appearing in a stochastic Taylor expansion with constant integrands cannot be chosen completely arbitrarily. Rather, the set of corresponding multi-indices must form an hierarchical set, i.e., a nonempty subset A of Mm with sup l(α) < ∞
and
−α ∈ A
for each
α ∈ A \ {∅}.
α∈A
The multi-indices of the remainder terms in a stochastic Taylor expansion for a given hierarchical set A belong to the corresponding remainder set B(A) of A defined by B(A) = {α ∈ Mm \ A : −α ∈ A}, i.e., consisting of all of the “next following” multi-indices with respect to the given hierarchical set. Then the Itô–Taylor expansion corresponding to the hierarchical set A and remainder set B(A) is f (t, X(t)) = Iα [fα (t0 , X(t0 ))]t0 ,t + Iα [fα (·, X· )]t0 ,t , α∈A
α∈B(A)
i.e., with constant integrands (hence constant coefficients) in the first sum and timedependent integrands in the remainder sum. Remark 3. A similar setup holds for vector valued SODEs with the differential operators appropriately modified. See Chapter 5 of Kloeden & Platen [82] for details.
3.3
Itô–Taylor numerical schemes for SODEs
Itô–Taylor numerical schemes are obtained by applying Itô–Taylor expansion to the identity function f = id on a subinterval [tn , tn+1 ] at a starting point (tn , Yn ) and discarding the remainder term. This will be illustrated for the case m = 1 of a single Wiener process. The simplest nontrivial Itô–Taylor expansion (3.9) gives the Euler–Maruyama scheme (3.3), which is the simplest nontrivial stochastic Taylor scheme and has strong order γ = 0.5 and weak order β = 1. The Itô–Taylor expansion (3.11) gives the Milstein scheme Yn+1 = Yn + a(tn , Yn ) n + b(tn , Yn ) Wn tn+1 s ∂b dWu dWs . + b(tn , Yn ) (tn , Xn ) ∂x tn tn
(3.14)
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3.3. Itô–Taylor numerical schemes for SODEs
33
∂b and the iterated Here the coefficient functions are f(0) = a, f(1) = b, f(1,1) = b ∂x integrals
I(0) [1]tn ,tn+1 =
tn+1 tn
dWs0 = n ,
I(1) [1]tn ,tn+1 =
tn+1 tn
dWs1 = Wn ,
and tn+1 s
I(1,1) [1]tn ,tn+1 =
tn
tn
dWτ1 dWs1 =
1 (Wn )2 − n . 2
(3.15)
The Milstein scheme converges with strong order γ = 1 and weak order β = 1. It has a higher order of convergence in the strong sense than the Euler–Maruyama scheme, but gives no improvement in the weak sense. In general, applying this idea to the Itô–Taylor expansion corresponding to the hierarchical set A gives the A-stochastic Taylor scheme A = Yn+1
α∈A
Iα id α (tn , YnA )
tn ,tn+1
= YnA +
id α (tn , YnA ) Iα [1]tn ,tn+1 . (3.16)
α∈A\∅
The strong order γ stochastic Taylor scheme, which converges with strong order γ , corresponds to the hierarchical set γ = α ∈ Mm : l(α) + n(α) ≤ 2γ
or
l(α) = n(α) = γ +
1 , 2
where n(α) denotes the number of components of a multi-index α which are equal to 0, while the weak order β stochastic Taylor scheme, which converges with weak order β, corresponds to the hierarchical set β = {α ∈ Mm : l(α) ≤ β} . For example, for m = 1 the hierarchical sets 1/2 = 1 = {∅, (0), (1)} give the Euler–Maruyama scheme, which is both strongly and weakly convergent, while the strongly convergent Milstein scheme corresponds to the hierarchical set 1 = {∅, (0), (1), (1, 1)}. Note that the Milstein scheme does not correspond to a stochastic Taylor scheme for a hierarchical set β . See Figure 3.1 for a graphical representation of the corresponding stochastic Taylor trees. Remark 4. Convergence theorems for stochastic Taylor schemes assume that the coefficients of the SODE are sufficiently often differentiable, so that all terms in these schemes make sense. Moreover, they assume that the coefficient functions of the Taylor schemes are globally Lipschitz continuous (see Theorem 10.6.3 in [82]
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34
Chapter 3. SODEs
0
0
0
1
0
0
1
1
0
0 1
0
1
1
0
1
Figure 3.1. Stochastic Taylor trees for the stochastic Euler scheme (left) and the Milstein scheme (right). The multi-indices are formed by concatenating indices along a branch from the right back toward the root ∅ of the tree. Dashed line segments correspond to remainder terms [78].
in the case of strong convergence and Theorem 14.5.1 in [82] in the case of weak convergence for the precise description of the used assumptions). These will be called the standard assumptions. They are obviously not satisfied by some SODEs that arise in many interesting applications, as will be seen later in this chapter. Remark 5. There are usually no simple expressions like (3.15) for multiple stochastic integrals involving different, independent Wiener processes. How to approximate such integrals is a major issue in stochastic numerics. The difficulty and cost of approximating stochastic integrals of higher multiplicity restricts the practical usefulness of higher order strong schemes. However, the work of Wiktorsson [124] on the simulation of the second iterated integral is very promising. Remark 6. The Itô–Taylor schemes are the “basic” schemes for the weak and strong approximation of stochastic differential equations. Based on these schemes, numerous other numerical schemes such as derivative-free Runge–Kutta-like schemes and multistep schemes have been constructed in recent years. See, e.g., Kloeden & Platen [82], Milstein [95], Milstein & Tretyakov [96], Rößler [107], and the references therein. See also Burrage, Burrage & Tian [13] for the construction of Runge– Kutta-like schemes for SODEs in terms of the Butcher B-series used for Runge–Kutta schemes for ODEs. Debrabant & Kværnø [29] provide a comparison of the B-series and the Itô–Taylor expansions used above.
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3.4. Pathwise convergence
3.4
35
Pathwise convergence
Pathwise convergence was already considered for RODEs in Chapter 2. It is also interesting for SODEs because numerical calculations of the random variables Yn in the numerical scheme above are carried out path by path. Itô stochastic calculus is, however, an L2 or a mean-square calculus and not a pathwise calculus. Nevertheless, some results for pathwise approximation of SODE are known. For example, in 1983 Talay [118] showed that the Milstein scheme SODE with a scalar Brownian motion has the pathwise error estimate 1 (Mil) (Mil) sup Xtn (ω) − Yn (ω) ≤ K,T (ω) 2 − n=0,...,NT
for all > 0 and almost all ω ∈ , i.e., he showed that the Milstein scheme converges pathwise with order 12 − . Later Gyöngy [41] and Fleury [33] showed that the Euler–Maruyama scheme has the same pathwise convergence order. Note that the error constants here depend on ω, so they are in fact random variables. The nature of their statistical properties is an interesting question, about which little is known theoretically so far and which requires further investigation. Plots of some empirical distributions of the random error constants are given in Figure 4.1 of Chapter 4. See also Kloeden & Neuenkirch [81] and Jentzen, Kloeden & Neuenkirch [70]. Given that the sample paths of a Wiener process are Hölder continuous with exponent 12 − one may ask: Is the convergence order 12 − “sharp” for pathwise approximation? The answer is no! Kloeden & Neuenkirch [81] showed that an arbitrarily high order of pathwise convergence is possible. Theorem 3.3. Under the standard assumptions, the Itô–Taylor scheme of strong order γ > 0 converges pathwise with order γ − for all > 0, i.e., (γ ) (γ ) sup Xtn (ω) − Yn (ω) ≤ K,T (ω) · γ − n=0,...,NT
for almost all ω ∈ . Thus, for example, the Milstein scheme has pathwise order 1 − rather than the lower order 12 − obtained in [118] (which was a consequence of the proof used there). The proof of Theorem 3.3, which will not be given here, is based on the Burkholder–Davis–Gundy inequality s p t p/2 E sup Xτ dWτ ≤ Cp · E Xτ2 dτ s∈[0,t]
0
0
and a Borel–Cantelli argument in the following lemma (see Lemma 2.1 in [81]).
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36
Chapter 3. SODEs
Lemma 3.4. Let γ > 0 and cp ≥ 0 for p ≥ 1. If {Zn }n∈N is a sequence of random variables with 1/p E|Zn |p ≤ cp · n−γ for all p ≥ 1 and n ∈ N, then for each > 0 there exists a nonnegative random variable K such that |Zn (ω)| ≤ K (ω) · n−γ +ε
a.s.
for all n ∈ N.
3.4.1
Numerical schemes for RODEs applied to SODEs
The Doss–Sussmann theory allows an SODE to be transformed to an RODE, at least for SODEs with additive or linear noise for which explicit transformations are known. For example, the SODE with additive noise, dXt = −Xt3 dt + dWt ,
(3.17)
can be transformed to the RODE dz = −(z + Ot )3 + Ot , dt
(3.18)
where z(t, ω) := Xt (ω) − Ot (ω) and Ot is the Ornstein–Uhlenbeck stochastic stationary process satisfying the linear SDE dOt = −Ot dt + dWt .
(3.19)
For linear noise it is better to start with a Stratonovich SODE (scalar here for simplicity) dXt = f (Xt ) dt + σ Xt ◦ dWt , (3.20) for which the corresponding Itô SODE is 1 dXt = f (Xt ) − σ 2 Xt dt + σ Xt dWt . 2 The transformation z(t, ω) := e−Ot (ω) Xt (ω) leads to the RODE dz = e−Ot f eOt z + Ot z. dt
(3.21)
(3.22)
The RODEs (3.18) and (3.22) can be solved numerically using one of the pathwise convergent numerical schemes for RODEs in Chapter 2. The Ornstein– Uhlenbeck stochastic stationary process can be simulated directly since its statistical properties are well known. Note that the cubic drift coefficient of the SODE (3.17) does not satisfy the standard assumptions for the convergence of Itô–Taylor schemes.
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3.5. Restrictiveness of the standard assumptions
3.5
37
Restrictiveness of the standard assumptions
Proofs in the literature of the above convergence orders, e.g., in the monographs Kloeden & Platen [82] and Milstein [95], assume that the coefficient functions fα in the Itô–Taylor schemes are uniformly bounded on Rd , i.e., the partial derivatives of appropriately high order of the SODE coefficient functions a, b1 , . . . , bm are uniformly bounded on Rd . This assumption is not satisfied for many SODEs in important applications such as the stochastic Ginzburg–Landau equation (the constants here are all positive), 1 2 3 (3.23) ν + σ Xt − λXt dt + σ Xt dWt , dXt = 2 or even the simpler SODE (3.17) with additive noise and a cubic nonlinearity. Matters are even worse for the Fisher–Wright equation dXt = [κ1 (1 − Xt ) − κ2 Xt ] dt + Xt (1 − Xt ) dWt , (3.24) the Feller diffusion with logistic growth SODE dXt = λXt (K − Xt ) dt + σ Xt dWt ,
(3.25)
and the Cox–Ingersoll–Ross equation dVt = κ (λ − Vt ) dt + θ Vt dWt ,
(3.26)
since the square-root function is not differentiable at zero and requires the expression under it to remain nonnegative for the SODE to make sense.
3.5.1
Counterexamples for the Euler–Maruyama scheme
The scalar SODE (3.17) with cubic drift and additive noise has a globally pathwise asymptotically stable stochastic stationary solution, since the drift satisfies a onesided dissipative Lipschitz condition. Its solution on the time interval [0, 1] for initial value X0 = 0 satisfies the stochastic integral equation t Xt = − Xs3 ds + Wt (3.27) 0
for every t ∈ [0, 1] and has finite first moment E |X1 | < ∞ (see, e.g., Theorem 2.4.1 in [91]). This SODE was first investigated numerically by Mattingly, Stuart & Higham [93] and Petersen [102]. The corresponding Euler–Maruyama scheme with constant step size = N1 is given by (N)
(N)
Yk+1 = Yk
(N) 3 − Yk + Wk (ω)
(3.28)
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38
Chapter 3. SODEs (N)
with Y0
= 0 and
Wk = W(k+1) − Wk .
The next theorem, due to Hutzenthaler, Jentzen & Kloeden [58], shows that the Euler–Maruyama scheme does not converge strongly for the SODE (3.27). In addition, the remark following the proof shows that it also does not converge weakly. Theorem 3.5. The solution Xt of (3.27) and its Euler–Maruyama approximation (N) Yk satisfy (N) (3.29) lim E X1 − YN = ∞. N→∞
Proof. Let N ∈ N be arbitrary, define rN := max{3N , 2}, and consider the event sup |Wk (ω)| ≤ 1, |W0 (ω)| ≥ rN . N := ω ∈ : k=1,...,N−1
Then it follows by induction that k−1 (N) Yk (ω) ≥ rN2
for all ω ∈ N
for every k = 1, 2, . . . , N. Clearly, for k = 1, (N) Y1 (ω) = |W0 (ω)| ≥ rN
(3.30)
(3.31)
for each ω ∈ N from the definition of the Euler–Maruyama scheme (3.28) and the definition of the set N . Therefore, assume that (3.30) holds for some k ∈ {1, 2, . . . , N − 1}. Then k−1 (N) (3.32) Yk (ω) ≥ rN2 ≥ rN ≥ 1 for every ω ∈ N . Now, by the definition of the set N , |Wk (ω)| ≤ 1
(3.33)
for every ω ∈ N , so by (3.28) it follows that 3 (N) (N) (N) Yk+1 (ω) = Yk (ω) − Yk (ω) + Wk (ω) (N) 3 (N) ≥ Yk (ω) − Yk (ω) − |Wk (ω)| (N) (N) 3 ≥ Yk (ω) − Yk (ω) − 1
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3.5. Restrictiveness of the standard assumptions
39
for every ω ∈ N . Hence the inequality (3.32) implies that (N) (N) 3 (N) 2 Yk+1 (ω) ≥ Yk (ω) − 2 Yk (ω) (N) 2 (N) ≥ Yk (ω) Yk (ω) − 2 (N) 2 (N) 2 ≥ Yk (ω) (rN − 2) ≥ Yk (ω) for every ω ∈ N , since rN − 2 ≥ 1 by definition. Thus, the induction hypothesis, i.e., the assumption that (3.30) holds for k, implies that k−1 2 k (N) = rN2 Yk+1 (ω) ≥ rN2 for all ω ∈ N , which shows that (3.30) holds for k + 1. Hence, inequality (3.30) holds for every k = 1, 2, . . . , N . In particular, N−1 (N) YN (ω) ≥ rN2 for every ω ∈ N and N ∈ N and it follows that (N) (N) E YN = YN (ω) P(dω) ≥
N
(N −1) (N) YN (ω) P(dω) ≥ P (N ) S · 22
for every N ∈ N. Now P (N ) = P
sup
sup |Wt | ≤
≥P
0≤t≤1
sup |Wt | ≤
=P ≥P
|Wk | ≤ 1 · P (|W0 | ≥ rN )
k=1,...,N−1
0≤t≤1
sup |Wt | ≤ 0≤t≤1
1 ≥ ·P 4
1 · P (|W0 | ≥ rN ) 2 √ √ 1 ·P N W1/N ≥ N rN 2 √ 1 1√ 2 · N rN e−( NrN ) 2 4
sup |Wt | ≤ 0≤t≤1
1 2 · e−NrN 2
for every N ∈ N. The next step needs the following lemma, which was proved in [58].
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40
Chapter 3. SODEs
Lemma 3.6. Let (, F , P) be a probability space and let Z : → R be an F /B(R)measurable mapping which is standard normally distributed. Then 1 1 2 2 P |Z| ∈ [x, 2x] S ≥ x e−2x (3.34) P |Z| ≥ x ≥ x e−x , 4 2 for every x ∈ [0, ∞). It follows from Lemma 3.6 that 1 2 N−1 (N) 1 E YN ≥ · P sup |Wt | ≤ · e−NrN · 22 4 2 0≤t≤1 for every N ∈ N. Hence (N) 1 lim E YN ≥ · P N→∞ 4 =
1 ·P 4
sup |Wt | ≤
0≤t≤1
sup |Wt | ≤ 0≤t≤1
1 2 N−1 · lim e−NrN · 22 2 N→∞ 1 3 N−1 · lim e−9N · 22 = ∞. 2 N→∞
Finally, since E |X1 | is finite, it follows that (N) (N) lim E X1 − YN ≥ lim E YN − E |X1 | = ∞, N→∞
N→∞
which is the assertion of the theorem. Remark 7. Theorem 3.5 implies, by Jensen’s inequality, that (N) p lim E X1 − YN = ∞ N→∞
(3.35)
for every p ∈ [1, ∞). Moreover, since E |X1 |p < ∞ for every p ∈ [1, ∞) (see, e.g., Theorem 2.4.1 in [91]), it follows that 1 p 1 (N) p p p (N) E YN = E YN − X1 + X1 p 1 1 p (N) ≥ E YN − X1 − E |X1 |p p → ∞ as N → ∞ by (3.35). Hence
(N) p lim E |X1 |p − E YN = ∞
N→∞
(3.36)
for every p ∈ [1, ∞).
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3.5. Restrictiveness of the standard assumptions
41
The above result is a special case of a more general result for scalar SODEs, dXt = a(Xt ) dt + b(Xt ) dWt ,
t ∈ [0, 1],
(3.37)
for which at least one coefficient function grows superlinearly, assuming that a solution exists, i.e., a predictable stochastic process Xt satisfying 1
P
|a(Xs )| + |b(Xs )|2 ds < ∞
=1
(3.38)
0
and
P Xt = ξ +
t
a(Xs ) ds +
0
t
g(Xs ) dWs
=1
(3.39)
0
for every t ∈ [0, 1], where the initial value is X0 = ξ . Theorem 3.7. Suppose that there exist constants C ≥ 1 and β > α > 1 such that max (|a(x)| , |b(x)|) ≥
|x|β , C
min (|a(x)| , |b(x)|) ≤ C |x|α
(3.40)
for all |x| ≥ C. If the solution Xt of the scalar SODE dXt = a(Xt ) dt + b(Xt ) dWt ,
X0 = ξ ,
(3.41)
for t ∈ [0, 1] satisfies E |X1 |p < ∞ for some p ∈ [1, ∞) and if the initial condition satisfies P ( g(ξ ) = 0 ) > 0, then (N) p (N) p and lim E |X1 |p − YN = ∞, (3.42) lim E X1 − YN = ∞ N→∞
N→∞
(N)
where YN denotes the Nth iterate of the corresponding Euler–Maruyama scheme with the constant time step = N1 . The assumption that the diffusion function does not vanish at the starting point ensures the presence of noise in the first time step. The proof of Theorem 3.7 can be found in Hutzenthaler, Jentzen & Kloeden [58]. The estimates in the proof require β > 1 and so do not apply to a superlinear drift function such as a(x) = x log x. However, the theorem does apply to the stochastic Ginzburg–Landau equation (3.23) and the Feller diffusion with logistic growth SODE (3.25), among others. Remark 8. The implicit Euler–Maruyama scheme applied to the SODE (3.27) converges in the strong sense; see, e.g., Szpruch & Mao [117].
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Chapter 4
Numerical Methods for SODEs with Nonstandard Assumptions
There are various ways to overcome the problems caused by nonstandard assumptions on the coefficients of an SODE. One way is to restrict attention to SODEs with special dynamical properties such as exponential ergodicity, e.g., by assuming that the coefficients satisfy certain dissipativity and nondegeneracy conditions; see Higham, Mao & Stuart [57], Mattingly, Stuart & Higham [93], and Milstein & Tretjakov [97]. This yields the appropriate order estimates without bounded derivatives of coefficients. However, several types of SODEs and, in particular SODEs with squareroot coefficients remain a problem. Many numerical schemes do not preserve the positivity of the solution of the SODE and hence may crash when implemented, which has led to various ad hoc modifications to prevent this happening. Sections 4.1 and 4.2 come from [70] and [71], and most of Section 4.3 comes from [59].
4.1
SODEs without uniformly bounded coefficients
Similar to Gyöngy [41], a localization argument was used by Jentzen, Kloeden & Neuenkirch [70] to show that Theorem 3.3 remains true for an SODE dXt = a(Xt ) dt +
m
j
bj (Xt ) dWt
j =1
when the coefficients satisfy a, b1 , . . . , bm ∈ C 2γ +1 (Rd ; Rd ) 43
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44
Chapter 4. SODEs with Nonstandard Assumptions
but do not necessarily have uniformly bounded derivatives, provided that a solution of the SODE exists on the time interval under consideration. This result is a special case of Theorem 4.1 below. The convergence obtained is pathwise. It applies, for example, to the stochastic Landau–Ginzburg equation (3.23) to give pathwise convergence of the Taylor schemes. Note that strong convergence of Itô–Taylor schemes does not in general hold under these assumptions according to [58] (see also Theorem 3.7 above). Empirical distributions for the random error constants of the Euler–Maruyama and Milstein schemes applied to the SODE dXt = −(1 + Xt )(1 − Xt2 ) dt + (1 − Xt2 ) dWt ,
X(0) = 0,
on the time interval [0, T ] for N = 104 sample paths and time step = 0.0001 are plotted in Figure 4.1.
(1.0)
(0.5)
Figure 4.1. Empirical distributions of K0.001 and K0.001 (sample size: N = 104 ) [71].
4.2
SODEs on restricted regions
The Fisher–Wright SODE (3.24), the Feller diffusion with logistic growth SODE (3.25), and the Cox–Ingersoll–Ross SODE (3.26) have square-root coefficients, which requires the solutions to remain in the region where the expression under the square root is nonnegative. However, numerical iterations may leave this restricted region, in which case the algorithm will terminate. One way to avoid this problem is to use an appropriately modified Itô–Taylor scheme proposed in Jentzen, Kloeden & Neuenkirch [70]. Consider an SODE dXt = a(Xt ) dt +
m
j
bj (Xt ) dWt ,
(4.1)
j =1
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4.2. SODEs on restricted regions
45
where Xt takes values in a domain D ⊂ Rd for t ∈ [0, T ]. Suppose that the coefficients a, b1 , . . . , bm are r-times continuously differentiable on D and that the SODE (4.1) has a unique strong solution. Define / D}. E := {x ∈ Rd : x ∈ Then choose auxiliary functions f , g 1 , . . . , g m ∈ C s (E; Rd ) for s ∈ N and define ! a (x) = a(x) · ID (x) + f (x) · IE (x),
x ∈ Rd ,
! bj (x) = bj (x) · ID (x) + g j (x) · IE (x),
(4.2)
x ∈ Rd ,
(4.3)
! bj (y)
(4.4)
for j = 1, . . . , m. In addition, for x ∈ ∂D define ! a (x) =
lim
y→x; y∈D
! bj (x) =
! a (y),
lim
y→x;∈y∈D
for j = 1, . . . , m, if these limits exist. Otherwise, define ! a (x) = 0 and ! bj (x) = 0 for x ∈ ∂D, respectively. Finally, define the “modified” derivative of a function h : Rd → Rd by ∂x l h(x) =
∂ h(x), ∂x l
x ∈ D ∪ E,
(4.5)
∂x l h(x)
(4.6)
and for x ∈ ∂D define ∂x l h(x) =
lim
y→x; y∈D
for l = 1, . . . , d, if this limit exists; otherwise set ∂x l h(x) = 0 for x ∈ ∂D. A modified Itô–Taylor scheme is the corresponding Itô–Taylor scheme for the SODE with modified coefficients a (Xt ) dt + dXt = !
m
j ! bj (Xt ) dWt ,
(4.7)
j =1
!0 , L !1 , . . . , L !m with the above modified coefficients and using differential operators L derivatives. Note that this method is well-defined as long as the coefficients of the equation are (2γ + 1)-times differentiable on D and the auxiliary functions are (2γ − 1)-times differentiable on E. The purpose of the auxiliary functions is twofold: to obtain a well-defined approximation scheme and to “reflect” the numerical scheme back to D if it leaves D. In particular, the auxiliary functions can always be chosen to be affine or even constant. It was shown in Jentzen, Kloeden & Neuenkirch [70] that Theorem 3.3 in Chapter 3 adapts to modified Itô–Taylor schemes for SODEs on domains in Rd . Readers are referred to [70] for the proof.
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46
Chapter 4. SODEs with Nonstandard Assumptions
" bm ∈ C 2γ +1 (D; Rd ) C 2γ −1 (E; Rd ). Theorem 4.1. Assume that ! a, ! b1 , . . . , ! Then for every > 0 and γ = 12 , 1, 32 , . . . there exists a nonnegative random (f ,g)
variable Kγ ,
such that (mod,γ ) sup Xtn (ω) − Yn (ω) ≤ Kγ(f,,g) (ω) · γ −
n=0,...,NT
(mod,γ )
for almost all ω ∈ , where = T /NT and the Yn Itô–Taylor scheme applied to the SODE (4.7).
correspond to the modified
The convergence rate does not depend on the choice of the auxiliary functions, but the random constant in the error bound clearly does. The next examples of Theorem 4.1 come from [70].
4.2.1
Examples
Consider the Cox–Ingersoll–Ross SODE dXt = κ (λ − Xt ) dt + θ Xt dWt
(4.8)
with κλ ≥ θ 2 /2 and x(0) = x0 > 0. Here D = (0, ∞) and the coefficients √ a(x) = κ (λ − x) , b(x) = θ x, x ∈ D, satisfy a, b ∈ C ∞ (D; R1 ). As auxiliary functions on E = (−∞, 0) choose, e.g., f (x) = g(x) = 0, or
f (x) = κ (λ − x) ,
x ∈ E,
g(x) = 0,
x ∈ E.
The first choice of auxiliary functions “kills” the numerical approximation as soon as it takes a negative value. However, the second is more appropriate, since if the scheme takes a negative value, the auxiliary functions force the numerical scheme to be positive again after the next steps, which recovers better the positivity of the exact solution. The coefficients of the d-dimensional stochastic Volterra–Lotka SODE dXt = diag(Xt1 , . . . , Xtd ) [(a + BXt ) dt + CXt dWt ]
(4.9)
are infinitely differentiable on Rd . Here superscripts index the components of the state vector x = (x 1 , . . . , x d ). Hence there is no need for a modification of the coefficients, and one can use the standard Itô–Taylor schemes. On the other hand, it was shown by Mao, Marion & Renshaw [92] that the solution never leaves the set D = (0, ∞)d if the initial value x0 ∈ (0, ∞)d and if the entries of the diffusion matrix C are positive
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4.3. Another type of weak convergence
47
with strictly positive diagonal components. It is, thus, also useful to apply modified Itô–Taylor methods for this SODE with D = (0, ∞)d and, e.g., f k (x) = −2|a k | x k I{x k <0} (x),
g k,j (x) = 0,
x ∈ E,
for k = 1, . . . , d and j = 1, . . . , m, where a k is the kth element of the vector a. A general version of the Fisher–Wright SODE dXt = a(Xt ) dt + Xt (1 − Xt ) dWt (4.10) typically has a polynomial drift coefficient a. Its solution should take values in the interval [0, 1], but, in general, depending on the structure of the function a, the solution can attain the boundary {0, 1} in finite time, i.e., / (0, 1)} < T τ{0,1} = inf {t ∈ [0, T ] : Xt ∈ almost surely. Thus, Theorem 4.1 cannot be applied directly to the SODE (4.10). (mod,γ ) with the auxiliary functions Nevertheless, a modified Itô–Taylor method Yn f = 0 and g = 0 to (4.10) can be used up to the first hitting time of the boundary of D. The error bound then takes the form (mod,γ ) (ω) ≤ ηγ , (ω) · n−γ + sup Yti (ω) − Yi i=0,...,n
for almost all ω ∈ and all n ∈ N, where Yti (ω) = Xt (ω) I{Xt (ω)∈(0,1)} ,
t ≥ 0, ω ∈ .
4.3 Another type of weak convergence The concepts of strong and weak convergence of a numerical scheme defined in (3.4) and (3.5), respectively, are theoretical discretization concepts. In practice, one has to estimate the expectations by a finite random sample. In the weak case, neglecting roundoff and other computational errors, one uses in fact the convergence N2 (N) 1 g YN (ωk ) = 0 (4.11) lim E g XT − 2 N→∞ N k=1 for smooth functions g : R → R with derivatives having at most polynomial growth. By the triangle inequality N2 (N) 1 E g XT − g YN (ωk ) (4.12) N2 k=1 N2 1 (N) (N) (N) − 2 g YN (ωk ) , ≤ E g XT − E g YN + E g YN N k=1
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48
Chapter 4. SODEs with Nonstandard Assumptions
where the first summand on the right-hand side of (4.12) is the weak discretization error due to approximating the exact solution with the numerical solution and the second summand is the statistical error due to approximating an expectation with the arithmetic average of finitely many independent samples. The time discretization error is usually the more difficult part to estimate and has been studied intensively in the literature under the standard assumption that the coefficients of the SODE are globally Lipschitz continuous. For superlinearly growing coefficients the situation is subtler and, as was seen in the previous chapter, weak convergence may not hold at all. However, the above convergence often holds, at least for the Euler–Maruyama scheme applied to a scalar SODE dX t = a(Xt ) dt + b(Xt ) dWt
(4.13)
with the drift coefficient a satisfying a global one-sided Lipschitz condition and the diffusion coefficient b a global Lipschitz condition, when the coefficients and the function g are sufficiently smooth, as will be shown in Theorem 4.2 below. 2 (N) The sample mean N12 N k=1 g YN (ωk ) in (4.11) is, in fact, a random variable, so the convergence (4.11) should be interpreted as holding almost surely. To formulate this in a succinct way, the sample paths of M independent Wiener processes (M) (1) Wt (ω), . . . , Wt (ω) for the same ω will be considered instead of M different sample paths Wt (ω1 ), . . . , Wt (ωM ) of the same Wiener process Wt . The choice M = N 2 ensures that such convergence for the Euler–Maruyama scheme has the same order as that for weak convergence under standard assumptions, namely, 1, since the Monte # (N) $ Carlo simulation of E g YN with M independent Euler approximations has convergence order 12 − for an arbitrarily small > 0. With these modifications, this convergence takes the form N2 (N,k) 1 lim E g XT − 2 g YN (ω) = 0, a.s., (4.14) N→∞ N k=1 (N,k)
where YN (ω) is the ω-realization of the N th iterate of the Euler–Maruyama scheme applied to the SODE with the ω-realization of the kth Wiener process (k) Wt (ω) and constant time step size := T /N . The following theorem and its explanations and outline below come from Hutzenthaler & Jentzen [59]. Theorem 4.2. Suppose that a, b, g : R → R are four times continuously differentiable functions with derivatives satisfying (n) (n) (n) (4.15) a (x) + b (x) + g (x) ≤ L 1 + |x|δ for all x ∈ R
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4.3. Another type of weak convergence
49
for n = 0,1,. . . , 4, where L ∈ (0, ∞) and δ ∈ (1, ∞) are fixed constants. Moreover, suppose that the drift coefficient a satisfies the global one-sided Lipschitz condition (x − y) · (a(x) − a(y)) ≤ L (x − y)2
for all x, y ∈ R
(4.16)
and that the diffusion coefficient satisfies the global Lipschitz condition |b(x) − b(y)| ≤ L |x − y|
for all x, y ∈ R.
(4.17)
Then there are F -measurable mappings Cε : → [0, ∞) for each ε ∈ (0, 1) and ˜ ∈ F with P[] ˜ = 1 such that an event 2 N 1 1 (N,k) E g(XT ) − g(YN (ω)) ≤ Cε (ω) · 1−ε (4.18) 2 N N m=1
˜ N ∈ N, and ε ∈ (0, 1), where Xt is the solution of the SODE (4.13) and for every ω ∈ , (N,k) YN the Nth iteration of the Euler–Maruyama scheme applied to the SODE (4.13) (k) with the Wiener process Wt for k = 1, . . . , N 2 . The assumptions of Theorem 4.2 ensure the existence of a solution Xt with continuous sample paths of the SODE (4.13) which satisfies * ) sup |Xt |p < ∞
E
(4.19)
0≤t≤T
$ # for all p ∈ [1, ∞). Hence, the expression E g(XT ) in (4.18) in Theorem 4.2 is well-defined. The proof of Theorem 4.2 can be found in [59]. The main step is to show the uniform boundedness of the restricted moments of the Euler–Maruyama approximations (N) sup E YN 1N < ∞, (4.20) N∈N
where {N }N∈N is a sequence of events whose probabilities converge to 1 sufficiently fast. Why this holds is easier to see in the special case of the SODE dXt = −Xt3 dt + Xt dWt
(4.21)
with the initial value X0 = 1, i.e., with coefficients a(x) = −x 3 and b(x) = x. The corresponding Euler–Maruyama approximation is given by (N)
(N)
Yk+1 = Yn
(N)
for k = 0, 1, . . . , N − 1 with Y0
(N) 3 (N) (N) − Yn + Yk Wk
= 1 and =
(4.22)
T N.
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50
Chapter 4. SODEs with Nonstandard Assumptions (N)
Fix N ∈ N and assume that the Euler–Maruyama iterates Yk (ω) for a given ω do not change sign up to and including the nth one for some n ∈ {1, . . . , N }, which (N) will be held fixed for now. Without loss of generality suppose that Yk (ω) ≥ 0 for x k = 0, 1, . . . , n. Then, by the inequality 1 + x ≤ e for all x ∈ R, it follows that (N) 3 (N) (N) (N) (N) Yk (ω) = Yk−1 (ω) − Yk−1 (ω) + Yk−1 Wk (ω) (N) (N) (N) (N) ≤ Yk−1 (ω) 1 + Wk (ω) ≤ Yk−1 (ω) exp Wk−1 (ω) . (4.23) Iterating this inequality then gives
(N) (N) (N) (N) Yk (ω) ≤ Yk−2 (ω) exp Wk−2 (ω) exp Wk−1 (ω) (N)
≤ Y0 (ω) exp
k−1
(N)
Wj
(ω)
j =0
(N) = exp Wk (ω) =: Dk (ω)
(4.24)
for k = 0, 1, . . . , n. (N) The Euler–Maruyama iterates Yk (ω) for k ∈ {0, 1, . . . , n} are thus bounded (N) above by those of the dominating process Dk , which has uniformly bounded absolute moments. Hence the absolute moments of the Euler–Maruyama approximation can become unbounded only if the iterates of the Euler–Maruyama (N) approximation change sign. Now, if Yk (ω) is very large for some k, then the next (N) iterate Yk+1 (ω) has the opposite sign and is very much larger in absolute value because of the negative cubic drift. The absolute values of the Euler–Maruyama approximation increases more and more through such a sequence of changes in sign. This can be avoided by restricting attention to ω in an event N for which the drift alone cannot change the sign of the Euler–Maruyama approximation. Such a sign change can come only from the diffusion term, but its coefficient grows at most linearly and such changes of sign can be controlled. Between consecutive sign changes, the Euler–Maruyama iterates are again bounded by a dominating process as above. The reader is referred to [59] for more details. Remark 9. A similar idea was used by Katsoulakis, Kossioris & Lakkis [75]. In particular, [75] motivates that one could define quasi-strong convergence with the expectations in the strong convergence criterion (3.4) restricted to events N with P(N ) → 1 sufficiently fast as N → ∞, i.e., (N) (4.25) E 1N XT − YN → 0 as N → ∞. If one would only assume that P(N ) → 1 as N → ∞ without any restriction on the speed of convergence, then quasi-strong convergence in the sense of (4.25) would be equivalent to convergence in probability.
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Chapter 5
Stochastic Partial Differential Equations
The stochastic partial differential equations (SPDEs) considered here are stochastic evolution equations of the parabolic type. The theory of such SPDEs is complicated by different types of solution concepts and function spaces depending on the spatial regularity of the driving noise process. There is an extensive literature containing existence and uniqueness results as well as many other results. For details the reader is referred to the monographs Chow [19], Da Prato & Zabczyk [24, 25, 26], Grecksch & Tudor [39], Hairer [45], Prévot & Röckner [104], Rozovskii [108], and Walsh [123]. The main result of this chapter, Theorem 5.1 in Section 5.4, establishes existence, uniqueness, and (maximal) regularity of solutions of SPDEs with globally Lipschitz continuous coefficients. It is strongly related to the results of Kruse & Larsson [87] and van Neerven, Veraar & Weis [121]. For completeness its proof is given in the appendix.
5.1
Random and stochastic PDEs
As with RODEs and SODEs, one can distinguish between random and stochastic PDEs. For simplicity of exposition, attention will be restricted here to parabolic reaction–diffusion-type equations on a bounded spatial domain D in Rd with a smooth boundary ∂D and a Dirichlet boundary condition. An example of a random PDE (RPDE) is ∂u = u + f (ζt , u), ∂t
u∂D = 0,
(5.1)
for t ≥ 0, where ζt , t ≥ 0, is a stochastic process (possibly infinite dimensional). This is interpreted and analyzed pathwise as a deterministic PDE. 53
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54
Chapter 5. Stochastic Partial Differential Equations An example of an Itô SPDE is dXt = [Xt + f (Xt )] dt + b(Xt ) dWt ,
Xt ∂D = 0,
(5.2)
for t ≥ 0, where Wt , t ≥ 0, is an infinite dimensional Wiener process of the form Wt (x, ω) =
∞
j
cj Wt (ω)φj (x),
t ≥ 0, x ∈ D,
(5.3)
j =1 j
with independent scalar Wiener processes Wt , j ∈ N. Here the family φj , j ∈ N, is an orthonormal basis in, e.g., L2 (D, R). As for SODEs, the theory of SPDEs is a mean-square theory and requires an infinite dimensional version of Itô stochastic calculus. The Doss–Sussmann theory is not as well developed for SPDEs as for SODEs, but in simple cases an SPDE can be transformed to an RPDE. For example, the SPDE (5.2) with additive noise dXt = [Xt + f (Xt )] dt + dWt , Xt = 0, ∂D
is equivalent to the RPDE ∂u u∂D = 0, = u + f (u + Ot ) + Ot , ∂t with u(t) = Xt − Ot , t ≥ 0, where Ot , t ≥ 0, is the (infinite dimensional) Ornstein– Uhlenbeck stochastic stationary solution of the linear SPDE dOt = [Ot − Ot ] dt + dWt , Ot = 0. (5.4) ∂D
As a specific example, the RPDE ∂u ∂ 2 u u∂D = 0, = 2 − u − (u + Ot )3 , ∂t ∂x on the domain D = (0, 1) is equivalent to the SPDE with additive noise 2 ∂ 3 dXt = X − X − X Xt ∂D = 0, t t t dt + dWt , 2 ∂x
(5.5)
(5.6)
on the domain D = (0, 1).
5.1.1
Mild solutions of SPDEs
Let A be an (in general) unbounded linear operator (e.g., the Laplace operator with Dirichlet boundary conditions) and (Wt )t≥0 an infinite dimensional Wiener process. An SPDE of the form dXt = [AXt + F (Xt )] dt + B(Xt ) dWt
(5.7)
on a Hilbert space H is not as easily interpreted as a stochastic integral equation as in the finite dimensional case for SODEs. In fact, there are several different
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5.2. Functional analytical preliminaries
55
interpretations in the literature as to how this should be done. The mild form of an SPDE will be used here since it is better suited for the derivation of Taylor expansions and numerical schemes. The mild form of the SPDE (5.7) is a stochastic integral equation t t A(t−s) At eA(t−s) B(Xs ) dWs a.s. (5.8) e F (Xs ) ds + X t = e X0 + 0
0
in H . Here : H → H , t ≥ 0, is a semigroup of solution operators of the deterministic ODE (really a PDE) dX = AX (5.9) dt on H , i.e., eAt = St for t ≥ 0, where St (x0 ) = X(t) is the solution of (5.9) with the initial value X(0) = x0 . In the finite dimensional case, H = Rd , the SPDE is an SODE, A is a d × d matrix, and eAt is a matrix exponential. What exactly (5.8) and the expressions in it are will be explained below. eAt
5.2
Functional analytical preliminaries
The precise mathematical setting for the types of SPDEs considered here is formulated in Section 5.3. This requires some terminology and basic concepts from infinite dimensional stochastic analysis, which will be reviewed briefly here. The reader is referred to Prévot & Röckner [104] for further details.
5.2.1
Hilbert–Schmidt and trace-class operators
Let (H , ·, ·H , · H ) and (U , ·, ·U , · U ) be two separable real Hilbert spaces and denote by (L(U , H ), · L(U ,H ) ) the real Banach space of all bounded linear operators from U to H . A bounded linear operator S ∈ L(U , H ) is called a Hilbert–Schmidt operator if there exists a set I and an orthonormal basis (ui )i∈I ⊂ U of U such that 2 i∈I Sui H < ∞. Let H S(U , H ) ⊂ L(U , H ) denote the set of all Hilbert–Schmidt operators from U to H . The set H S(U , H ) equipped with the norm 1/2 2 S H S(U ,H ) := Sui H (5.10) i∈I
for all Hilbert–Schmidt operators S ∈ H S(U , H ) and all orthonormal bases (ui )i∈I ⊂ U of U and the corresponding scalar product is a separable real Hilbert space. It can be shown that the norm · H S(U ,H ) is indeed well-defined through (5.10) (see Appendix B in Prévot & Röckner [104]). An operator S ∈ L(H ) = L(H , H ) is called a trace-class operator if there exists a set I, a family of nonnegative real numbers (λi )i∈I ⊂ [0, ∞), and an orthonormal
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56
Chapter 5. Stochastic Partial Differential Equations
basis (gi )i∈I ⊂ H of H with λi < ∞ i∈I
and
Sv =
λi gi , vH gi
(5.11)
i∈I
for all v ∈ H . Clearly, a trace-class operator is also a Hilbert–Schmidt operator.
5.2.2
Hilbert space valued random variables
Let (, F , P) be a probability space and let (H , · H , ·, ·H ) be a separable real Hilbert space. Denote by B(E) = σ (E ) the Borel sigma-algebra of a topological space (E, E ). An F /B(H )-measurable mapping Y : → H is called an (H -valued) random variable. A random variable Y : → H is said to be normal distributed if the real valued random variable v, Y H : → R is normal distributed for every v ∈ H . This chapter concentrates on SPDEs on real Hilbert spaces. For SPDEs on real Banach spaces, the reader is referred to van Neerven, Veraar & Weis [121, 122], and to Chapter 7.
5.2.3
Hilbert space valued stochastic processes
Let a, b ∈ [0, ∞) with a < b and let (, F , P) be a probability space with a normal filtration (Ft )t∈[a,b] . In addition, let (H , · H , ·, ·H ) be a separable real Hilbert space. A mapping Y : [a, b] × → H is called an (H -valued) stochastic process if Yt : → H given by Yt (ω) = Y (t, ω) for all ω ∈ is F /B(H )-measurable for every t ∈ [a, b]. A (B([a, b]) ⊗ F )/B(H )-measurable mapping Y : [0, T ] × → H is called a product measurable stochastic process. Moreover, a stochastic process Y : [a, b] × → H is said to be adapted (with respect to the filtration (Ft )t∈[a,b] ) if Yt : → H is Ft /B(H )-measurable for every t ∈ [a, b]. Let P[a,b] denote the predictable sigma-algebra on [a, b] × (with respect to the filtration (Ft )t∈[a,b] ) which is generated by {a} × A for A ∈ Fa and (s, t] × A for A ∈ Fs , a ≤ s < t ≤ b, i.e., P[a,b] = σ {a} × A : A ∈ Fa ∪ (s, t] × A : s, t ∈ [a, b], s ≤ t, A ∈ Fs . (5.12) A P[a,b] /B(H )-measurable mapping Y : [a, b] × → H is called a predictable stochastic process.
5.2.4
Infinite dimensional Wiener processes
Let T ∈ (0, ∞), let (, F , P) be a probability space with a normal filtration (Ft )t∈[0,T ] , and let (U , · U , ·, ·U ) be a separable real Hilbert space. In addition,
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5.3. Setting and assumptions
57
let Q : U → U be a trace-class operator, so there exists a set I, a summable family of nonnegative real numbers (λi )i∈I ⊂ [0, ∞), and an orthonormal basis (ui )i∈I ⊂ U of U such that Qui = λi ui for all i ∈ I. In the next step let W i : [0, T ] × → R, i ∈ I0 := {j ∈ I : λj = 0}, be a family of independent scalar Wiener processes with respect to the filtration (Ft )t∈[0,T ] . Then there exists an adapted stochastic process W : [0, T ] × → U with W0 = 0 and with P-a.s. continuous sample paths which satisfies Wt = λi Wti ui (5.13) i∈I0
P-a.s. for all t ∈ [0, T ] (see Proposition 2.1.10 in Prévot & Röckner [104]). The stochastic process W : [0, T ] × → U is called a standard Q-Wiener process with respect to the filtration (Ft )t∈[0,T ] (see Definition 2.1.12 in [104]) and Q : U → U is called its covariance operator. The stochastic process W : [0, T ] × → U has the properties # $ E Wt = 0, E v, Wt U w, Wt U = v, tQwU (5.14) for all v, w ∈ U and all t ∈ [0, T ]. Further details on standard Q-Wiener processes can be found in Section 2.1 in [104]. In many situations one is interested in an infinite dimensional Wiener process with a covariance operator that is a nonnegative symmetric bounded linear operator but not a trace-class operator (such as the identity operator on an infinite dimensional Hilbert space). This can be achieved by the concept of a cylindrical Wiener process. ˜ : U → U be a nonnegative symmetric bounded linear To illustrate this concept, let Q operator (but not necessarily a trace-class operator), let U1 , · U1 , ·, ·U1 be a separable real Hilbert space with U ⊂ U1 continuously, and let Q1 : U1 → U1 be a trace 1/2 ˜ 1/2 (U ) and with Q−1/2 (u) U1 = Q ˜ −1/2 (u) U class operator with Q1 (U1 ) = Q 1 1/2 −1/2 1/2 −1/2 ˜ (U ) (here Q ˜ for all u ∈ Q1 (U1 ) = Q and Q denote the pseudoinverses 1 1/2 1/2 of Q1 and Q , respectively; see Appendix C in [104]). A standard Q1 -Wiener process W˜ : [0, T ] × → U1 with respect to the filtration (Ft )t∈[0,T ] is then called ˜ a cylindrical Q-Wiener process with respect to the filtration (Ft )t∈[0,T ] (see Propo˜ sition 2.5.2 in [104]). Thus the cylindrical Q-Wiener process (W˜ )t∈[0,T ] in general does not take values in U but in the larger space U1 with a weaker topology. The ˜ : U → U is called the covariance nonnegative symmetric bounded linear operator Q ˜ ˜ operator of the cylindrical Q-Wiener process (Wt )t∈[0,T ] . More results on cylindrical ˜ Q-Wiener processes can be found in Section 2.5 in [104].
5.3
Setting and assumptions
The following setting will be used throughout this chapter. Let T ∈ (0, ∞) and let (, F , P) be a probability space with a normal filtration (Ft )t∈[0,T ] . Let (H , ·, ·H , · H ) and (U , ·, ·U , · U ) be two separable real
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58
Chapter 5. Stochastic Partial Differential Equations
Hilbert spaces. Moreover, let Q : U → U be a nonnegative symmetric bounded linear operator and let (Wt )t∈[0,T ] be a cylindrical Q-Wiener process with respect to the filtration (Ft )t∈[0,T ] . Assumption 1 (linear operator A). Let I be a finite or countable set and let (λi )i∈I ⊂ R be a family of real numbers with inf i∈I λi > −∞. In addition, let (ei )i∈I ⊂ H be an orthonormal basis of H and let A : D(A) ⊂ H → H be a linear operator with Av = −λi ei , vH ei (5.15) i∈I
for all v ∈ D(A) and with D(A) = w ∈ H :
2 2 i∈I |λi | |ei , wH |
<∞ .
Let η ∈ [0, ∞) be a nonnegative real +number with+ η > − inf i∈I λi . For r ∈ R let Hr := D (η − A)r with norm · Hr := +(η − A)r (·)+H and corresponding scalar product ·, ·Hr denote the real Hilbert spaces of domains of fractional powers of the linear operator η − A : D(A) ⊂ H → H . (See Chapter 8 for more details on this family of Hilbert spaces.) Assumption 2 (drift term F ). Let α, δ ∈ R be real numbers with δ − α < 1 and let F : Hδ → Hα be a globally Lipschitz continuous mapping. 1
Let (U0 , ·, ·U0 , · U0 ) denote the separable real Hilbert space U0 := Q 2 (U ) 1
1
with v, wU0 = Q− 2 v, Q− 2 wU for all v, w ∈ U0 (see, for example, Section 2.3.2 in [104]). For an arbitrary bounded linear operator S ∈ L(U ), let S −1 : im(S) ⊂ U → U denote the pseudoinverse of S (see Appendix C in [104]). Assumption 3 (diffusion term B). Let β ∈ R be a real number with δ − β < let B : Hδ → H S(U0 , Hβ ) be a globally Lipschitz continuous mapping.
1 2
and
Assumption 4 (initial value ξ ). Let γ ∈ [δ, min(α + 1, β + 12 )] and p ∈ [2, ∞) be real numbers and let ξ : → Hγ be an F0 /B Hγ -measurable mapping with # $ p E ξ Hγ < ∞.
5.4
Existence, uniqueness, and regularity of solutions of SPDEs
The literature contains many existence and uniqueness theorems for mild solutions of SPDEs. Theorem 5.1 below provides an existence, uniqueness, and regularity result for solutions of SPDEs with globally Lipschitz continuous coefficients in the setting of Section 5.3.
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5.4. Existence, uniqueness, and regularity of solutions of SPDEs
59
Some additional notation is needed to formulate this result. For real numbers T ∈ (0, ∞), r ∈ (0, 1) and a normed real vector space (V , · V ), define f C r ([0,T ],V ) := sup f (t) V + t∈[0,T ]
f (t2 ) − f (t1 ) V ∈ [0, ∞], |t2 − t1 |r t1 ,t2 ∈[0,T ] sup
(5.16)
t1 =t2
f C([0,T ],V ) := f C 0 ([0,T ],V ) := sup f (t) V ∈ [0, ∞]
(5.17)
t∈[0,T ]
for all mappings f : [0, T ] → V from [0, T ] to V , let C 0 ([0, T ], V ) := C([0, T ], V ) denote the set of all continuous functions from [0, T ] to V , and let (5.18) C r ([0, T ], V ) := f : [0, T ] → V : f C r ([0,T ],V ) < ∞ denote the space of all r-Hölder continuous functions from [0, T ] to V . Theorem 5.1 (existence, uniqueness, and regularity of solutions of SPDEs). Assume that the setting in Section 5.3 is fulfilled. Then there exists a unique (up-tomodifications) stochastic process X : [0, T ] × → Hγ satisfying # predictable p $ supt∈[0,T ] E Xt Hγ < ∞ and Xt = eAt ξ +
t
eA(t−s) F (Xs ) ds +
t
eA(t−s) B(Xs ) dWs
(5.19)
0
0
P-a.s. for all t ∈ [0, T ]. In addition, 1
X ∈ ∩r∈(−∞,γ ] C min(γ −r, 2 ) ([0, T ], Lp (; Hr )). A predictable stochastic process X : [0, T ] × → Hγ satisfying (5.19) with # p $ supt∈[0,T ] E Xt Hγ < ∞ is called a mild solution of the SPDE dXt = AXt + F (Xt ) dt + B(Xt ) dWt ,
X0 = ξ ,
t ∈ [0, T ].
(5.20)
Theorem 5.1 thus proves existence, uniqueness (up to modifications), and regularity of mild solutions of the SPDE (5.20). The proof of Theorem 5.1 is given in the appendix. Similar existence, uniqueness, and regularity results for mild solutions of SPDEs can be found in Kruse & Larsson [87] and van Neerven, Veraar & Weis [121]. In addition to mild solutions of SPDEs (see, e.g., Da Prato & Zabczyk [24, 25]; see also Walsh [123] for another type of mild solution for SPDEs), weak solutions of SPDEs are also frequently studied in the literature (see, e.g., Prévot & Röckner [104] and Rozovskii [108]).
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60
5.5
Chapter 5. Stochastic Partial Differential Equations
Examples
The setting in Section 5.3 covers several types of examples, including finite dimensional SODEs and SPDEs with second and fourth order spatial differential operators. SPDEs with time-dependent coefficients are also included.
5.5.1
Finite dimensional SODEs
The abstract setting for SPDEs of evolutionary type in Section 5.3 also covers finite dimensional SODEs. The consequences of Theorem 5.1 are illustrated here for this finite dimensional context. Let T ∈ (0, ∞) and let (, F , P) be a probability space with a normal filtration (Ft )t∈[0,T ] . For d, m ∈ N let H = Rd with v, wH = v, wRd , v H = v Rd for all v, w ∈ Rd and let U = Rm with v, wU = v, wRm , v U = v Rm for all v, w ∈ Rm . In addition, let Q = I : Rm → Rm be the identity operator on the Rm and let W : [0, T ] × → Rm be an m-dimensional standard (Ft )t∈[0,T ] -Wiener process. Finally, let I = {1, 2, . . . , d}, λ1 = λ2 = · · · = λd = 0 and let e1 = (1, 0, . . . , 0), e2 = (0, 1, 0, . . . , 0), . . . , ed = (0, . . . , 0, 1) ∈ H . The linear operator A : D(A) ⊂ H → H in Assumption 1 thus satisfies Av = 0 for all v ∈ D(A) = H = Rd . In the next step, let F : Rd → Rd and B : Rd → H S(Rm , Rd ) be two globally Lipschitz continuous functions. In addition, for some real number p ∈ [2, ∞), let p ξ : → Rd be an F0 /B(Rd )-measurable mapping with E[ ξ Rd ] < ∞. The setting in Section 5.3 is thus satisfied with η ∈ (0, ∞) arbitrary and α = β = δ = γ = 0. Theorem 5.1 therefore implies the existence of a unique (up-to-modifications) 1 predictable stochastic process X : [0, T ] × → Rd ∈ C 2 ([0, T ], Lp (; Rd )) satisfying the SODE t t Xt = X0 + F (Xs ) ds + B(Xs ) dWs (5.21) 0
0
P-a.s. for all t ∈ [0, T ]. For further existence and uniqueness results for finite dimensional SODEs see Klenke [76] and Theorem 2.9 in Chapter 5 in Karatzas & Shreve [74].
5.5.2
Second order SPDEs
Some SPDEs involving spatial derivatives of second order are formulated here as examples of the setting in Section 5.3. In order to make the presentations as explicit as possible, SPDEs on the simple domain D = (0, 1)d for d ∈ N are described here, although SPDEs on much more complicated domains in Rd also fit into this setting. Let T ∈ (0, ∞) and let (, F , P) be a probability space with a normal filtration (Ft )t∈[0,T ] . Denote by H = U = L2 (D, R) the real Hilbert space of equivalence
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5.5. Examples
61
classes of Lebesgue square integrable functions from D ⊂ Rd to R equipped with the norm and the scalar product 1
v H = v U =
D
|v(x)|2 dx
2
,
v, wH = v, wU =
D
v(x) · w(x) dx
(5.22) for all v, w ∈ H = U . As a prominent example of the linear operator in Assumption 1 let ϑ ∈ (0, ∞) be a real number and for I = Nd let the eigenfunctions (ei )∈I ⊂ H and the eigenvalues (λi )∈I ⊂ R of the linear operator −A : D(A) ⊂ H → H be given by d (5.23) ei (x) = 2 2 sin(i1 πx1 ) · . . . · sin(id π xd ), λi = ϑπ 2 |i1 |2 + · · · + |id |2 for all x = (x1 , . . . , xd ) ∈ D = (0, 1)d and all i = (i1 , . . . , id ) ∈ I = Nd . The linear operator A : D(A) ⊂ H → H in Assumption 1 thus reduces to the Laplacian equipped with Dirichlet boundary conditions times the constant ϑ > 0. For r ∈ R denote by Hr := D((−A)r ) the real Hilbert spaces of domains of fractional powers of the linear operator −A : D(A) ⊂ H → H with scalar product ·, ·Hr and norm · Hr := (−A)r (·) H . In the next step, let c ∈ [0, ∞) be a real number and let f , b : D × R → R be two Borel measurable functions with |f (x, 0)|2 dx < ∞, |b(x, 0)|2 dx < ∞, D
D
and |f (x, y1 ) − f (x, y2 )| + |b(x, y1 ) − b(x, y2 )| ≤ c |y1 − y2 |
(5.24)
for all x ∈ D and all y1 , y2 ∈ R. Condition (5.24) ensures that f and b are uniformly globally Lipschitz continuous in their last variable. Finally, set α = δ = 0 and observe that the (in general nonlinear) mapping F : H → H given by F (v) (x) = f (x, v(x)) (5.25) for all x ∈ D and all v ∈ H satisfies Assumption 2. The operator F is known as a Nemytskii operator in the literature (see, e.g., Runst & Sickel [109]). Assumption 3 will be verified separately for the two cases of space–time white noise and trace–class noise. Space–time white noise Let d = 1, so D = (0, 1), and let Q = I be the identity operator on H = U = L2 ((0, 1), R). Let β ∈ (− 12 , − 14 ) be arbitrary and consider the (in general nonlinear)
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62
Chapter 5. Stochastic Partial Differential Equations
mapping B : H → H S(H , Hβ ) given by 1 b(x, v(x)) · u(x) · w(x) dx B(v)u (w) =
(5.26)
0
for all u, v ∈ H and all w ∈ H−β . Then B is well-defined, since it follows from the embedding H−β ⊂ C([0, 1], R) continuously and by the Cauchy–Schwarz inequality that 1 |b(x, v(x)) · u(x) · w(x)| dx 0 1
≤ 0
≤
1
|b(x, v(x)) · u(x)| dx w C([0,1],R) (5.27)
1 2
1
|b(x, v(x))|2 dx
0
1 2
|u(x)|2 dx
0
w C([0,1],R)
= b(·, v(·)) H u H w C([0,1],R) < ∞ for all u, v ∈ H and all w ∈ H−β ⊂ C([0, 1], R). Inequality (5.27) implies, in particular, that 1 |b(x, v(x)) · u(x) · w(x)| dx ≤ b(·, v(·)) H u C([0,1],R) w H < ∞ (5.28) 0
for all v, w ∈ H and all u ∈ H−β ⊂ C([0, 1], R) and this shows B(v)(H−β ) ⊂ H for all v ∈ H . In addition, observe that w1 , B(v)w2 H = B(v)w1 , w2 H
(5.29)
for all v ∈ H and all w1 , w2 ∈ H−β . Equation (5.29) then implies that + +2 B(v) − B(w) 2H S(H ,Hβ ) = +(−A)β (B(v) − B(w))+H S(H ) , - ei , (−A)β (B(v) − B(w)) ej 2 = H i,j ∈I
, - ej , (B(v) − B(w)) (−A)β ei 2
=
H
i,j ∈I
=
(B(v) − B(w)) ei 2H |λi |2β
i∈I
and B(v) − B(w) H S(H ,Hβ ) ≤
√
2
i∈I
1 2
|λi |
1 |b(x, v(x)) − b(x, w(x))| dx
2β
2
2
D
√ + + ≤ c 2 +(−A)β +H S(H ) v − w H < ∞
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5.5. Examples
63
+ + for all v, w ∈ H due to inequality (5.24). Note that +(−A)β +H S(H ) < ∞ since β < − 14 . The mapping B : H → H S(H , Hβ ) in (5.26) thus fulfills Assumption 3. Finally, assume that γ = β + 12 in Assumption 4. Then Theorem 5.1 implies that the SPDE 2 ∂ dXt (x) = ϑ X (x) + f (x, X (x)) dt + b(x, Xt (x)) dWt (x) (5.30) t t ∂x 2 with X0 (x) = ξ (x) and Xt (0) = Xt (1) = 0 for x ∈ (0, 1) and t ∈ [0, T ] has a unique (up-to-modifications) mild solution X : [0, T ] × → Hβ+ 1 . Here (Wt )t∈[0,T ] is a 2 cylindrical I -Wiener process on H . For example, the SPDE with additive noise 2 ∂ X (x) + f (x, X (x)) dt + dWt (x) dXt (x) = ϑ t t ∂x 2
(5.31)
with X0 (x) = ξ (x) and Xt (0) = Xt (1) = 0 for x ∈ (0, 1), t ∈ [0, T ], where b(x, y) = 1 for all x ∈ (0, 1), y ∈ R, and the stochastic heat equation with linear multiplicative noise 2 ∂ dXt (x) = ϑ X (x) dt + Xt (x) dWt (x) (5.32) t ∂x 2 with X0 (x) = ξ (x) and Xt (0) = Xt (1) = 0 for x ∈ (0, 1), t ∈ [0, T ], where f (x, y) = 0, b(x, y) = y for all x ∈ (0, 1), y ∈ R, both have unique (up-to-modifications) mild solutions. Remark 10. It was shown by Walsh [123] that mild solutions do not exist for space– time white noise in spatial domains of dimension higher than one. Trace–class noise In this subsection there is no restriction on the dimension d ∈ N of the spatial domain. Let J be a countable set and let gj : D¯ = [0, 1]d → R, j ∈ J, be a family of 2 continuous functions + + which form an orthonormal basis in H = L (D, R) and which + + satisfy supj ∈J gj C(D, ¯ R) < ∞. In addition, let rj j ∈J ⊂ [0, ∞) be a family of nonnegative real numbers with j ∈J rj < ∞ and let Q : H → H be a bounded linear operator given by Qu =
j ∈J
, rj gj , u H gj
(5.33)
for all u ∈ H . Note that Q : H → H is a trace–class operator.
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64
1 2
Chapter 5. Stochastic Partial Differential Equations Next let (U0 , · U0 , ·, ·U0 ) denote the separable real Hilbert space U0 := 1
Q (U ) with u U0 = Q− 2 u H for all v ∈ U0 and consider the mapping B : H → H S(U0 , H ) given by (B(v)u) (x) = b(x, v(x)) · u(x) (5.34) 1 ¯ R) confor all x ∈ D, v ∈ H , and u ∈ U0 . The embedding U0 = Q 2 (U ) ⊂ C(D, tinuously then demonstrates that B in (5.34) is indeed well-defined. Moreover, the inequality 1 + +2 2 1 + + B(v) − B(w) = (B(v) − B(w)) Q 2 gi
H S(U0 ,H )
H
i∈I
+ 1 +2 +Q 2 gi + ¯ ≤ C(D,R) i∈I
1 2
1 |b(x, v(x)) − b(x, w(x))| dx 2
D
2
+ 1+ ≤ c +Q 2 +H S(H ) sup gj C(D, ¯ R) v − w H < ∞ j ∈J
(5.35) holds for all v, w ∈ H , which shows that B satisfies Assumption 3. Finally, assume γ = 12 in Assumption 4. Then Theorem 5.1 implies that the SPDE ) * ∂2 ∂2 + · · · + 2 Xt (x) + f (x, Xt (x)) dt + b(x, Xt (x)) dWt (x) dXt (x) = ϑ ∂x12 ∂xd (5.36) and with X0 (x) = ξ (x) and Xt |∂D ≡ 0 for x = (x1 , . . . , xd ) ∈ D = t ∈ [0, T ] has a unique (up to modifications) mild solution X : [0, T ] × → H 1 , 2 where W : [0, T ] × → H is a standard Q-Wiener process here. For example, the SPDE with additive noise ) * ∂2 ∂2 + · · · + 2 Xt (x) + f (x, Xt (x)) dt + dWt (x) (5.37) dXt (x) = ϑ ∂x12 ∂xd (0, 1)d
with X0 (x) = ξ (x) and Xt |∂D ≡ 0 for x = (x1 , . . . , xd ) ∈ D = (0, 1)d and t ∈ [0, T ], where b(x, y) = 1 for all x ∈ (0, 1)d , y ∈ R, and the stochastic heat equation with linear multiplicative noise ) * ∂2 ∂2 + · · · + 2 Xt (x) dt + Xt (x) dWt (x) (5.38) dXt (x) = ϑ ∂x12 ∂xd with X0 (x) = ξ (x) and Xt |∂D ≡ 0 for x = (x1 , . . . , xd ) ∈ D = (0, 1)d and t ∈ [0, T ], where f (x, y) = 0, b(x, y) = y for all x ∈ (0, 1)d , y ∈ R, both have unique (up-tomodifications) mild solutions.
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5.5. Examples
5.5.3
65
Fourth order SPDEs
Some SPDEs involving spatial derivatives of up to fourth order are formulated here as an example of the setting in Section 5.3. For simplicity only SPDEs on the onedimensional domain (0, 1) are considered. To be more precise, let H = U = L2 ((0, 1), R) be the real Hilbert space of equivalence classes of Lebesgue square integrable functions from (0, 1) to R equipped with the norm and the scalar product v H = v U =
1
1 2
|v(x)| dx 2
,
0
v, wH = v, wU =
1
v(x) · w(x) dx
0
(5.39) for all v, w ∈ H = U . In addition, let Q = I : H → H be the identity operator on H . For the linear operator A : D(A) ⊂ H → H in Assumption 1, let I = N and let (ei )∈I ⊂ H and (λi )∈I ⊂ R be given by √ ei (x) = 2 sin(iπ x), λi = π 4 i 4 (5.40) for all x ∈ (0, 1) and all i ∈ N. The linear operator A : D(A) ⊂ H → H in Assump ∂4 tion 1 thus fulfills (Av)(x) = − ∂x 4 v(x) for all v ∈ D(A) with D(A) = f ∈ H 4 ((0, 1), R) : f (0) = f (1) = f (0) = f (1) = 0 . (5.41) Here H 4 ((0, 1), R) = W 4,2 ((0, 1), R) is the Sobolev space of all four times weakly differential functions from (0, 1) to R with Lebesgue square integrable derivatives (see, e.g., Section 6.4 in Renardy & Rogers [106]). For r ∈ R denote by Hr := D((−A)r ) the real Hilbert spaces of domains of fractional powers of the linear operator −A : D(A) ⊂ H → H with scalar product ·, ·Hr and norm · Hr := (−A)r (·) H . Next let c ∈ [0, ∞) be a real number and let f , b : (0, 1) × R2 → R be two Borel measurable functions with |f (x, 0, 0)|2 dx < ∞, |b(x, 0, 0)|2 dx < ∞ D
D
and |f (x, y1 , z1 ) − f (x, y2 , z2 )| + |b(x, y1 , z1 ) − b(x, y2 , z2 )| ≤ c (|y1 − y2 | + |z1 − z2 |) (5.42) for all x ∈ (0, 1) and all y1 , z1 , y2 , z2 ∈ R. Then let α = − 12 , let δ = 14 , and consider the (in general nonlinear) operator F : H 1 → H− 1 given by 4
2
F (v) (w) =
1
f (x, v (x), v(x)) · w (x) dx
(5.43)
0
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66
Chapter 5. Stochastic Partial Differential Equations
for all v ∈ H 1 and all w ∈ H 1 . Observe that
4
2
.
1
F (v) (w) = f (·, v (·), v(·)), −(−A) 2 w
/ H
1 = − (−A) 2 f (·, v (·), v(·)) (w)
1 for all v ∈ H 1 and all w ∈ H 1 . It thus follows that F (v) = −(−A) 2 f (·, v (·), v(·)) 4 2 ∈ H−1/2 for all v ∈ H 1 . This and inequality (5.42) then imply that 4
F (v) − F (w) H−1/2 =
1
f (x, v (x), v(x)) − f (x, w (x), w(x))2 dx
1 2
0
+ + + + + + ≤ c +v − w +H + v − w H = c v − w H1/4 + +(−A)−1/4 (v − w)+ H1/4 ≤ c 1 + (−A)−1/4 L(H ) v − w H1/4 < ∞ for all v, w ∈ H1/4 . This demonstrates that F given by (5.43) indeed satisfies Assumption 2 with α = − 12 and δ = 14 . In the next step, let β ∈ (− 14 , − 18 ) be arbitrary and consider the (in general nonlinear) mapping B : Hδ → H S(H , Hβ ) given by b(x, v (x), v(x)) · u(x) · w(x) dx (5.44) B(v)u (w) = D
for all u ∈ H , v ∈ Hδ and all w ∈ H−β . Then B is well-defined, since, by the embedding H−β ⊂ C([0, 1], R) continuously and the Cauchy–Schwarz inequality, 1 1 b(x, v (x), v(x)) · u(x) dx w C([0,1],R) |b(x, v(x)) · u(x) · w(x)| dx ≤ 0
0
≤
b(x, v (x), v(x))2 dx
1
1 2
0
1
1 2
|u(x)|2 dx
0
w C([0,1],R)
(5.45)
+ + = +b(·, v (·), v(·))+H u H w C([0,1],R) < ∞ for all u ∈ H , v ∈ Hδ and all w ∈ H−β ⊂ C([0, 1], R). As in the space–time white noise case in Section 5.5.2, observe that , - ei , (−A)β (B(v) − B(w)) ej 2 B(v) − B(w) 2H S(H ,Hβ ) = H i,j ∈I
, - ej , (B(v) − B(w)) (−A)β ei 2 = (B(v) − B(w)) ei 2H |λi |2β = H i,j ∈I
i∈I
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5.5. Examples
67
for all v, w ∈ H 1 . Therefore 4
B(v) − B(w) H S(H ,Hβ ) 1 1 2 √ 2 2β b(x, v (x), v(x)) − b(x, w (x), w(x))2 dx |λi | ≤ 2 i∈I
D
≤ c 2 (−A)β H S(H ) 1 + (−A)−1/4 L(H ) v − w H1/4 < ∞ √
for all v, w ∈ H 1 due to inequality (5.42). Note that (−A)β H S(H ) < ∞ since 4
β < − 18 . The mapping B : Hδ → H S(H , Hβ ) in (5.44) thus fulfills Assumption 3. Finally, assume γ = β + 12 in Assumption 4. Then Theorem 5.1 implies that the Cahn–Hilliard–Cook-type SPDE 4 2 ∂ ∂ Xt (x) + f (x, Xt (x), Xt (x)) dt dXt (x) = − ∂x 4 ∂x 2 + b(x, Xt (x), Xt (x)) dWt (x)
(5.46)
with X0 (x) = ξ (x) and Xt (0) = Xt (1) = Xt (0) = Xt (1) = 0 for x ∈ (0, 1) and t ∈ [0, T ] has a unique (up-to-modifications) mild solution X : [0, T ]× → Hβ+ 1 . Here 2 (Wt )t∈[0,T ] is a cylindrical I -Wiener process on H . For more results on this type of fourth order SPDE see Blömker [8].
5.5.4
SPDEs with time-dependent coefficients
The setting in Section 5.3 also applies to SPDEs with time-dependent coefficients. Let T ∈ (0, ∞), let (, F , P) be a probability space with a normal filtration (Ft )t∈[0,T ] , and let (H˜ , ·, ·H˜ , · H˜ ) and (U , ·, ·U , · U ) be two separable real Hilbert spaces. Then let Q : U → U be a nonnegative symmetric bounded linear operator and let (Wt )t∈[0,T ] be a cylindrical Q-Wiener process with respect to (Ft )t∈[0,T ] . In addition, let I˜ be a finite or countable set, let (λi )i∈I˜ ⊂ R be a family of real numbers with inf i∈I˜ λi > −∞, and let (e˜i )i∈I˜ ⊂ H˜ be an orthonormal basis of H˜ . ˜ ⊂ H˜ → H˜ be a linear operator with Then let A˜ : D(A) ˜ = Av −λi e˜i , vH˜ e˜i (5.47) i∈I˜
2 2 <∞ . ˜ and with D(A) ˜ = w ∈ H˜ : for all v ∈ D(A) i∈I˜ |λi | e˜i , wH˜ Let η˜ ∈ [0, ∞) be a nonnegative real number with η˜ > − inf i∈I˜ λi and for r ∈ R ˜ r the real Hilbert spaces of domains of fractional powers denote by H˜ r := D (η − A)
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68
Chapter 5. Stochastic Partial Differential Equations
+ + ˜ r (·)+ ˜ ˜ ⊂ H˜ → H˜ with norm · ˜ := +(η − A) of the linear operator η˜ − A˜ : D(A) Hr H and corresponding scalar product ·, ·H˜ r . In the next step let α, δ ∈ R be real numbers with δ −α < 1 and let F˜ : R× H˜δ → H˜ α be globally Lipschitz continuous. In addition, denote by U0 , ·, ·U0 , · U0 the 1
1
1
separable real Hilbert space U0 := Q 2 (U ) with v, wU0 = Q− 2 v, Q− 2 wU for all v, w ∈ U0 . Let β ∈ R be a real number with δ − β < 12 and let B˜ : R × H˜ δ → H S(U0 , H˜ β ) be globally Lipschitz continuous. Finally, let γ ∈ [δ, min(α + 1, β + 12 )] and p ∈ [2, ∞) be numbers and let ξ˜ : → H˜ γ be an F0 /B(H˜ γ )-measurable # real p $ mapping with E ξ ˜ < ∞. Hγ
Now consider the separable real Hilbert space (H := R × H˜ , ·, ·H , · H ) with 1 ˜ ∪ I˜ (t, v) H = (|t|2 + v 2˜ ) 2 for all t ∈ R and all v ∈ H˜ . Moreover, define I := {I} H and λI˜ := 0, and consider the family (ei )i∈I ⊂ H given by eI˜ = (1, 0) and ei = (0, e˜i ) ˜ Note that the family (ei )i∈I ⊂ H is an orthonormal basis of H . for all i ∈ I. ˜ for all t ∈ R The linear operator in Assumption 1 thus fulfills A(t, v) = (0, Av) and all v ∈ H˜ . Furthermore, let η ∈ [η, ˜ ∞) ∩ (0, ∞) and for r ∈ R denote by Hr := D((−A)r ) the real Hilbert spaces of domains of fractional powers of the linear operator η − A : D(A) ⊂ H → H with scalar product ·, ·Hr and norm · Hr := (η − A)r (·) H . Next note that F : Hδ → Hα given by F (t, v) = (1, F˜ (t, v)) for all t ∈ R, v ∈ H˜ δ ˜ v)u) satisfiesAssumption 2, while B : Hδ → H S(U0 , Hβ ) given by B(t, v)u = (0, B(t, for all t ∈ R, v ∈ H˜ δ , u ∈ U0 satisfies Assumption 3. Finally, let ξ : → Hγ be given by ξ (ω) = (0, ξ˜ (ω)) for all ω ∈ . Clearly, ξ satisfies Assumption 4. The setting in Section 5.3 is thus fulfilled. Theorem 5.1 therefore implies the existence of a unique (up-to-modifications) predictable stochastic process X : [0, T ] × → Hγ ∈ ∩r∈(−∞,γ ] C min(γ −r,1/2) ([0, T ], Lp (; Hr )), which satisfies (5.19). Define X˜ : [0, T ] × → H˜ γ by X˜ t (ω) = i∈I˜ ei , Xt (ω)H e˜i for all t ∈ [0, T ] and all ω ∈ . Note that X : [0, T ] × → H˜ γ is a predictable stochastic process in ∩r∈(−∞,γ ] C min(γ −r,1/2) ([0, T ], Lp (; H˜ r )) satisfying ˜ X˜ t = eAt ξ˜ +
t
e 0
˜ A(t−s)
F˜ (s, X˜ s ) ds +
t
˜ ˜ X˜ s ) dWs eA(t−s) B(s,
(5.48)
0
P-a.s. for all t ∈ [0, T ].
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Chapter 6
Numerical Methods for SPDEs
The numerical approximation of SPDEs, specifically stochastic evolution equations of the parabolic or hyperbolic type, encounters all the difficulties that arise in the numerical solution of both deterministic PDEs and finite dimensional SODEs as well as many more due to the infinite dimensional nature of the driving noise processes. The state of development of numerical schemes for SPDEs compares with that for SODEs in the early 1970s. There is now a large number of papers on the numerical approximation of SPDEs. An extensive list can be found in the review paper [68]. However, most of the numerical schemes that have been proposed to date have a low order of convergence, especially in terms of an overall computational effort, and only recently has it been shown how to construct higher order schemes. The breakthrough for SODEs started with the Milstein scheme and continued with the systematic derivation of stochastic Taylor expansions and the numerical schemes based on them. These stochastic Taylor schemes are based on an iterated application of the Itô formula. The crucial point is that the multiple stochastic integrals which they contain provide more information about the noise processes within discretization subintervals, and this allows an approximation of higher order to be obtained. This theory was briefly recalled in Chapter 3. There is, however, no general Itô formula for the solutions of stochastic PDEs in Hilbert spaces or Banach spaces (see Chapter 7 for more details). Nevertheless, it has recently been shown that Taylor expansions for the solutions of such equations can be constructed by taking advantage of the mild-form representation of the solutions. These new results will be presented in the subsequent chapters. This chapter is an extract from [68].
69
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70
Chapter 6. Numerical Methods for SPDEs
6.1 An early result To fix ideas consider a parabolic SPDE with a Dirichlet boundary condition on a bounded domain D in Rd of the form dXt = [AXt + f (Xt )] dt + g(Xt ) dWt
(6.1)
and suppose for now that the coefficients satisfy the assumptions of the previous chapter. In addition, assume that the eigenvalues λj and the corresponding eigenfunctions φj ∈ H01,2 (D) of the operator −A, i.e., with −Aφj = λj φj ,
j = 1, 2, . . . ,
form an orthonormal basis in L2 (D) with λj → ∞ as j → ∞. Assume also that Wt is a standard scalar Wiener process. Projecting the SPDE (6.1) onto the N -dimensional subspace HN of L2 (D) spanned by {φ1 , . . . , φN } gives an N -dimensional Itô–Galerkin SODE in RN of the form (N) (N) (N) (N) (6.2) dXt = AN Xt + fN (Xt ) dt + gN (Xt ) dWt , N,j φ ∈ H or (X N,1 , . . . , X N,N ) where X (N) is written synonymously for N N j =1 X j N ∈ R according to the context. Moreover, fN = PN f H and gN = PN g H , where N
N
f and g are now interpreted as mappings of L2 (D) or H01,2 (D) into itself, and PN is the projection of L2 (D) or H01,2 (D) onto HN , while AN = PN AH is the diagonal N # $ matrix diag λj , . . . , λN . Grecksch & Kloeden [38] showed in 1996 that the combined truncation and global discretization error for a strong order γ stochastic Taylor scheme applied to (6.2) with constant time step has the form γ + 12 +1 γ (N,) −1/2 ≤ KT λN+1 + λN (6.3) , max E Xk − Yk k=0,1,...,NT
L2 (D)
where x denotes the integer part of the real number x and the constant KT depends on the initial value and bounds on the coefficient functions f and g (and their derivatives) of the SPDE (6.1) as well as on the length of the time interval [0, T ] under (N,) consideration. Here and below Yk is written Yk to emphasize the dependence on the time step size . Since λj → ∞ as j → ∞, a very small time step is needed in high dimensions for convergence, i.e., the Itô–Galerkin SODE (6.2) is stiff and explicit schemes such as strong stochastic Taylor schemes are not really appropriate. Obviously, an implicit scheme should be used here, but the special structure of the SODE (6.2) suggests a simpler linear-implicit scheme, since it is the matrix AN in the linear part of the
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6.2. Other results
71
drift coefficient that causes the troublesome growth with respect to the eigenvalues, so only this part of the drift coefficient needs to be made implicit. For example, the linear-implicit Euler scheme for the SODE (6.2) is (N)
(N)
Yk+1 = Yk
(N)
(N)
(N)
+ AN Yk+1 + fN (Yk ) + gN (Yk ) Wk ,
(6.4)
(N)
which is easily solved for Yk+1 because its matrix IN − AN is diagonal. Kloeden & Shott [84] showed that for a linear-implicit strong order γ stochastic Taylor scheme the combined error has the form (N,) −1/2 max E Xk − Yk ≤ KT λN+1 + γ . (6.5) L2 (D)
k=0,1,...,NT
The time step can thus be chosen independently of the dimension N of the Itô– Galerkin SODE (6.2). The above results are of limited use because Wt is only one-dimensional and the proofs of the convergence of Taylor schemes for SODEs in the monographs [82, 95] assume that partial derivatives of the coefficient functions of the Galerkin SODE are uniformly bounded on RN .
6.2
Other results
Much of the literature is concerned with a semilinear stochastic heat equation with additive space–time white noise W˙ t on the one-dimensional domain D = (0, 1) over the time interval [0, T ], i.e., ∂u ∂ 2 u = 2 + f (u) + W˙ t ∂t ∂x
(6.6)
with the Dirichlet boundary condition. These papers also use an overall convergence rate which combines the two components of the error bound due, respectively, to the temporal and spatial discretization. The overall convergence rate is usually expressed in terms of the computational cost of the scheme. For one-dimensional domains this is defined by K = N · M, where N arithmetical operations, random number, and function evaluations per time (N,M) of the scheme (this is related to the dimension step to calculate the next iterate Yk T . (Here in a Galerkin approximation) for M iterations with a constant time step = M (N,M)
Yk is now written Yk to emphasize the dependence on the number of time steps M.) If the scheme has error bound (N,M) 2 E Xtk − Yk sup
k=0,...,M
L2 (D)
1 2
≤ KT
1 1 + β α N M
(6.7)
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72
Chapter 6. Numerical Methods for SPDEs
for α, β > 0, then the optimal overall rate is cost, i.e., (N,M) 2 max E Xtk − Yk
L2 (D)
k=0,...,M
β
αβ α+β
α
with respect to the computational
1 2
≤ KT · K
for N = K α+β and M = K α+β . For example, if α = rate is 13 .
1 2
αβ − α+β
and β = 1, then the overall
The following papers are a representative selection of many others in the literature dealing with the SPDE (6.6). Gyöngy & Nualart [44] introduced an implicit numerical scheme for this SPDE in 1995 and showed that it converges strongly to the exact solution without giving a rate. In 1998 and 1999 Gyöngy [42] also applied finite differences to an SPDE driven by space–time white noise and then used several temporal implicit and explicit schemes, in particular, the linear-implicit Euler scheme. He showed that these schemes converge with order 12 in the space and with order 14 in time (assuming a smooth initial value) and, hence, he obtained an overall convergence rate of 16 with respect to the computational cost in space and time. In 1999 Shardlow [111] applied finite differences to the SPDE (6.6) to obtain a spatial discretization, which he then discretized in time with a θ -method. This had an overall convergence rate of 16 with respect to the computational cost. In a seminal paper published in 2001, Davie & Gaines [27] showed that any numerical scheme applied to the SPDE (6.6) with f = 0 which uses only values of the noise Wt cannot converge faster than the rate of 16 with respect to the computational cost. Müller-Gronbach & Ritter [99] showed in 2007 that this is also a lower bound for the convergence rate. They even showed that one cannot improve this rate of convergence by choosing nonuniform time steps; see [100, 101]. Higher rates were obtained for smoother types of noise. For example, in 2003 Hausenblas [51] applied the linear-implicit and explicit Euler schemes and the Crank–Nicholson scheme to an SPDE of the form (6.6). For trace–class noise, she obtained the order 14 with respect to the computational cost, but in the general case of space–time white noise the convergence rate was no better than the Davie–Gaines barrier rate 16 . Similarly, in 2004 Lord & Rougemont [90] discretized in time the Galerkin– SODE obtained from the SPDE (6.6) with the numerical scheme (N,M)
Yk+1
(N,M) (N,M) = eAN h Yk + hfN (Yk ) + WkN ,
(6.8)
which they showed to be useful when the noise is very smooth in space, in particular with Gévrey regularity. However, in the general case of space–time white noise the scheme (6.8) converges at the Davie–Gaines barrier rate 16 .
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6.3. The exponential Euler scheme
73
6.3 The exponential Euler scheme Davie & Gaines [27, page 129], remarked that it may be possible to improve the convergence rate by using suitable linear functionals of the noise. This suggestion was used by Jentzen & Kloeden [67], who considered a parabolic SPDE with additive noise dXt = [AXt + F (Xt )] dt + dWt ,
X0 = x 0 ,
(6.9)
in a Hilbert space (H , | · |) with inner product ·, ·, where A is an (in general) unbounded operator (for example, A = ), F : H → H is a nonlinear continuous function, and Wt is a cylindrical Wiener process. They interpreted this in the mild sense, t
Xt = eAt x0 +
t
eA(t−s) F (Xs ) ds +
0
eA(t−s) dWs ,
(6.10)
0
and used the fact that the solution of the N -dimensional Itô–Galerkin SODE in the space HN := PN H (or, equivalently, in RN ) (6.11) dXtN = AN XtN + FN (XtN ) dt + dWtN has an analogous “mild” representation t t AN (t−s) N + e F (X ) ds + eAN (t−s) dWsN . XtN = eAN t uN N s 0 0
(6.12)
0
This motivated what they called the exponential Euler scheme (N,M) (N,M) (N,M) AN e + A−1 − I fN (Yk ) Yk+1 = eAN Yk N +
tk+1
tk
eAN (tk+1 −s) dWsN
(6.13)
T for some M ∈ N and discretization times tk = k for k = with time step = M 0, 1, . . . , M. The exponential Euler scheme is, in fact, easier to simulate than may seem at (N,M) first sight. Denoting the components of Yk and FN by . / (N,M) (N,M) i = 1, . . . , N , Yk,i = ei , Yk , FNi = ei , FN ,
the numerical scheme (6.13) can be rewritten as (N,M) Yk+1,1
=
.. . (N,M)
Yk+1,N
=
1 2 (1 − e−λ1 ) 1 (N,M) q1 −2λ1 FN (Yk )+ (1 − e ) Rk1 , λ1 2λ1 .. .. . . 1 −λN ) 2 (1 − e qN (N,M) (N,M) e−λN Yk,N + FNN (Yk )+ (1 − e−2λN ) RkN , λN 2λN (N,M) e−λ1 Yk,1 +
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74
Chapter 6. Numerical Methods for SPDEs
where the Rki for i = 1, . . . , N and k = 0, 1, . . . , M − 1 are independent, standard normal distributed random variables.
6.3.1
Convergence
The convergence of the exponential Euler scheme will be proved under the following assumptions, which the reader should compare with Assumptions 1–4 of Theorem 5.1 in Chapter 5. Assumption 5 (linear operator A). There exist sequences of real eigenvalues 0 < λ1 ≤ λ2 ≤ · · · and orthonormal eigenfunctions (en )n≥1 of −A such that the linear operator A : D(A) ⊂ H → H is given by Av =
∞
−λn en , v en
n=1
2 2 for all v ∈ D(A) with D(A) = v ∈ H : ∞ n=1 |λn | |en , v| < ∞ . Assumption 6 (nonlinearity F ). The nonlinearity F : H → H is two times continuously Fréchet differentiable and its derivatives satisfy + + +F (x) − F (y)+ ≤ L |x − y|H , (−A)(−r) F (x)(−A)r v ≤ L |v|H H
and r = 0, 1, and for all x, y ∈ H , v ∈ 1 1 −1 A F (x)(v, w) ≤ L (−A)− 2 v (−A)− 2 w 1 2,
D((−A)r ),
H
H
H
for all v,w,x ∈ H , where L > 0 is a positive constant. Assumption 7 (cylindrical Q-Wiener process Wt ). There exist a sequence (qn )n≥1 of positive real numbers and a real number γ ∈ (0, 1) such that ∞
2γ −1
λn
qn < ∞
n=1
and pairwise independent scalar Ft -adapted Wiener processes Wtn t≥0 for n ≥ 1. The cylindrical Q-Wiener process Wt is given formally by Wt =
∞ √
qn Wtn en .
(6.14)
n=1
Assumption 8 (initial value). The random variable x0 : → D((−A)γ ) satisfies E |(−A)γ x0 |4H < ∞, where γ > 0 is given in Assumption 7.
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6.3. The exponential Euler scheme
75
Under Assumptions 5–8 the SPDE (6.9) has a unique mild solution Xt on the time interval [0, T ], where Xt is the predictable stochastic process in D((−A)γ ) given by (6.10). See, e.g., Theorem 5.1 above as well as Da Prato & Zabczyk [24] or Prévot & Röckner [104]. Since Assumption 6 also applies to FN , the Itô–Galerkin SODE (6.11) has a unique solution on [0, T ], which is given implicitly by (6.12). The above series (6.14) for the cylindrical Wiener process may not converge in H but in another space into which H can be continuously embedded. Thus, the formalism here includes space–time white noise (in one-dimensional domains only) as well as trace–class noise. Next the convergence theorem of [67] is presented. Theorem 6.1. Suppose that Assumptions 5–8 are satisfied. Then there is a constant CT > 0 such that 12 log(M) (N,M) 2 −γ sup E Xtk − Yk ≤ C T λN + H M k=0,...,M
(6.15) (N,M)
holds for all N , M ∈ N, where Xt is the solution of SPDE (6.9), Yk is the k numerical solution given by (6.13), tk = T M for k = 0, 1, . . . , M, and γ > 0 is the constant given in Assumption 8. In fact, the exponential Euler scheme (6.13) converges in time with a strong order 1 − ε for an arbitrary small ε > 0 since log(M) can be estimated by M ε , so 1 log(M) ≈ (1−ε) . M M Importantly, the error coefficient CT does not depend on the dimension N of the Itô–Galerkin SDE. t An essential point is that the integral tkk+1 eAN (tk+1 −s) dWsN includes more information about the noise on the discretization interval. Such additional information was the key to the higher order of stochastic Taylor schemes for SODE but is included there in terms of simple multiple stochastic integrals rather than integrals weighted by an exponential integrand.
6.3.2
Numerical results
To illustrate Theorem 6.1 consider the semilinear stochastic heat equations (6.6) on the one-dimensional domain (0, 1) with f (u) = 12 u and f (u) = −u3 , i.e., ∂u ∂ 2 u 1 = 2 + u + W˙ t , ∂t 2 ∂x
∂u ∂ 2 u = 2 − u3 + W˙ t , ∂t ∂x
(6.16)
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76
Chapter 6. Numerical Methods for SPDEs
with the Dirichlet boundary condition and the initial value u0 (x) =
∞
√ n−0.6 2 sin(nπ x)
for
x ∈ (0, 1).
n=1
The first example is based on the function f (u) = 12 u, which satisfies the assumptions used in Theorem 6.1, while the second with the function f (u) = −u3 does not. For the SPDE (6.16) the linear-implicit Euler scheme is (N,M) (N,M) (N,M) Yk+1 = (I − AN )−1 Yk + · f (Yk ) + WkN , the Lord–Rougemont scheme is (N,M) (N,M) (N,M) Yk+1 = eAN Yk + · f (Yk ) + WkN , and the exponential Euler scheme is (N,M)
Yk+1
(N,M)
= eAN Yk
(N,M) AN e + A−1 − I f (Y ) + k N
tk+1
tk
eAN (tk+1 −s) dWsN
for k = 0, 1, . . . , M − 1 and N , M ≥ 1. From Figure 6.1 it is clear that the linear-implicit Euler and Lord–Rougemont schemes converge with the rate 16 , while the exponential Euler scheme converges with the rate 13 .
6.3.3
Restrictiveness of the assumptions
Assumptions 5–8 in Theorem 6.1 are quite restrictive and Theorem 6.1 has several serious shortcomings. First, the eigenvalues and eigenfunctions of the operator A are rarely known except in very simple domains. Finite element methods are a possible way around this difficulty. Some papers applying finite element methods to SPDEs are Allen, Novosel & Zhang [1], Katsoulakis, Kossioris & Lakkis [75], Kovács, Larsson & Lindgren [85], Kovács, Larsson & Saedpanah [86], Walsh [123], and Yan [125]. More seriously, Assumption 6 on the nonlinearity F , which is understood as the Nemytskii operator of some function f : R → R, is very restrictive and excludes Nemytskii operators for functions like f (u) =
u , 1 + u2
f (u) = u − u3 ,
u ∈ R.
These are important in applications involving stochastic reaction–diffusion equations. One reason is the Fréchet differentiability of the function F when considered as a
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6.3. The exponential Euler scheme
77
0
10
error
Linear Implicit Euler Lord−Rougemont Exponential Euler Orderlines 1/6 and 1/3
−1
10
−2
10
2
10
3
10 computational effort
4
10
Figure 6.1. Mean-square error vs. computational effort as log-log plot with the function f (u) = 12 u [68]. mapping between the Hilbert space H and also the boundedness of the derivatives of F as expressed in Assumption 6. This restriction is weakened with the order 12 in u time in Jentzen [63], where also nonlinearities such as f (u) = 1+u 2 are considered, by the observation that the solution process in fact takes values not only in H , but also in a smaller subspace V , e.g., the space of continuous functions on which F is indeed Fréchet differentiable. The other problem is the global Lipschitz estimate on F . This difficulty also arises for the finite dimensional Itô SODEs, for which it can be overcome by using pathwise convergence rather than strong convergence as discussed in Chapter 3. In particular, the results there for SODEs have been generalized to SPDEs in Jentzen [64], where polynomial nonlinearities of the form f (u) = u − u3 are also considered.
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Chapter 7
Taylor Approximations for SPDEs with Additive Noise
Taylor expansions of solutions of SPDEs in Banach spaces are the basis for deriving higher order numerical schemes for SPDEs, just as for SODEs. However, there is a major difficulty for SPDEs. Although an SPDE dXt = [AXt + F (Xt )] dt + B(Xt ) dWt
(7.1)
is driven by a Wiener process, which is a martingale, the solution process is usually not a semimartingale. See [37] for a clear discussion of this problem. In particular, a general Itô formula does not exist for the solutions of the SPDE (7.1). (In the special case of the squared norm as a test function, a standard Itô formula can be found in [88, 104] and in the references therein. In addition, in more general situations appropriately modified Itô-type formulas can be found in [37, 23] and in the references therein.) Hence stochastic Taylor expansions for the solutions of the SPDE (7.1) cannot be derived as in Chapter 3 for the solutions of finite dimensional SODEs. Apart from the exponential Euler scheme (6.13), only temporal approximations of low order for the solutions of SPDEs (7.1) were known until recently. For example, the stochastic convolution of the operator A = (Laplacian with Dirichlet boundary conditions) and B(U ) ≡ I in one spatial dimension has sample paths which are only ( 14 − ε)-Hölder continuous, and previously considered approximations did not attain a higher order than this in time. The reason is that the infinite dimensional noise process, in general, has only minimal spatial regularity, and the convolution of the semigroup and the noise is only as smooth in time as in space. This comparable regularity in time and space is a fundamental property of the dynamics of semigroups; see, e.g., [110]. To overcome these problems, robust Taylor expansions are needed for SPDEs driven by infinite dimensional Wiener processes. In this chapter it will be explained
79
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80
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
how this can be done for SPDEs with additive noise of the form dXt = [AXt + f (Xt )] dt + B dWt ,
(7.2)
which will be interpreted in mild form. The Taylor expansions will then be derived in essentially the same way as for RODEs in Chapter 2. The multiplicative noise case will be considered in Chapter 8. The additive noise case is, of course, a special case of the multiplicative noise case, but its special structure allows much stronger results to be obtained. In particular, pathwise as well as strong convergence results can be obtained and the driving noise process need not be a Wiener process; for example, it could be a fractional Brownian motion. This chapter is based on Chapter 2 in [62].
7.1 Assumptions Throughout this chapter the following assumptions are used. Let T ∈ (0, ∞) be a real number, let (, F , P) be a probability space, and let (V , · V ) be a separable real Banach space. Assumption 9 (semigroup S). Let S : [0, ∞) → L(V ) be a mapping satisfying S0 = I , St1 St2 = St1 +t2 ,
sup St L(V ) < ∞,
t∈[0,T ]
sup
St − Ss L(V ) · s
s,t∈[0,T ] s
for all t1 , t2 ∈ [0, ∞), where I is the identity operator on V .
(t − s)
<∞ (7.3)
Assumption 10 (nonlinearity F ). Let F : V → V be an infinitely often Fréchet differentiable mapping. Assumption 11 (stochastic process O). Let O : [0, T ] × → V be a stochastic process with θ -Hölder continuous sample paths. Assumption 12 (initial value ξ ). Let ξ : → V be an F /B(V )-measurable mapping with supt∈(0,T ] 1t St ξ (ω) − ξ (ω) V < ∞ for all ω ∈ . Assumption 13 (existence of a solution). There exists a stochastic process X : [0, T ] × → V with continuous sample paths, which satisfies the integral equation t Xt (ω) = St ξ (ω) + St−s F (Xs (ω)) ds + Ot (ω) (7.4) 0
for all t ∈ [0, T ] and all ω ∈ . These assumptions will now be discussed and explained in some detail. Examples of SPDEs satisfying them will be given in Section 7.3.
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7.1. Assumptions
81
Comments on Assumption 9. The semigroup S : [0, ∞) → L(V ) in Assumption 9 is not assumed to be strongly continuous as is usual in the literature; see, e.g., [32, p. 83]. This greater generality covers some important examples in Section 7.3. The last condition in (7.3) implies that St is locally Lipschitz continuous on (0, T ] and also indicates the size of the local Lipschitz constant. If St = eAt , t ≥ 0, is a strongly continuous semigroup on V with the generator A :+D(A) ⊂ V → V , then the last condition in (7.3) follows from the estimate + +AeAt + ≤ c/t for all t ∈ (0, T ] and some constant c > 0, since L(V ) + + + + + At2 + + + +e v − eAt1 v + = + eA(t2 −t1 ) − I eAt1 v + V
V
+ + + (t2 −t1 ) + + + As At1 e Ae v ds + =+ + 0 +
(t2 −t1 )
≤
V
eAs AeAt1 v V ds
0
≤
sup eAt L(V ) 0≤t≤T
c (t2 − t1 ) v V t1
for all 0 < t1 ≤ t2 ≤ T and all v ∈ V . Comments on Assumption 10. The smoothness of the drift coefficient provided by Assumption 10 is required for the Taylor expansions that will be derived below. The infinitely often differentiability is assumed only for convenience, and it would suffice to assume that F is k-times Fréchet differentiable for some k ∈ N, which is sufficiently large. The type of the nonlinearity F in Assumption 10 covers stochastic reaction– diffusion equations to be considered in Section 7.3. Comments on Assumption 11. This assumption means that the mappings Ot : → V ,
ω → Ot (ω) := O(t, ω),
ω ∈ ,
are F /B(V )-measurable for all t ∈ [0, T ] and that + + +Ot (ω) − Ot (ω)+ 2 1 V sup <∞ θ |t | − t t1 ,t2 ∈[0,T ] 2 1 t1 =t2
holds for all ω ∈ . Pathwise Hölder continuity is typical of noise processes in applications. Assumption 11 does not require the noise process to be generated by a Wiener process. For example, a stochastic convolution involving a fractional Brownian motion is also possible.
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82
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
Note also that nothing has been said about the predictability of the stochastic processes. Comments on Assumption 12. The condition in Assumption 12 states that the initial random variable ξ : → V is smooth in some sense. This may not be satisfied in some applications but may possibly be overcome with the following observation. Remark 11. Let Assumption 9 be satisfied and let O˜ : [0, T ] × → V be a stochastic process with θ-Hölder continuous sample paths, where θ ∈ (0, 1). In addition, suppose that ξ˜ : → V is an F /B(V )-measurable mapping which does not satisfy Assumption 12 but nevertheless satisfies + 1+ + + sup θ +St ξ˜ (ω) − ξ˜ (ω)+ < ∞ V t t∈(0,T ] for all ω ∈ . Then the stochastic process O : [0, T ] × → V given by Ot (ω) := St ξ˜ (ω) + O˜ t (ω) for all ω ∈ and all t ∈ [0, T ] and the F /B(V )-measurable mapping ξ : → V defined by ξ (ω) = 0 for all ω ∈ satisfy Assumption 11 and Assumption 12. Comments on Assumption 13. The existence of a solution of the SPDE provided by Assumption 13 is a minimal assumption to do numerical analysis. This will be provided by more specific assumptions on the nonlinearity F in SPDEs such as (7.2). The following result is needed in what follows. Proposition 1. Suppose that Assumptions 9 and 10 are satisfied and let X : [0, T ] × → V be a stochastic process with continuous sample paths. Then Y : [0, T ] × → V given by t
Yt (ω) :=
S(t−s) F (Xs (ω)) ds 0
for all t ∈ [0, T ] and all ω ∈ is a well-defined stochastic process with continuous sample paths. Proof. The integrand function [0, t] → V ,
s → S(t−s) F (Xs (ω)),
s ∈ [0, t],
is continuous on [0, t) by Assumptions 9 and 10 and is therefore strongly measurable (see Definition A.1.1 in [104]) for all t ∈ [0, T ] and ω ∈ . Moreover, + + sup +S(t−s) F (Xs (ω))+V < ∞ 0≤s≤t
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7.1. Assumptions
83
for all t ∈ [0, T ] and all ω ∈ . The F /B(V )-measurability of Yt : → V for each t ∈ [0, T ] follows immediately, so it is a well-defined stochastic process. Finally, the sample paths of Y : [0, T ] × → V are continuous, since + + +Yt (ω) − Yt (ω)+ 2 1 V + + =+ +
t2 t1
S(t2 −s) F (Xs (ω)) ds +
t2 +
+S(t
≤
2
t1
+ + −s)
+ + S(t2 −s) − S(t1 −s) F (Xs (ω)) ds + +
t1
0
F (Xs (ω)) V ds + L(V )
t1 +
+S(t
0
2 −s)
V
+ − S(t1 −s) +L(V ) F (Xs (ω)) V ds
≤
sup Ss L(V ) · sup F (Xs (ω)) V (t2 − t1 )
0≤s≤T
0≤s≤T
+
sup F (Xs (ω)) V
0≤s≤T
t1 +
+S(t
2 −t1 +s)
0
+ − Ss +L(V ) ds
holds for all tt , t2 ∈ [0, T ] with t1 ≤ t2 and all ω ∈ .
7.1.1
Properties of the solutions
The pathwise uniqueness and Hölder continuity of the solution process, the existence of which is provided by Assumption 13, follow from Assumptions 9–12. Lemma 7.1. Suppose that Assumptions 9–13 are satisfied and let X, Y : × [0, T ] → V be two stochastic processes with continuous sample paths, which satisfy (7.4), i.e., t Xt (ω) = St ξ (ω) + S(t−s) F (Xs (ω)) ds + Ot (ω), 0
Yt (ω) = St ξ (ω) +
t
S(t−s) F (Ys (ω)) ds + Ot (ω)
0
for all t ∈ [0, T ] and ω ∈ . Then Xt (ω) = Yt (ω) for all t ∈ [0, T ] and ω ∈ .
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84
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
Proof. Let E : × [0, T ] → [0, ∞) be given by Et (ω) := Xt (ω) − Yt (ω) V for all t ∈ [0, T ] and ω ∈ , and let R : → [0, ∞) be the F /B([0, ∞))-measurable mapping given by + + R(ω) := sup sup +F (rXt (ω) + (1 − r) Yt (ω))+L(V ) t∈[0,T ] r∈[0,1]
for all ω ∈ , which is finite due to Assumption 10. Then + + 1 + + + ≤ R Xt − Yt V , F (Xt ) − F (Yt ) V = + (X )) (X ) (Y + r − Y − Y dr F t t t t t + + 0
V
and hence + + t + + + (F S ) − F (Y )) ds (X Et = + s s (t−s) + + 0
≤
sup Ss L(V )
0≤s≤T
0≤s≤T
=R
F (Xs ) − F (Ys ) V ds
0
sup Ss L(V )
≤R
t
t
Xs − Ys V ds
0
sup Ss L(V )
0≤s≤T
V
t
Es ds < ∞
0
for all t ∈ [0, T ]. Gronwall’s inequality then gives Et = 0 for all t ∈ [0, T ]. The unique solution process in Assumption 13 will henceforth be denoted by X. Its sample paths are not only continuous, but are in fact Hölder continuous as well. Lemma 7.2. Suppose that Assumptions 9–13 hold. Then the unique solution process X : [0, T ] × → V has θ -Hölder continuous sample paths; i.e., + + +Xt (ω) − Xt (ω)+ 2 1 V sup <∞ (t2 − t1 )θ 0≤t1
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7.1. Assumptions
85
Proof. Let R : → [0, ∞) be the F /B([0, ∞))-measurable mapping given by 1 St ξ (ω) − ξ (ω) V + sup St L(V ) t∈[0,T ] t∈(0,T ] t + + +St − St + 2 1 L(V ) t1 + sup F (Xt (ω)) V + sup t2 − t1 0≤t1
R(ω) := sup
for all ω ∈ , which is finite due to Assumptions 9–13. Then + + + + + + +St ξ − St ξ + = +St S(t −t ) ξ − ξ + ≤ R +S(t −t ) ξ − ξ + ≤ R 2 (t2 − t1 ) (7.5) 2 1 1 2 1 2 1 V V V for all 0 ≤ t1 < t2 ≤ T . Moreover, + + + +
+ t1 + S(t2 −s) F (Xs ) ds − S(t1 −s) F (Xs ) ds + + 0 0 V + + t t1 + + 2 + S F (X S F (X ) ds + − S ) ds =+ s s (t2 −s) (t1 −s) (t2 −s) + + t2
≤
0
t1 t2 +
+ + + +S(t −s) + F (Xs ) V ds + + 2 + L(V )
0
t1
t1
S(t2 −s) − S(t1 −s)
V
+ + F (Xs ) ds + +
V
and therefore + + + +
0
t2
S(t2 −s) F (Xs ) ds −
0
t1
+ + S(t1 −s) F (Xs ) ds + +
t1 +
≤ R 2 (t2 − t1 ) + 0
V
+ +S(t −s) − S(t −s) + F (Xs ) V ds 2 1 L(V )
≤ R (t2 − t1 ) + R 2
t1 +
+S(t
0
2 −s)
+1−θ + +θ − S(t1 −s) +L(V ) +S(t2 −s) − S(t1 −s) +L(V ) ds
for all 0 ≤ t1 < t2 ≤ T . Hence, + + + +
+ + S(t2 −s) F (Xs ) ds − S(t1 −s) F (Xs ) ds + + 0 0 V θ t1 (t2 − t1 ) R ≤ R 2 (t2 − t1 ) + R (2R)1−θ ds (t1 − s) 0 t1 (t1 − s)−θ ds, ≤ R 2 (t2 − t1 ) + 2R 2 (t2 − t1 )θ t2
t1
0
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86
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
which yields + + + +
+ + S(t1 −s) F (Xs ) ds + S(t2 −s) F (Xs ) ds − + 0 0 V t1 s −θ ds ≤ R 2 (t2 − t1 ) + 2R 2 (t2 − t1 )θ 0 1−θ 2 2 T (t2 − t1 )θ ≤ R (t2 − t1 ) + 2R (1 − θ)
t2
t1
(7.6)
for all 0 ≤ t1 < t2 ≤ T . Combining (7.5), (7.6), and Assumption 11 yields the assertion. Subtracting the noise process Ot from the solution process Xt in integral equation (7.4) gives the equation Xt − Ot = St ξ +
t
S(t−s) F (Xs ) ds
(7.7)
0
for all t ∈ [0, T ]. The following lemma shows that the process Xt − Ot is smoother than the original solution process Xt . This additional regularity of the stochastic process Xt − Ot will play an important role in the derivation Taylor expansions. Lemma 7.3. Suppose that Assumptions 9–13 hold. Then the stochastic process X − O : [0, T ] × → V has Lipschitz continuous sample paths; i.e., sup
+ + + Xt (ω) − Ot (ω) − Xt (ω) − Ot (ω) + 2 2 1 1 V t2 − t1
0≤t1
<∞
holds for all ω ∈ , where O and X are given in Assumptions 11 and 13. Proof. First, + + + 1 + + + + +F Xt − F Xt + = + F Xt1 + r Xt2 − Xt1 Xt2 − Xt1 dr + + 2 1 V + 0 + ≤
V
+ + + + sup +F Xt1 + r Xt2 − Xt1 +L(V ) +Xt2 − Xt1 +V
r∈[0,1]
for all t1 , t2 ∈ [0, T ]. In addition, the F /B([0, ∞))-measurable mapping R : → [0, ∞) defined by
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7.1. Assumptions
87
1 St ξ (ω) − ξ (ω) V + sup St L(V ) + sup F (Xt (ω)) V t∈[0,T ] t∈[0,T ] t∈(0,T ] t
R(ω) := sup
+
+ +
+ St2 − St1 +L(V ) t1
sup
t2 − t1
0≤t1
+
+ + +F (Xt (ω)) − F (Xt (ω))+ 2 1 V
sup
(t2 − t1 )θ
0≤t1
for all ω ∈ is finite due to Assumptions 9–13 and Lemma 7.2. Moreover, + + + + + + +St ξ − St ξ + = +St S(t −t ) ξ − ξ + ≤ R +S(t −t ) ξ − ξ + ≤ R 2 (t2 − t1 ) 1 1 2 2 1 2 1 V V V for all 0 ≤ t1 < t2 ≤ T . Thus, + + S(t1 −s) F (Xs ) ds + S(t2 −s) F (Xs ) ds − + 0 0 V + + t t1 + + 2 + S(t2 −s) F (Xs ) ds + S(t2 −s) − S(t1 −s) F (Xs ) ds + =+ +
+ + + +
t2
t1
0
t1 t2 +
V + + t + 1 + + + + + S(t2 −s) − S(t1 −s) F (Xs ) − F (Xt1 ) ds + ≤ S(t2 −s) L(V ) F (Xs ) V ds + + + 0 t1 V + t + + 1 + S(t2 −s) − S(t1 −s) F (Xt1 ) ds + ++ + +
0
V
and therefore + t + 2 + S(t2 −s) F (Xs ) ds − + 0
t1
0
+ + S(t1 −s) F (Xs ) ds + +
t1 +
V
+ + + +S(t −s) − S(t −s) + +F (Xs ) − F (Xt )+ ds ≤ R 2 (t2 − t1 ) + 1 2 1 L(V ) V 0 + t + 1 + + +R + S(t2 −t1 )+s − Ss ds + + + 0
L(V )
for all 0 ≤ t1 < t2 ≤ T . Hence, + + + +
0
t2
S(t2 −s) F (Xs ) ds −
0 t1 +
t1
+ + S(t1 −s) F (Xs ) ds + +
V
+ +S(t −s) − S(t −s) + |s − t1 |θ ds ≤ R 2 (t2 − t1 ) + R 2 1 L(V ) 0 + t + t1 + 1 + + +R + S(t2 −t1 )+s ds − Ss ds + + 0
0
L(V )
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88
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
and + + + +
0
t2
S(t2 −s) F (Xs ) ds −
t1
0 t1 +
+ + S(t1 −s) F (Xs ) ds + +
V
+ +S(t −s) − S(t −s) + |s − t1 |θ ds ≤ R 2 (t2 − t1 ) + R 2 1 L(V ) 0 + t + t1 + 2 + + +R + Ss ds − Ss ds + + (t2 −t1 )
0
L(V )
for all 0 ≤ t1 < t2 ≤ T . This yields + + t t1 + + 2 + + S F (X ) ds − S F (X ) ds s s (t1 −s) (t2 −s) + + 0 0 V t1 t2 − t1 2 2 θ |s − t1 | ds ≤ R (t2 − t1 ) + R 0 t1 − s + + t t2 −t1 + + 2 + S ds − S ds +R + s s + + 0
t1
L(V )
and + + S(t2 −s) F (Xs ) ds − S(t1 −s) F (Xs ) ds + + 0 0 V t1 ≤ R 2 (t2 − t1 ) + R 2 (t2 − t1 ) (t1 − s)θ−1 ds + 2R 2 (t2 − t1 ) 0 t1 2 2 = R (t2 − t1 ) + R (t2 − t1 ) s θ−1 ds + 2R 2 (t2 − t1 ) 0 ≤ R 2 + R 2 (T + 1)θ −1 + 2R 2 (t2 − t1 )
+ + + +
t2
t1
for all 0 ≤ t1 < t2 ≤ T , which gives the assertion.
7.2 Autonomization The coefficients of SPDEs in many applications often depend explicitly on time, e.g., instead of (5.2) the SPDE has the form dXt = [Xt + f (t, Xt )] dt + g(t, Xt ) dWt , X ∂D = 0. (7.8) Existence and uniqueness theorems such as Theorem 5.1 as well as the Taylor expansions to be derived in this and the next chapter can also be applied to such SPDEs
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7.2. Autonomization
89
through a procedure known as autonomization, by means of which the SPDE (7.8) with solution Xt is replaced by an SPDE with solution (t, Xt ). The resulting SPDE in the augmented state space now has coefficients that seemingly do not depend explicitly on time. For ODEs this would lead to an autonomous ODE, hence the term autonomization, although SPDEs are, strictly speaking, never autonomous due to the time variation of the forcing noise. This method will be explained and justified here in the context of the additive noise SPDEs under consideration in this chapter. Let (V˜ , · V˜ ) be a real Banach space, let S˜ : [0, ∞) → L(V˜ ) be a mapping with S˜0 = I , S˜t1 S˜t2 = S˜(t1 +t2 ) ,
+ + + + sup +S˜t +
L(V˜ )
0≤t≤T
< ∞,
+ + + + s · +S˜t − S˜s +
L(V˜ )
sup
t −s
0≤s
<∞
for all t1 , t2 ∈ [0, T ], and let F˜ : R × V˜ → V˜ be an infinitely often Fréchet differentiable mapping. In addition, let O˜ : [0, T ] × → V˜ and X˜ : [0, T ] × → V˜ be two mappings such that the mappings O˜ t : → V˜ ,
˜ ω), ω → O˜ t (ω) := O(t,
ω ∈ ,
X˜ t : → V˜ ,
˜ ω), ω → X˜ t (ω) := X(t,
ω ∈ ,
are F /B(V˜ )-measurable for all t ∈ [0, T ], the mappings [0, T ] → V˜ ,
t → X˜ t (ω),
t ∈ [0, T ],
are continuous for all ω ∈ , and
sup
+ + + +˜ +Ot2 (ω) − O˜ t1 (ω)+ ˜ V
(t2 − t1 )θ
0≤t1
<∞
holds for all ω ∈ , where θ ∈ (0, 1). Also let ξ˜ : → V˜ be F /B(V˜ )-measurable with + 1+ + + sup +S˜t ξ˜ (ω) − ξ˜ (ω)+ < ∞ V˜ t∈(0,T ] t for all ω ∈ . Finally, assume that X˜ : [0, T ] × → V˜ satisfies X˜ t (ω) = S˜t ξ˜ (ω) +
t
S˜(t−s) F˜ (s, X˜ s (ω)) ds + O˜ t (ω)
0
for all t ∈ [0, T ] and all ω ∈ .
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90
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
Now consider the separable real Banach space V := R × V˜ equipped with the norm 0 (t, v) V := |t|2 + v 2˜ for all t ∈ R, v ∈ V˜ , V
and consider the mapping S : [0, ∞) → L(V ) defined by for all s ∈ R, t ∈ [0, ∞), v ∈ V˜ . St (s, v) := s, S˜t v In addition, let F : V → V and O : [0, T ] × → V be defined by F (t, v) := 1, F˜ (t, v) for all t ∈ R, v ∈ V˜ and let O : [0, T ] × → V be defined by Ot (ω) := 0, O˜ t (ω) for all t ∈ [0, T ], v ∈ V˜ , and ω ∈ . Furthermore, define ξ : → V and X : [0, T ] × → V by ξ (ω) := 0, ξ˜ (ω) , Xt (ω) := t, X˜ t (ω) , for all t ∈ [0, T ] and ω ∈ . Then it follows that t t t = Xt (ω) = X˜ t (ω) S˜t ξ˜ (ω) + 0 S˜(t−s) F˜ (s, X˜ s (ω)) ds + O˜ t (ω) =
0 S˜t ξ˜ (ω)
+
t ˜(t−s) F˜ (s, X˜ s (ω)) ds S 0
t
+
0 O˜ t (ω)
.
Hence Xt (ω) = St
0 ˜ξ (ω)
+
t 0
= St ξ (ω) +
t
S(t−s) 0
= St ξ (ω) +
t
1 ˜ ˜ S(t−s) F (s, X˜ s (ω)) 1 F˜ (s, X˜ s (ω))
ds + Ot (ω)
ds + Ot (ω)
S(t−s) F (Xs (ω)) ds + Ot (ω)
0
for all t ∈ [0, T ] and all ω ∈ . Assumptions 9–13 are satisfied by V , S, F , O, ξ , and X defined in this way. Hence “nonautonomous” SPDEs can be handled in the “autonomous” setting that is being used here.
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7.3. Examples
7.3
91
Examples
Let H be the real Hilbert space of equivalence classes of square integrable functions from D = (0, 1)d to R for some d ∈ N, i.e., H = L2 ((0, 1)d , R), and let V = C([0, 1]d , R) be the real Banach space of continuous functions from [0, 1]d to R. The scalar product and the norm in H are given by
u, vH =
u(x) v(x) dx, (0,1)d
u H =
1 2
2
u(x) dx (0,1)d
for all u, v ∈ H , while for continuous functions in V the norm is given by v V = sup |v(x)| x∈[0,1]d
for all v ∈ V , where |·| is the absolute value of a real number. Note that V ⊂ H densely and continuously. The semigroup to be introduced in Section 7.3.1 is generated by the Laplace operator on (0, 1)d with the Dirichlet boundary condition. The eigenfunctions and eigenvalues of minus this operator are, respectively, given by d
ei (x) = 2 2 sin(i1 π x1 ) . . . sin(id π xd ),
x ∈ [0, 1]d ,
λi = π 2 i12 + · · · + id2
and
(7.9) (7.10)
with indices i = (i1 , . . . , id ) ∈ I = Nd . By x 2 := (|x|2 + · · · + |xd |2 )1/2 for all x ∈ Rd the Euclidean norm on Rd is denoted. Moreover, note that the family ei ∈ V , i ∈ Nd , is an orthonormal basis of H . Some important examples satisfying Assumptions 9–11 on the space V will be constructed in the following three subsections. A similar setting is used in Section 7.2 of Da Prato & Zabczyk [24].
7.3.1
Semigroup generated by the Laplacian
The next proposition gives an example of a semigroup satisfying Assumption 9. Proposition 2. The mapping S : [0, ∞) → L(V ) defined by St v = e−λi t ei , vH ei = lim e−λi t ei , vH ei S0 v = v, i∈Nd
N→∞
i∈Nd i 2 ≤N
for all t ∈ (0, ∞) and all v ∈ V satisfies Assumption 9. The functions (ei )i∈Nd and the real numbers (λi )i∈Nd are given by (7.9) and (7.10).
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92
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
Of course, this is simply the semigroup generated by the Laplacian with Dirichlet boundary conditions (see, e.g., Section 3.8.1 in [110]). Other boundary conditions such as Neumann or periodic boundary conditions could also be considered here. It is easy to see that the semigroup given in Proposition 2 is not strongly continuous. The mapping [0, T ] → V ,
t → St v,
t ∈ [0, T ],
is not continuous at t = 0 if, for example, v(x) = 1 for all x ∈ [0, 1]d , although the mapping [0, T ] → H ,
t → St v,
t ∈ [0, T ],
is continuous for all v ∈ V with respect to the H -norm · H .
7.3.2 The drift as a Nemytskii operator The drift mapping F : V → V can often be defined as the Nemytskii operator (see, e.g., Section 9.3.4 in [106] or the beginning of Chapter 5 in [16]) of a real valued function of real variables. Let f : [0, 1]d × R → R be a continuous function such that the mappings R → R,
y → f (x, y),
y ∈ R,
(7.11)
are infinitely often differentiable for each x ∈ [0, 1]d and let ∂n f : [0, 1]d × R → R ∂y n be the nth derivative of f with respect to its last variable in [0, 1]d × R ⊂ Rd+1 . The next proposition defines the Nemytskii operator F : V → V corresponding to the mapping f and gives its corresponding smoothness properties. Proposition 3. Let f : [0, 1]d × R → R be a continuous function which is smooth ∂n in its last variable in the sense of (7.11) such that the partial derivatives ∂y nf : d [0, 1] × R → R are continuous for all n ∈ N. Then the corresponding Nemytskii operator F :V →V,
(F (v)) (x) := f (x, v(x)),
x ∈ [0, 1]d ,
v ∈V,
is infinitely often Fréchet differentiable, and its Fréchet derivatives F (n) : V → L(n) (V , V ) are given by n ∂ F (n) (v)(v1 , . . . , vn ) (x) = f (x, v(x)) · v1 (x) · . . . · vn (x) (7.12) ∂y n
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7.3. Examples
93
for all x ∈ [0, 1]d , v, v1 , . . . , vn ∈ V , and n ∈ N. Moreover, these derivatives satisfy n + + ∂ + (n) + = sup f (x, v(x)) (7.13) +F (v)+ (n) n L (V ,V ) ∂y x∈[0,1]d for all v ∈ V and all n ∈ N.
7.3.3
Stochastic process as stochastic convolution
An example of a stochastic process O : × [0, T ] → V satisfying Assumption 11 is constructed here. In principle, O could be an arbitrary stochastic process with Hölder continuous sample paths such as a fractional Brownian motion. For comparability, and since it is a very important case, the example here involves stochastic convolutions of the semigroup S constructed in Proposition 2 and a cylindrical Wiener process. The following result provides an appropriate version of such a process (see Lemma 4.3 in [9]). Proposition 4. Let ρ > 0, let Wti t∈[0,T ] , i ∈ Nd , be a family of pairwise independent standard scalar Wiener processes, and let b : Nd → R be a given function with 2ρ−2 |b(i)|2 < ∞. Then there exists a stochastic process O : [0, T ] × i∈Nd i 2 → V which satisfies + + +Ot (ω) − Ot (ω)+ 2 1 V sup <∞ θ (t2 − t1 ) 0≤t1
i∈{1,...,N }d
0
(7.14) where the functions (ei )i∈Nd and the real numbers (λi )i∈Nd are given by (7.9) and (7.10). In particular, the stochastic process O satisfies Assumption 11. The stochastic process O constructed in Proposition 4 is unique up to indistinguishability. More precisely, if O, O˜ : [0, T ] × → V are two stochastic processes with continuous sample paths that satisfy (7.14), then P Ot = O˜ t ; ∀ t ∈ [0, T ] = 1. In view of Proposition 4 t t −λi (t−s) i b(i) e dWs ei = P-a.s., S˜(t−s) B dWs , Ot = i∈Nd
0
0
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94
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
for all t ∈ [0, T ] and an appropriate cylindrical I -Wiener process Wt on H , where the bounded linear operator B : H → H is given by b(i) ei , vH ei Bv = i∈Nd
for all v ∈ H , and S˜ : [0, ∞) → L(H ) is given by S˜t v = e−λi t ei , vH ei S˜0 v = v, i∈Nd
for all v ∈ H and all t ∈ (0, ∞). In this sense, O is indeed a stochastic convolution of the semigroup with a cylindrical Wiener process as it is usually considered in the literature (see, e.g., Section 5 in [24]). The stochastic convolutions of the above S˜t and B commute for all t ∈ [0, ∞). However, Assumption 11 does not require this condition to be satisfied.
7.3.4
Concrete examples of SPDEs with additive noise
Following the above preparation, some examples of specific SPDEs of the form t S(t−s) F (Xs ) ds + Ot (7.15) Xt = St ξ + 0
on the real Banach space V = C [0, 1]d , R in the sense of Assumption 13 can now be given. Here S, F , and O are as constructed in Propositions 2–4 and ξ satisfies Assumption 12.
Remark 12. There are many possible choices for the state space V for PDEs and SPDEs of evolutionary type. The choice V = C [0, 1]d , R will allow the difference of a Taylor approximation and the exact solution to be measured in the supremum norm v V = supx∈[0,1]d |v(x)| for v ∈ V , which yields very strong results. The same setting is also considered in Section 7.2 in [24]; see equation (7.35)(ii) there. In principle, the Dirichlet boundary condition could be incorporated in the space V so that the semigroup becomes strongly continuous. This could, however, restrict the nonlinearity in some sense. Further numerical approximation results in such Banach spaces can be found in [43, 103, 9, 64, 22] and in the references therein. A solution of the stochastic integral equation (7.15) in the sense of Assumption 13 exists if, e.g., the function f : [0, 1]d × R → R used in Proposition 3 satisfies an appropriate one-sided linear growth condition such as y f (x, y) < ∞; 2 x∈[0,1]d y∈R 1 + y sup sup
see, for instance, Proposition 5.3.3 in [16].
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7.3. Examples
95
In addition, let B : H → H be a bounded linear operator of the form
Bv =
b(i) ei , vH ,
v ∈ H,
i∈Nd
where b : Nd → R is the function used in Proposition 4. Finally, with t ∈ [0, T ] and x ∈ [0, 1]d , write ) dXt =
* ∂2 ∂2 + · · · + 2 Xt + f (x, Xt ) dt + B dWt ∂x12 ∂xd
(7.16)
with Xt |∂(0,1)d = 0,
X0 = ξ ,
(7.17)
for the SPDEs (7.4) and (7.15) on C [0, 1]d , R . The three particular choices for the parameters d ∈ N and T > 0 and functions f : [0, 1]d × R → R, b : Nd → R, and ξ : → V below provide concrete examples of SPDEs that will be used later in numerical computations. Example 7.4. Let d = 1 and T = 1. In addition, let f (x, y) = y − y 3 for all x ∈ [0, 1] and y ∈ R, let b(i) = 12 for each i ∈ N, and, finally, let ξ = 14 e1 . The corresponding SPDE (7.16)–(7.17) reduces to dXt =
1 ∂2 3 X + X − X t t t dt + dWt 2 ∂x 2
(7.18)
with boundary and initial conditions √ Xt (0) = Xt (1) = 0,
X0 (x) =
2 sin(π x), 4
on C ([0, 1], R) for t, x ∈ [0, 1]. It follows from Proposition 4 that Assumption 11 is valid for all θ ∈ (0, 14 ). Example 7.5. As before set d = 1 and T = 1, but now choose the functions f (x, y) =
y , 1 + y2
x ∈ [0, 1], y ∈ R,
and b(i) = 1 for all i ∈ N and the initial value ξ = 0.
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96
Chapter 7. Taylor Approximations for SPDEs with Additive Noise The corresponding SPDE (7.16)–(7.17) on C ([0, 1], R) reduces to ) * Xt ∂2 dt + dWt Xt + dXt = ∂x 2 1 + Xt2
(7.19)
with boundary and initial conditions Xt (0) = Xt (1) = 0,
X0 = 0,
for t, x ∈ [0, 1]. It follows from Proposition 4 that Assumption 11 is satisfied for all θ ∈ (0, 14 ). Example 7.6. Now let d = 2 with T = 1 and ξ = 0 as before and consider the function x ∈ [0, 1]2 , y ∈ R,
f (x, y) = −y 3 , and b(i) = 0
1
,
i12 + i22
i = (i1 , i2 ) ∈ N2 .
The SPDE (7.16)–(7.17) on C [0, 1]2 , R reduces to * ) ∂2 ∂2 3 Xt − Xt dt + B dWt + dXt = ∂x12 ∂x22
(7.20)
with boundary and initial conditions Xt |∂(0,1)2 = 0,
X0 = 0,
for t ∈ [0, 1] and x ∈ [0, 1]2 . Proposition 4 yields that Assumption 11 is satisfied for all θ ∈ (0, 12 ).
7.4 Taylor expansions Taylor expansions of the solution of the SPDE (7.4) will be derived here in much the same way as was done for RODEs in Chapter 2, i.e., using Taylor expansions of the drift function and inserting lower order Taylor expansions of the right-hand side of higher order ones to obtain a closed form expression for the solution. This will involve special integral operators acting on a space C. These will be defined below and in Section 7.4.1 with some important properties given in Propositions 5 and 6.
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7.4. Taylor expansions
97
Various Taylor expansions on an abstract level will then be derived in Section 7.5 and applied to some specific SPDEs in Section 7.6. Let t0 ∈ [0, T ) be fixed and let C be the space of all stochastic processes Y : [t0 , T ] × → V with continuous sample paths. More precisely, let
C := Y : [t0 , T ]× → V (∀ t ∈ [0, T ] : Yt : → V is F /B(V )-measurable) and (∀ ω ∈ : [t0 , T ] t → Yt (ω) ∈ V is continuous ) . This is an R-vector space since V is assumed to be separable. Define t := t − t0 ∈ R for all t ∈ [t0 , T ] and define the stochastic processes X, O : [t0 , T ] × → V ∈ C by Xt (ω) := Xt (ω) − Xt0 (ω)
Ot (ω) := Ot (ω) − Ot0 (ω)
and
for all t ∈ [t0 , T ] and all ω ∈ , where O is the stochastic process given in Assumption 11 and X is the unique solution process of the SPDE (7.4). Since X t = St ξ +
t
S(t−s) F (Xs ) ds + Ot t0 t = St St0 ξ + S(t−s) F (Xs ) ds + S(t−s) F (Xs ) ds + Ot 0
= St St0 ξ +
0 t0
0
t0
S(t0 −s) F (Xs ) ds +
= St Xt0 − Ot0 +
t
t
S(t−s) F (Xs ) ds + Ot
t0
S(t−s) F (Xs ) ds + Ot
t0
for all t ∈ [t0 , T ], the increment X satisfies t Xt = St Xt0 − Ot0 + S(t−s) F (Xs ) ds + Ot − Xt0 t0
and hence Xt = (St − I ) Xt0 − Ot0 +
t
S(t−s) F (Xs ) ds + Ot
(7.21)
t0
for all t ∈ [t0 , T ]. The basic formula (7.21) is the starting point for derivation of Taylor expansions of the solution process X and its increment X.
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98
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
7.4.1
Integral operators
To begin, define the stochastic process I 0 ∈ C by I 0 (t, ω) := (St − I ) Xt0 (ω) − Ot0 (ω) +
t
S(t−s) F (Xt0 (ω)) ds
t0
= (St − I ) Xt0 (ω) − Ot0 (ω) +
0
t
Ss ds F (Xt0 (ω))
for all t ∈ [t0 , T ] and all ω ∈ . In addition, define the stochastic process I∗0 ∈ C by t S(t−s) F (Xs (ω)) ds I∗0 (t, ω) := (St − I ) Xt0 (ω) − Ot0 (ω) + t0
for all t ∈ [t0 , T ] and all ω ∈ . In particular, note that the subscript ∗ means that the solution process X is included in the integrand, while the absence of this subscript means that only the constant value Xt0 is present. The basic formula (7.21) for the solution increment X can be written in terms of these integral operators as Xt = I∗0 (t) + Ot for all t ∈ [t0 , T ] and, symbolically, in the space C as X = I∗0 + O.
(7.22)
In the next step, define n-multilinear symmetric mappings I n : C n → C and I∗n : C n → C for n ∈ N by 1 t S(t−s) F (n) (Xt0 (ω)) (g1 (s, ω), . . . , gn (s, ω)) ds I n [g1 , . . . , gn ](t, ω) := n! t0 and I∗n [g1 , . . . , gn ](t, ω) 1
t
:=
S(t−s) t0
0
F (n) (Xt0 (ω) + rXs (ω)) (g1 (s, ω), . . . , gn (s, ω))
(1 − r)n−1 dr ds (n − 1)!
for all t ∈ [t0 , T ], ω ∈ , g1 , . . . , gn ∈ C, and n ∈ N. The presence and absence of the subscript ∗ here has a similar meaning as in the previous situation. It follows as in Proposition 1 from Assumptions 9–13 that the stochastic processes I 0 , I∗0 ∈ C and the mappings I n , I∗n : C n → C for n ∈ N are well-defined.
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7.4. Taylor expansions
99
A Taylor approximation of Xt with t ∈ (t0 , T ] about Xt0 can depend only on the value of the process X at time t0 , i.e., only on Xt0 . The stochastic processes I 0 and I n [g1 , . . . , gn ] ∈ C for g1 , . . . , gn ∈ C and n ∈ N depend on the solution process X at time t0 only and are therefore useful Taylor approximations for the solution process X and its increment X. However, the stochastic processes I∗0 and I∗n [g1 , . . . , gn ] ∈ C for g1 , . . . , gn ∈ C and n ∈ N depend on the whole solution Xs with s ∈ [t0 , T ] and can thus represent remainder terms that could then be further expanded to give a better approximation using the following proposition. Proposition 5. Let Assumptions 9–13 hold. Then we obtain I∗0 = I 0 + I∗1 [I∗0 ] + I∗1 [O]
(7.23)
I∗n [g1 , . . . , gn ] = I n [g1 , . . . , gn ] + I∗n+1 [I∗0 , g1 , . . . , gn ]
(7.24)
and
+ I∗n+1 [O, g1 , . . . , gn ] for all g1 , . . . , gn ∈ C and all n ∈ N. Proof. Consider first (7.23). Applying the Fundamental Theorem of Calculus for Banach space valued functions (see, e.g., Proposition A.2.3 in [104]) to the function [0, 1] r → F Xt0 + r Xs − Xt0 ∈ V yields F (Xs ) = F (Xt0 ) +
0
= F (Xt0 ) +
1
0
= F (Xt0 ) +
1
F (Xt0 + r(Xs − Xt0 ))(Xs − Xt0 ) dr F (Xt0 + rXs )(Xs ) dr
1
F
0
(Xt0 + rXs )(I∗0 (s)) dr +
0
1
F (Xt0 + rXs )(Os ) dr
for all s ∈ [t0 , T ], where (7.22) has been used. Hence, t 0 I∗ (t) = (St − I ) Xt0 − Ot0 + S(t−s) F (Xs ) ds
= (St − I ) Xt0 − Ot0 + + +
t
1
S(t−s) 0
t0 t
S(t−s)
t0
0
1
t0 t
t0
S(t−s) F (Xt0 ) ds
F (Xt0 + rXs )(I∗0 (s)) dr ds F (Xt0 + rXs )(Os ) dr ds,
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100
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
which implies that I∗0 (t) = I 0 (t) + I∗1 [I∗0 ](t) + I∗1 [O](t) for all t ∈ [t0 , T ]. Then, for (7.24), note that
1
0
F (n) (Xt0 + rXs )(g1 (s), . . . , gn (s))
(1 − r)n−1 dr (n − 1)!
(1 − r)n r=1 = −F (n) (Xt0 + rXs )(g1 (s), . . . , gn (s)) n! r=0 1 (1 − r)n (n+1) dr + F (Xt0 + rXs )(Xs , g1 (s), . . . , gn (s)) n! 0 1 = F (n) (Xt0 )(g1 (s), . . . , gn (s)) n! 1 (1 − r)n F (n+1) (Xt0 + rXs )(I∗0 (s), g1 (s), . . . , gn (s)) dr + n! 0 1 (1 − r)n dr F (n+1) (Xt0 + rXs )(Os , g1 (s), . . . , gn (s)) + n! 0
holds for all s ∈ [t0 , T ] and all g1 , . . . , gn ∈ C, n ∈ N, by integration by parts and (7.22). Finally, I∗n [g1 , . . . , gn ](t) t 1 (1 − r)n−1 = S(t−s) F (n) (Xt0 + rXs )(g1 (s), . . . , gn (s)) dr ds (n − 1)! t0 0 =
1 t S(t−s) F (n) (Xt0 ) (g1 (s), . . . , gn (s)) ds n! t0 t 1 (1 − r)n + S(t−s) F (n+1) (Xt0 + rXs )(I∗0 (s), g1 (s), . . . , gn (s)) dr ds n! t0 0 t 1 (1 − r)n + S(t−s) F (n+1) (Xt0 + rXs )(Os , g1 (s), . . . , gn (s)) dr ds n! t0 0 (n+1)
= I n [g1 , . . . , gn ](t) + I∗
(n+1)
[I∗0 , g1 , . . . , gn ](t) + I∗
[O, g1 , . . . , gn ](t)
for all t ∈ [t0 , T ] and all g1 , . . . , gn ∈ C, n ∈ N.
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7.4. Taylor expansions
101
An iterated application of Proposition 5 then allows Taylor expansions to be derived step by step, as will be seen below. The corresponding Taylor approximation is obtained by omitting the remainder terms in the Taylor expansion. Write Yt = O((t)r ) for a stochastic process Y ∈ C and some real number r ∈ (0, ∞) to denote that Yt (ω) V <∞ r t∈(t0 ,T ] (t − t0 ) sup
holds for all ω ∈ . The following proposition will allow the order of a Taylor approximation to be estimated. Proposition 6. Let Assumptions 9–13 hold. Then I 0 (t) = O (t) and I∗0 (t) = O (t) . Moreover, if g1 , . . . , gn ∈ C for n ∈ N satisfy g1 (t) = O ((t)α1 ) , . . . , gn (t) = O ((t)αn ) with α1 , . . . , αn ∈ (0, ∞), then I n [g1 , . . . , gn ](t) = O (t)1+α1 +···+αn , I∗n [g1 , . . . , gn ](t) = O (t)1+α1 +···+αn . (7.25) Proof. Define the F /B([0, ∞))-measurable mapping R : → [0, ∞) by R(ω) := 1 + sup F (Xt (ω)) V + sup St L(V ) 0≤t≤T
+
0≤t≤T
sup
+ + + Xt (ω) − Ot (ω) − Xt (ω) − Ot (ω) + 2 2 1 1 V t2 − t1
0≤t1
for all ω ∈ , which is indeed finite and hence well-defined by Assumptions 9–13 and Lemma 7.3. It follows from (7.7) that
Xt2 − Ot2 − Xt1 − Ot1 t1 t2 = St2 − St1 ξ + S(t2 −s) F (Xs ) ds − S(t1 −s) F (Xs ) ds 0 0 t2 t1 = S(t2 −t1 ) − I St1 ξ + S(t1 −s) F (Xs ) ds + S(t2 −s) F (Xs ) ds 0
= S(t2 −t1 ) − I Xt1 − Ot1 +
t1
t2
t1
S(t2 −s) F (Xs ) ds
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102
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
for all 0 ≤ t1 < t2 ≤ T . Hence + + S(t
2 −t1 )
−I
+ Xt1 − Ot1 +V
+ + − X − =+ X − O − O t2 t1 t1 + t2
t2
t1
+ + S(t2 −s) F (Xs ) ds + +
+ + + + + + ≤ Xt2 − Ot2 − Xt1 − Ot1 V + + +
t2
t1
V
+ + S(t2 −s) F (Xs ) ds + +
V
≤ R (t2 − t1 ) + R 2 (t2 − t1 ) ≤ 2R 2 (t2 − t1 ) for all 0 ≤ t1 < t2 ≤ T . It then follows that + + t + + + + + 0 + + S(t−s) F (Xt0 ) ds + +I (t)+ = + S(t−t0 ) − I Xt0 − Ot0 + + V t0
V
+ + t + + + + + + + ≤ S(t−t0 ) − I Xt0 − Ot0 V + + S(t−s) F (Xt0 ) ds + + t0
t
≤ 2R (t − t0 ) + 2
t0
V
+ + +S(t−s) +
+ + +F (Xt )+ ds 0 L(V ) V
≤ 2R 2 (t − t0 ) + R 2 (t − t0 ) ≤ 3R 2 (t − t0 ) and + + t + + + + + 0 + + S(t−s) F (Xs ) ds + +I∗ (t)+ = + S(t−t0 ) − I Xt0 − Ot0 + + V t0
V
+ t + + + + + + ≤ + S(t−t0 ) − I Xt0 − Ot0 +V + + S F (X ) ds s (t−s) + + t0
≤ 2R 2 (t − t0 ) +
t
t0
+ + +S(t−s) +
L(V )
V
F (Xs ) V ds
≤ 2R 2 (t − t0 ) + R 2 (t − t0 ) ≤ 3R 2 (t − t0 ) for all t ∈ [t0 , T ], which shows I 0 (t) = O(t) and I∗0 (t) = O(t).
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7.4. Taylor expansions
103
Now g1 , . . . , gn ∈ C and α1 , . . . , αn ∈ (0, ∞) for some n ∈ N can be given with g1 = O ((t)α1 ), . . . , gn = O ((t)α1 ). Then the F /B([0, ∞))-measurable mapping Dn : → [0, ∞) defined by
Dn (ω) := sup
sup
0≤r≤1 0≤t1 ≤t2 ≤T
+ sup t∈(t0 ,T ]
+ + + + (n) rXt1 (ω) + (1 − r)Xt2 (ω) + +F
L(n) (V ,V )
|g1 (t, ω)| |gn (t, ω)| α1 + · · · + sup αn (t − t0 ) t∈(t0 ,T ] (t − t0 )
for all ω ∈ is well-defined in view of the O (·)-notation. Therefore, + + n +I [g1 , . . . , gn ](t)+ ∗ + V + t 1 + = + S(t−s) F (n) Xt0 + r Xs − Xt0 (g1 (s), . . . , gn (s)) + t0 0 + + (1 − r)n−1 + · dr ds + + (n − 1)! V
≤
+ +S(t−s) +
t+
L(V )
t0
0
·
1+
+ + (n) + Xt0 + r Xs − Xt0 (g1 (s), . . . , gn (s))+ +F
V
(1 − r)n−1 dr ds (n − 1)!
t 1 + + + (n) + ≤R Xt0 + r Xs − Xt0 + +F t0
L(n) (V ,V )
0
·
· g1 (s) V · . . . · gn (s) V
(1 − r)n−1 dr ds (n − 1)!
t 1 + + + (n) + ≤ R (Dn )n Xt0 + r Xs − Xt0 + +F t0
0
·
L(n) (V ,V )
(s − t0 )α1 +···+αn
(1 − r)n−1 dr ds (n − 1)!
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104 and
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
+ + n +I [g1 , . . . , gn ](t)+ ∗ V t 1 (1 − r)n−1 (s − t0 )α1 +···+αn dr ds ≤ R (Dn )n+1 (n − 1)! 0 t0 = R (Dn )
n+1
t
α1 +···+αn
(s − t0 )
0
t0
1 = R (Dn )n+1 n! ≤
1
t
(1 − r)n−1 dr ds (n − 1)!
(s − t0 )α1 +···+αn ds
t0
1 R (Dn )n+1 (t − t0 )1+α1 +···+αn = n!
1 R (Dn )n+1 (t)1+α1 +···+αn n!
for all t ∈ [t0 , T ], which shows that I∗n [g1 , . . . , gn ] = O((t)1+α1 +···+αn ). It can be shown in the same way that I n [g1 , . . . , gn ] = O((t)1+α1 +···+αn ).
7.5 Abstract examples of Taylor expansions Proposition 5 will be used here to derive various Taylor expansions of the solution X of the SPDE (13), and Proposition 6 will be used to determine their orders of convergence.
Taylor expansion of order 1 The simplest Taylor expansion of the increment X is given by (7.22), i.e., X = I∗0 + O
(7.26)
or Xt = I∗0 (t) + Ot for t ∈ [t0 , T ]. If the solution X is known at time t0 only, the expression I∗0 (t) cannot be used to determine Xt = Xt − Xt0 for t ∈ (t0 , T ] since it contains the unknown solution path Xs with s ∈ [t0 , t]. (See (7.23) for the definition of I∗0 .) Omitting I∗0 gives the approximation X ≈ O, which can also be written as Xt ≈ Xt0 + Ot − Ot0 ,
t ∈ [t0 , T ].
(7.27)
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7.5. Abstract examples of Taylor expansions
105
By Proposition 6 the remainder term I∗0 of the Taylor approximation (7.27) can be estimated by I∗0 (t) = O(t), so Xt = Xt0 + Ot − Ot0 + O(t)
(7.28)
gives the simplest Taylor approximation of the solution process X.
Taylor expansion of order 1 + θ Proposition 5 can now be used to derive a higher order expansion of (7.26). More precisely, formula (7.23) is inserted into the remainder stochastic process I∗0 , i.e., I∗0 = I 0 + I∗1 [I∗0 ] + I∗1 [O] into (7.26), to give X = I 0 + I∗1 [I∗0 ] + I∗1 [O] + O, which can also be written as
X = I 0 + O + I∗1 [I∗0 ] + I∗1 [O] .
(7.29)
Omitting the double integral terms I∗1 [I∗0 ] and I∗1 [O] gives the Taylor approximation X ≈ I 0 + O, (7.30) or, using the definition of the stochastic process I 0 , t Ss ds F (Xt0 ) + Ot Xt ≈ (St − I ) Xt0 − Ot0 + 0
for t ∈ [t0 , T ]. Hence Xt ≈ St Xt0 +
0
t
Ss ds F (Xt0 ) + Ot − St Ot0
(7.31)
is another Taylor approximation for the solution of the SPDE (7.4). By Proposition 6 I∗0 (t) = O (t), and by Assumption 11 Ot = O (t)θ , so it follows from (7.25) that I∗1 [I∗0 ](t) = O (t)2 , I∗1 [O](t) = O (t)1+θ . These imply that
I∗1 [I∗0 ](t) + I∗1 [O](t) = O (t)1+θ ,
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106
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
since 1 + θ < 2. Hence, from (7.29),
Xt = I 0 (t) + Ot + O (t)1+θ ,
and it then follows from the definition of the stochastic process I 0 ∈ C that t Xt = St Xt0 + Ss ds F (Xt0 ) + Ot − St Ot0 + O (t)1+θ . (7.32) 0
The Taylor approximation (7.32) plays an analogous role to the strong order γ = 0.5 Taylor expansion in Chapter 3 on which the Euler–Maruyama scheme for finite dimensional SODEs is based. It will be called the exponential Euler approximation, since it gives the exponential Euler scheme (6.12) for SPDEs with additive noise, as will be seen in Section 7.7.
Taylor expansion of order 1 + min(1, 2θ) Further expansions of the remainder terms in a Taylor expansion give a Taylor expansion of higher order. The remainder term of (7.29) consists of two terms, I∗1 [I∗0 ] and I∗1 [O]. There are now two possibilities for obtaining a higher orderTaylor ex pansion: either expand I∗1 [I∗0 ](t) = O (t)2 or expand I∗1 [O](t) = O (t)1+θ . Since, by Assumption 11, 1 + θ < 2 the stochastic process I∗1 [O] is of lower order than I∗1 [I∗0 ]. Hence the term I∗1 [O] should be expanded to improve on the approximation order 1 + θ of the Taylor approximation (7.32). More precisely, from (7.24), I∗1 [O] = I 1 [O] + I∗2 [I∗0 , O] + I∗2 [O, O], which is inserted into (7.29) to yield X = I 0 + O + I 1 [O] + R,
(7.33)
where the remainder term R ∈ C is given by R = I∗1 [I∗0 ] + I∗2 [I∗0 , O] + I∗2 [O, O].
(7.34)
By Proposition 6 I∗1 [I∗0 ](t) = O (t)2 ,
I∗2 [I∗0 , O](t) = O (t)2+θ ,
and I∗2 [O, O](t) = O (t)1+2θ . Since min(2, 2 + θ , 1 + 2θ ) = 1 + min(1, 2θ ), it follows that R = O (t)1+min(1,2θ) . Thus Xt = I 0 (t) + Ot + I 1 [O](t) + O (t)1+min(1,2θ ) .
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7.5. Abstract examples of Taylor expansions
107
This can also be written as
t Ss ds F (Xt0 ) + Ot − St Ot0 Xt = St Xt0 + 0 t + S(t−s) F (Xt0 )Os ds + O (t)1+min(1,2θ )
(7.35)
t0
and is a Taylor approximation of order 1 + min(1, 2θ ). It will be used in Section 7.7 to derive the numerical scheme (7.68).
Taylor expansion of order 1 + min(1, 3θ) The remainder term (7.34) consists of three parts, namely, I∗2 [I∗0 , O](t) = O (t)2+θ , I∗1 [I∗0 ](t) = O (t)2 , I∗2 [O, O](t) = O (t)1+2θ . Since θ < 12 and hence min(2, 2 + θ , 1 + 2θ ) = 1 + 2θ in the examples in Section 7.6, the stochastic process I∗2 [O, O] will be expanded here. Applying Proposition 5 to this term yields I∗2 [O, O] = I 2 [O, O] + I∗3 [I∗0 , O, O] + I∗3 [O, O, O], and inserting this into (7.33) then gives X = I 0 + O + I 1 [O] + I 2 [O, O] + R,
(7.36)
where the remainder R ∈ C reads R = I∗1 [I∗0 ] + I∗2 [I∗0 , O] + I∗3 [I∗0 , O, O] + I∗3 [O, O, O].
(7.37)
By Proposition 6 I∗3 [I∗0 , O, O](t) = O (t)2+2θ ,
I∗3 [O, O, O](t) = O (t)1+3θ ,
so R = O (t)1+min(1,3θ ) since min(2, 2 + θ , 2 + 2θ , 1 + 3θ ) = 1 + min(1, 3θ ). Finally, the Taylor expansion written out fully is t t Ss ds F (Xt0 ) + Ot − St Ot0 + S(t−s) F (Xt0 )Os ds Xt = St Xt0 + 1 + 2
0
t
t0
t0
S(t−s) F (Xt0 ) (Os , Os ) ds + O (t)1+min(1,3θ ) .
(7.38)
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108
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
Taylor expansion of order 1 + min(1, 4θ) The remainder term (7.37) of the previous Taylor approximation consists of the following four terms: I∗2 [I∗0 , O](t) = O (t)2+θ , I∗1 [I∗0 ](t) = O (t)2 , I∗3 [I∗0 , O, O](t) = O (t)2+2θ ,
I∗3 [O, O, O](t) = O (t)1+3θ .
In the concrete examples (7.18) and (7.19) Assumption 11 is satisfied for all θ ∈ (0, 14 ). Since min(2, 2 + θ , 2 + 2θ, 1 + 3θ ) = 1 + 3θ in this case, the stochastic process I∗3 [O, O, O] will be expanded here. Then Proposition 5 yields I∗3 [O, O, O] = I 3 [O, O, O] + I∗4 [I∗0 , O, O, O] + I∗4 [O, O, O, O].
(7.39)
This is inserted into (7.36) to give X = I 0 + O + I 1 [O] + I 2 [O, O] + I 3 [O, O, O] + R
(7.40)
with the remainder R = I∗1 [I∗0 ] + I∗2 [I∗0 , O] + I∗3 [I∗0 , O, O] + I∗4 [I∗0 , O, O, O] + I∗4 [O, O, O, O]. Since I∗0 (t) = O (t) and Ot = O (t)θ , it follows from Proposition 6 that I∗4 [I∗0 , O, O, O](t) = O (t)2+3θ , I∗4 [O, O, O, O](t) = O (t)1+4θ . This and the fact thatmin (2, 2 + θ, 2 + 2θ , 2 + 3θ , 1 + 4θ ) = 1 + min(1, 4θ ) show R = O (t)1+min(1,4θ ) . Hence, by the definition of the integral operators in Section 7.4.1, (7.40) gives the Taylor expansion t Xt = St Xt0 + (7.41) Ss ds F (Xt0 ) + Ot − St Ot0 0
+
t
S(t−s) F (Xt0 )Os ds +
t0
1 + 6
t
t0
1 2
t
t0
S(t−s) F (Xt0 ) (Os , Os ) ds
S(t−s) F (3) (Xt0 ) (Os , Os , Os ) ds + O (t)1+min(1,4θ ) .
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7.5. Abstract examples of Taylor expansions
109
Taylor expansion of order 1 + min(1 + θ , 3θ) Assumption 11 is satisfied in example (7.20) for all θ ∈ (0, 12 ). Since min(2, 2 + θ , 2 + 2θ , 1 + 3θ ) = 2 for such θ , the stochastic process I∗1 [I∗0 ] in the remainder term (7.37) of the Taylor expansion (7.36) will be expanded here instead of the stochastic process I∗3 [O, O, O] above. Then Proposition 5 yields I∗1 [I∗0 ] = I 1 [I∗0 ] + I∗2 [I∗0 , I∗0 ] + I∗2 [O, I∗0 ] = I 1 [I 0 ] + I 1 [I∗1 [I∗0 ]] + I 1 [I∗1 [O]] + I∗2 [I∗0 , I∗0 ] + I∗2 [O, I∗0 ]. Inserting this into (7.36) gives X = I 0 + O + I 1 [I 0 ] + I 1 [O] + I 2 [O, O] + R
(7.42)
with the remainder R = I 1 [I∗1 [I∗0 ]] + I 1 [I∗1 [O]] + I∗2 [I∗0 , I∗0 ] + 2I∗2 [I∗0 , O]
(7.43)
+ I∗3 [I∗0 , O, O] + I∗3 [O, O, O]. From Proposition 6, I 1 [I∗1 [I∗0 ]](t) = O (t)3 ,
I 1 [I∗1 [O]](t) = O (t)2+θ ,
(7.44)
I∗2 [I∗0 , I∗0 ](t) = O (t)3 ,
I∗2 [O, I∗0 ](t) = O (t)2+θ .
(7.45)
Then R = O (t)1+min(1+θ ,3θ) since min(3, 2 + θ, 2 + 2θ , 1 + 3θ ) = 1 + min(1 + θ , 3θ). Thus (7.42) gives the Taylor expansion Xt = St Xt0 +
t
+
t0
+
t
t0
Ss ds F (Xt0 ) + Ot − St Ot0
t
0
S(t−s) F (Xt0 ) (Ss − I ) Xt0 − Ot0 + 1 S(t−s) F (Xt0 )Os ds + 2
t
t0
(7.46)
0
s
Su du F (Xt0 ) ds
S(t−s) F (Xt0 ) (Os , Os ) ds
+ O (t)1+min(1+θ,3θ) .
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110
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
Taylor expansion of order 1 + min(1 + θ, 4θ) The Taylor expansions (7.40) and (7.42) are combined here, i.e., the stochastic process I∗3 [O, O, O] in the remainder term (7.43) of the Taylor expansion (7.42) is expanded. Inserting (7.39) into (7.42) yields X = I 0 + O + I 1 [I 0 ] + I 1 [O] + I 2 [O, O] + I 3 [O, O, O] + R, (7.47) where the remainder R ∈ C is given by R = I 1 [I∗1 [I∗0 ]] + I∗1 [I∗1 [O]] + I∗2 [I∗0 , I∗0 ] + 2I∗2 [I∗0 , O] + I∗3 [I∗0 , O, O] + I∗4 [I∗0 , O, O, O] + I∗4 [O, O, O, O].
(7.48)
By Proposition 6 I∗4 [I∗0 , O, O, O] = O (t)2+3θ ,
I∗4 [O, O, O, O] = O (t)1+4θ ,
from which,with min(3, 2 + θ , 2 + 2θ , 2 + 3θ , 1 + 4θ ) = 1 + min(1 + θ , 4θ), it follows that R = O (t)1+min 1+θ ,4θ . Therefore, by (7.47), Xt = St Xt0 +
t0 t
+
t0
+
1 6
Ss ds F (Xt0 ) + Ot − St Ot0
(7.49)
S(t−s) F (Xt0 ) (Ss − I ) Xt0 − Ot0 +
t
+
0
t
1 S(t−s) F (Xt0 )Os ds + 2 t
t0
0
t
t0
s
Su du F (Xt0 ) ds
S(t−s) F (Xt0 ) (Os , Os ) ds
S(t−s) F (3) (Xt0 ) (Os , Os , Os ) ds + O (t)1+min(1+θ ,4θ) .
Taylor expansion of order 1 + min(1 + θ, 5θ) From the remainder term (7.48) above, min(3, 2 + θ , 2 + 2θ , 2 + 3θ , 1 + 4θ ) = 1 + min(1+θ , 4θ ), so for θ < 14 as in the concrete examples (7.18) and (7.19), the stochastic process I∗4 [O, O, O, O] has the lowest order of all stochastic processes in (7.48). Expanding this term yields I∗4 [O, O, O, O] = I 4 [O, O, O, O] + I∗5 [I∗0 , O, O, O, O] + I∗5 [O, O, O, O, O],
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7.5. Abstract examples of Taylor expansions
111
so the Taylor expansion is given by X = I 0 + O + I 1 [I 0 ] + I 1 [O] + I 2 [O, O] + I 3 [O, O, O] + I 4 [O, O, O, O] + R with the remainder R = I 1 [I∗1 [I∗0 ]] + I∗1 [I∗1 [O]] + I∗2 [I∗0 , I∗0 ] + I∗2 [O, I∗0 ] + I∗2 [I∗0 , O] + I∗3 [I∗0 , O, O] + I∗4 [I∗0 , O, O, O] + I∗5 [I∗0 , O, O, O, O] + I∗5 [O, O, O, O, O]. The corresponding Taylor approximation of the solution of the SPDE (7.4) is thus given by Xt = St Xt0 + + + +
t0 t
1 6
Ss ds F (Xt0 ) + Ot − St Ot0
S(t−s) F (Xt0 )Os ds +
1 + 24
S(t−s) F (Xt0 ) (Ss − I ) Xt0 − Ot0 +
t
t0
0
t
t
t0
1 2
t
t0
(7.50)
0
s
Su du F (Xt0 ) ds
S(t−s) F (Xt0 ) (Os , Os ) ds
S(t−s) F (3) (Xt0 ) (Os , Os , Os ) ds t
t0
S(t−s) F (4) (Xt0 ) (Os , Os , Os , Os ) ds + O (t)1+min(1+θ ,5θ) .
Taylor expansion of order 1 + min(1 + θ , 6θ) If again θ < 14 , then the stochastic process I∗5 [O, O, O, O, O] has the lowest order of all occurring terms in (7.50). Applying Proposition 5 then yields I∗5 [O, O, O, O, O] = I 5 [O, O, O, O, O] + I∗6 [I∗0 , O, O, O, O, O] + I∗6 [O, O, O, O, O, O].
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112
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
The corresponding Taylor expansion is X = I 0 + O + I 1 [I 0 ] + I 1 [O] + I 2 [O, O] + I 3 [O, O, O] + I 4 [O, O, O, O] + I 5 [O, O, O, O] + R with the remainder R ∈ C given by R = I 1 [I∗1 [I∗0 ]] + I∗1 [I∗1 [O]] + I∗2 [I∗0 , I∗0 ] + 2I∗2 [I∗0 , O] + I∗3 [I∗0 , O, O] + I∗4 [I∗0 , O, O, O] + I∗5 [I∗0 , O, O, O, O]
(7.51)
+ I∗6 [I∗0 , O, O, O, O, O] + I∗6 [O, O, O, O, O, O] and by Proposition 6 R = O (t)1+min(1+θ ,6θ) . The Taylor approximation here is Xt = St Xt0 + + +
t0 t
1 + 6
Ss ds F (Xt0 ) + Ot − St Ot0
S(t−s) F (Xt0 )Os ds +
1 + 24
S(t−s) F (Xt0 ) (Ss − I ) Xt0 − Ot0 +
t
t0
0
t
t
t0
1 2
t
t0
(7.52)
s
0
Su du F (Xt0 ) ds
S(t−s) F (Xt0 ) (Os , Os ) ds
S(t−s) F (3) (Xt0 ) (Os , Os , Os ) ds t
t0
S(t−s) F (4) (Xt0 ) (Os , Os , Os , Os ) ds
t 1 S(t−s) F (5) (Xt0 ) (Os , Os , Os , Os , Os ) ds 120 t0 + O (t)1+min(1+θ,6θ ) .
+
Taylor expansion of order 1 + min(2, 1 + 2θ, 6θ) From (7.51), when θ < 14 , the stochastic processes that have the lowest order are I∗1 [I∗1 [O]], I∗2 [O, I∗0 ], and I∗2 [I∗0 , O], so expanding these three terms will improve the Taylor approximation (7.52). Note that I∗2 [I∗0 , O] = I∗2 [O, I∗0 ]. Then
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7.5. Abstract examples of Taylor expansions
113
applying Proposition 5 yields I∗1 [I∗1 [O]] = I 1 [I∗1 [O]] + I∗2 [I∗0 , I∗1 [O]] + I∗2 [O, I∗1 [O]] = I 1 [I 1 [O]] + I 1 [I∗2 [I∗0 , O]] + I 1 [I∗2 [O, O]] + I∗2 [I∗0 , I∗1 [O]] + I∗2 [O, I∗1 [O]] and I∗2 [I∗0 , O] = I 2 [I∗0 , O] + I∗3 [I∗0 , I∗0 , O] + I∗3 [O, I∗0 , O] = I 2 [I 0 , O] + I 2 [I∗1 [I∗0 ], O] + I 2 [I∗1 [O], O] + I∗3 [I∗0 , I∗0 , O] + I∗3 [O, I∗0 , O]. Thus the Taylor expansion is given by X = I 0 + O + I 1 [I 0 ] + I 1 [O] + I 2 [O, O] + I 3 [O, O, O] +I 4 [O, O, O, O] + I 5 [O, O, O, O] + 2I 2 [I 0 , O] + I 1 [I 1 [O]] + R with remainder R = I 1 [I∗1 [I∗0 ]] + I∗1 [I∗2 [I∗0 , O]] + I∗1 [I∗2 [O, O]] + I∗2 [I∗0 , I 1 [O]] + I∗2 [O, I 1 [O]] + I∗2 [I∗0 , I∗0 ] + 2I∗2 [I∗1 [I∗0 ], O] + 2I∗2 [I∗1 [O], O] + 2I∗3 [I∗0 , I 0 , O] + 2I∗3 [O, I 0 , O] + I∗3 [I∗0 , O, O] + I∗4 [I∗0 , O, O, O] + I∗5 [I∗0 , O, O, O, O] + I∗6 [I∗0 , O, O, O, O, O] + I∗6 [O, O, O, O, O, O].
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114
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
The Taylor approximation here is Xt = St Xt0 +
t
+
t0
Ss ds F (Xt0 ) + Ot − St Ot0
S(t−s) F (Xt0 ) (Ss − I ) Xt0 − Ot0 +
+
t
t0
+
0
t
1 6
1 S(t−s) F (Xt0 )Os ds + 2 t
t0
s
0
t
t0
Su du F (Xt0 ) ds
S(t−s) F (Xt0 ) (Os , Os ) ds
S(t−s) F (3) (Xt0 ) (Os , Os , Os ) ds
1 t S(t−s) F (4) (Xt0 ) (Os , Os , Os , Os ) ds (7.53) 24 t0 t 1 + S(t−s) F (5) (Xt0 ) (Os , Os , Os , Os , Os ) ds 120 t0 s t + Su du F (Xt0 ), Os ds S(t−s) F (Xt0 ) (Ss − I ) Xt0 − Ot0 + +
+
t0 t t0
0
s
S(s−u) F (Xt0 )Ou du ds S(t−s) F (Xt0 ) t0 +O (t)1+min(2,1+2θ,6θ) .
Remark 13. The Taylor approximations above are clearly becoming increasingly cumbersome. The terms that need to be included can be characterized succinctly with stochastic trees and stochastic woods of appropriate indices. This is developed in some detail in Jentzen & Kloeden [69]. The corresponding notation for SPDEs with multiplicative noise will be explained in Chapter 8.
7.6
Examples of Taylor approximations
The above Taylor expansions will now be considered in the context of SPDEs with specific types of noise.
7.6.1
Space–time white noise
SPDEs with space–time white noise on a one-dimensional spatial domain are very prominent in the literature. They are included in the above theoretical developments when Assumption 11 is satisfied with θ = 14 − ε for an arbitrarily small ε ∈ (0, 14 ). The abstract Taylor approximations above with this parameter are now given below and will then be applied in the following subsection to the concrete SPDE (7.18).
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7.6. Examples of Taylor approximations
115
The Taylor approximations (7.32), (7.35), (7.38), and (7.41) are now, respectively, t 5 Xt = St Xt0 + Ss ds F (Xt0 ) + Ot − St Ot0 + O (t) 4 −ε , 0
t Ss ds F (Xt0 ) + Ot − St Ot0 Xt = St Xt0 + 0 t 3 + S(t−s) F (Xt0 )Os ds + O (t) 2 −ε , t0
Xt = St Xt0 + +
1 2
0
t0
and
Xt = St Xt0 +
t
+
1 6
t
t0
In addition,
+
t
t
0
1 2
t
t0
S(t−s) F (Xt0 ) (Os , Os ) ds
Ss ds F (Xt0 ) + Ot − St Ot0
1 S(t−s) F (Xt0 )Os ds + 2
1 24
S(t−s) F (Xt0 )Os ds
S(t−s) F (3) (Xt0 ) (Os , Os , Os ) ds + O (t)2−ε .
t0
1 + 6
t0
Ss ds F (Xt0 ) + Ot − St Ot0
S(t−s) F (Xt0 ) (Ss − I ) Xt0 − Ot0 +
t t0
+
0
Xt = St Xt0 + +
t
S(t−s) F (Xt0 )Os ds +
t0
t
7 S(t−s) F (Xt0 ) (Os , Os ) ds + O (t) 4 −ε ,
t
+
Ss ds F (Xt0 ) + Ot − St Ot0 +
t
t
t0
t
t0
0
s
Su du F (Xt0 ) ds
S(t−s) F (Xt0 ) (Os , Os ) ds
S(t−s) F (3) (Xt0 ) (Os , Os , Os ) ds t
t0
9 S(t−s) F (4) (Xt0 ) (Os , Os , Os , Os ) ds + O (t) 4 −ε
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116
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
and Xt = St Xt0 +
+
t
1 S(t−s) F (Xt0 )Os ds + 2
+ +
1 6
t0
1 24
+
t
t0
t t0
t
t0
S(t−s) F (Xt0 ) (Os , Os ) ds
S(t−s) F (3) (Xt0 ) (Os , Os , Os ) ds t
t0
1 120
+
t
Su du F (Xt0 ) ds
0
t0
+
s
t0
0
Ss ds F (Xt0 ) + Ot − St Ot0
S(t−s) F (Xt0 ) (Ss − I ) Xt0 − Ot0 +
t
+
t
S(t−s) F (4) (Xt0 ) (Os , Os , Os , Os ) ds t
t0
S(t−s) F (5) (Xt0 ) (Os , Os , Os , Os , Os ) ds
S(t−s) F (Xt0 ) (Ss − I ) Xt0 − Ot0 +
S(t−s) F (Xt0 )
s
t0
s
0
Su du F (Xt0 ), Os ds
5 S(s−u) F (Xt0 )Ou du ds + O (t) 2 −ε
are based on the Taylor approximations (7.50) and (7.53).
7.6.2 Taylor approximations for a nonlinear SPDE The above Taylor approximations will now be given for the concrete example with the nonlinear SPDE (7.18), i.e., 2 1 ∂ 3 dXt = X + X − X t t t dt + dWt , 2 2 ∂x on the function space V = C ([0, 1], R), i.e., V is the Banach space of continuous functions from [0, 1] to R with the supremum norm. These will involve terms of the form v · w ∈ V and v n ∈ V for v, w ∈ V and n ∈ N, which are defined by (v w) (x) := v(x) · w(x),
v n (x) := (v(x))n ,
x ∈ [0, 1].
Moreover, ε here is always an arbitrarily small real number (0, 14 ).
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7.6. Examples of Taylor approximations
117
The Taylor approximations in the previous subsection become, respectively, Xt = Xt0 + Ot + O (t) , Xt = St Xt0 +
(7.54)
5 Ss ds (Xt0 − Xt30 ) + Ot − St Ot0 + O (t) 4 −ε ,
t
0
(7.55) t Xt = St Xt0 + Ss ds (Xt0 − Xt30 ) + Ot − St Ot0 0 t 3 + S(t−s) 1 − 3Xt20 Os ds + O (t) 2 −ε ,
(7.56)
t0
Ss ds (Xt0 − Xt30 ) + Ot − St Ot0 Xt = St Xt0 + 0 t t 2 + S(t−s) (1 − 3Xt0 )Os ds − 3 S(t−s) Xt0 (Os )2 ds t0 t0 7 −ε +O (t) 4 , (7.57) t
t Xt = St Xt0 + Ss ds Xt0 − Xt30 + Ot − St Ot0 0 t t + S(t−s) 1 − 3Xt20 Os ds − 3 S(t−s) Xt0 (Os )2 ds t0 t
−
S(t−s) (Os )3 ds + O (t)2−ε
t0
t0
3 S(t−s) Xt0 + Os − Xt0 + Os ds t0 + Ot − St Ot0 + O (t)2−ε ,
= St Xt0 +
Xt = St Xt0 + +
t
t0
S(t−s)
1 − 3Xt20
(Ss − I ) Xt0 − Ot0 ds
s 2 3 + S(t−s) 1 − 3Xt0 Su du Xt0 − Xt0 ds t0 0 9 +O (t) 4 −ε ,
(7.58)
3 S(t−s) Xt0 + Os − Xt0 + Os ds + Ot − St Ot0
t
t0 t
t
(7.59)
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118
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
and
Xt = St Xt0 + +
3 ds + Ot − St Ot0 S(t−s) Xt0 + Os − Xt0 + Os
t
t0
t
S(t−s)
+
t0 t
2 S(t−s) 1 − 3Xt0 t
−6
t0 t
t0 t
S(t−s) Xt0 Os
S(t−s) 1 − 3Xt20 t0 5 +O (t) 2 −ε . +
s
Xt0 − Xt30
Su du
ds
ds S(t−s) Xt0 Os (Ss − I ) Xt0 − Ot0
(Ss − I ) Xt0 − Ot0 ds
0
t0
−6
1 − 3Xt20
s
Su du
0
Xt0 − Xt30
ds
s 2 Ss−u (1 − 3Xt0 )Ou du ds
t0
(7.60)
The linear-implicit Euler and Crank–Nicolson approximations Commonly used approximations in the literature are the linear-implicit Euler and the linear-implicit Crank–Nicolson approximations. For the nonlinear SPDE (7.18) they are, respectively, t (I − tA)−1 dWs Xt ≈ (I − tA)−1 Xt0 + t Xt0 − Xt30 (7.61) + t0
and
−1 −1 1 1 Xt0 − Xt30 I − tA Xt0 + t I − tA 2 2
1 Xt ≈ I + tA 2 +
t t0
−1
1 I − tA 2
dWs ,
(7.62)
where Wt is a cylindrical I -Wiener process and A is the Laplacian with Dirichlet boundary conditions. The expressions (7.61) and (7.62) approximate the exact solution locally in time with the order 14 − ε, while the above Taylor approximations approximate it locally in time with the orders 1, 54 − ε, 32 − ε, . . . , 52 − ε. Comparative simulations Numerical simulations indicate that the estimates for the remainders in the Taylor approximations above are sharp. In Figure 7.1, the pathwise approximation error
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7.6. Examples of Taylor approximations
119
−3
10
Taylor appr. order 1− Taylor appr. order 5/4− Taylor appr. order 7/4− Taylor appr. order 9/4− Orderlines 1, 5/4, 7/4, 9/4
−4
Pathwise maximum error
10
−5
10
−6
10
−7
10
−6
10
−5
−4
10
−3
−2
10 10 Time step t − t
10
−1
10
0
−4
10
Pathwise maximum error
Taylor appr. order 3/2− Taylor appr. order 2− Orderlines 3/2, 2
−5
10
−6
10
−7
10
−4
10
−3
−2
10
10
−1
10
Time step t − t
0
Figure 7.1. Pathwise approximation error of the Taylor approximations (7.54)–(7.59) vs. t = t − 12 for different t ∈ [ 12 , 1] and one random realization of (7.18).
of the Taylor approximations (7.54)–(7.59) with respect to the supremum norm on C ([0, 1], R) is plotted against the time step t = t − t0 for different t ∈ [t0 , T ] with t0 = 12 and T = 1 for a single random realization of the SPDE (7.18). The infinite dimensional space V = C ([0, 1], R) is approximated by its 40-dimensional subspace P40 (V ). This will be explained in more detail in Section 7.7. In Figure 7.2 the mean square approximation error of the Taylor approximations 1 (7.54)–(7.59) with respect to the supremum norm in C ([0, 1], R), i.e., E · 2V 2 , is plotted against time steps t = t − 12 for different t ∈ [ 12 , 1], where the expectation is approximated by 50 independent random realizations and the infinite dimensional space V is also approximated by its subspace P40 (V ).
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120
Chapter 7. Taylor Approximations for SPDEs with Additive Noise −3
10
Taylor appr. order 1− Taylor appr. order 5/4− Taylor appr. order 7/4− Taylor appr. order 9/4− Orderlines 1, 5/4, 7/4, 9/4
Mean square maximum error
−4
10
−5
10
−6
10
−7
10
−6
10
−5
−4
10
−3
−2
10 10 Time step t − t
10
−1
10
0
−4
10
Mean square maximum error
Taylor appr. order 3/2− Taylor appr. order 2− Orderlines 3/2, 2
−5
10
−6
10
−7
10
−4
10
−3
−2
10
10
−1
10
Time step t − t0
Figure 7.2. Mean square approximation error of the Taylor approximations (7.54)–(7.59) vs. t = t − 12 for different t ∈ [ 12 , 1] and (7.18).
Figures 7.1 and 7.2 show that the Taylor approximations (7.54)–(7.59) converge with their theoretically predicted order. Remark 14. The iterated integrals in the Taylor approximations need to be computed, and it is natural to ask if the extra computational burden is really worthwhile. To answer this, note first that Taylor approximations not only are useful for computational purposes but are also important from a theoretical point of view, i.e., for understanding local properties of the solution process. In addition, for example, the Taylor approximation in (7.55) approximates the exact solution with the order 5 4 − ε but with (up to a constant) the same computational effort! Moreover, other approximations, which attain a higher order and involve just a few more terms, can be derived quite easily (see Section 7.7). Finally, Taylor approximations such as
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7.7. Numerical schemes from Taylor expansions
121
(7.54)–(7.60) provide a theoretical basis for deriving various higher order one-step numerical schemes for reaction–diffusion-like SPDEs with additive noise.
7.6.3
Smoother noise
Solutions of SPDEs driven by space–time white noise in spatial dimensions higher than one such as the SPDE (7.20) may not exist or may take values in a function space. To handle smoother types of driving noise it will be supposed here that Assumption 11 holds with θ = 12 − ε for arbitrarily small ε ∈ (0, 12 ). The Taylor approximations (7.28) and (7.32) here are Xt = Xt0 + Ot + O (t) and
Xt = St Xt0 +
t
0
3 Ss ds F (Xt0 ) + Ot − St Ot0 + O (t) 2 −ε ,
(7.63)
(7.64)
while (7.35) is t Ss ds F (Xt0 ) + Ot − St Ot0 Xt = St Xt0 + 0 t + S(t−s) F (Xt0 )Os ds + O (t)2−ε
(7.65)
t0
and (7.46) is t Ss ds F (Xt0 ) + Ot − St Ot0 Xt = St Xt0 + 0 t 1 t + S(t−s) F (Xt0 )Os ds + S(t−s) F (Xt0 ) (Os , Os ) ds 2 t0 t0 s t + S(t−s) F (Xt0 ) (Ss − I ) Xt0 − Ot0 + Su du F (Xt0 ) ds t0 0 5 + O (t) 2 −ε . (7.66) In comparison with these Taylor approximations, the linear-implicit Euler and the linear-implicit Crank–Nicolson approximations above applied to the SPDE (7.20) approximate the solution locally with the order 12 − ε.
7.7
Numerical schemes from Taylor expansions
The numerical approximation of the SPDE (7.4) requires the discretization of the solution on both the time interval [0, T ] and the infinite dimensional space V . The
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122
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
above Taylor expansions provide the temporal discretization in the space V , while the space discretization is introduced in the following assumption, which is motivated by Galerkin approximations. Assumption 14 (projection operators). For each N ∈ N, let PN : V → V be a bounded linear operator with (PN )2 = PN and PN St = St PN for all t ∈ [0, ∞). Denote the image of PN in V by VN := PN (V ) = im(PN ) ⊂ V . Since the operator PN : V → V is linear, VN , · VN is a normed real vector subspace of V for each N ∈ N. In addition, VN is separable and its Borel σ -algebra satisfies B(VN ) = B(V ) ∩ VN . Moreover, by Assumption 14, PN (v) = v for all v ∈ VN and the spaces VN are invariant with respect to the semigroup St , i.e., St (VN ) ⊂ VN for all t ∈ [0, ∞). Usually the spaces VN are finite dimensional. Then PN projects the infinite dimensional SPDE (7.4) onto the finite dimensional space VN , where numerical computations can be done. For this the semigroup S, the nonlinearity F , the driving process O, and the initial value ξ will be approximated by mappings S N : [0, ∞) → L(VN ) and F N : VN → VN by a stochastic process O N : [0, T ] × → VN ⊂ V and an F /B(VN )measurable random variable ξ : → VN defined by, respectively, StN (v) := St (v), OsN (ω) := PN
FN (v) := PN (F (v)) ,
x0N (ω) := PN (ξ (ω)) for all v ∈ VN , t ∈ [0, ∞), s ∈ [0, T ], ω ∈ , and each N ∈ N. Here L(VN ), · L(VN ) is the real normed linear space of all bounded linear operators from VN to VN . This truncated mapping S N satisfies + N + +S − S N + + + ·s t s L(VN ) + + S0N = I , StN1 StN1 = S(tN1 +t2 ) , sup +StN + < ∞, sup <∞ L(VN ) t −s 0≤t≤T 0≤s
for all t1 , t2 ∈ [0, ∞) by Assumption 9, so it is a semigroup on VN for each N ∈ N. In addition, by Assumption 10, FN : VN → VN is infinitely often Fréchet differentiable with (n) FN (v0 ) (v1 , . . . , vn ) = PN F (n) (v0 ) (v1 , . . . , vn ) for all v0 , . . . , vn ∈ VN for all n and all N in N. Finally, O N : [0, T ] × → VN has θ -Hölder continuous sample paths for all N ∈ N, where θ ∈ (0, 1) is given in Assumption 11. With this setup one-step numerical schemes in the spaces VN , N ∈ N, can now be derived from the Taylor approximations in the previous sections to approximate the solution process Xt of the SPDE (7.4). The iterates of these schemes will be denoted by the F /B(VN )-measurable mappings YkN,M : → VN
with
Y0N,M := ξ N + O0N
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7.7. Numerical schemes from Taylor expansions
123
for k = 0, 1, . . . , M and N , M ∈ N. Here N ∈ N, which is usually the dimension of VN , indicates the accuracy of the spatial discretization and M ∈ N indicates the number of times steps on the interval [0, T ] with constant step size h = T /M. Remark 15. Note that the initial value of the driving process here is absorbed into that of the exact solution of the SPDE, i.e., X0 = ξ + O0 , and similarly for the initial value Y0N,M of the numerical scheme.
7.7.1 The exponential Euler scheme The global convergence order of the Taylor approximation (7.28) is too low to give a consistent scheme. The next higher order Taylor approximation (7.31) leads to the numerical scheme h N,M N N,M N N N Yk+1 = Sh Yk + Ss ds FN (YkN,M ) + O(k+1)h − ShN Okh (7.67) 0
for k = 0, 1, . . . , M − 1 for fixed N , M ∈ N. First, it is necessary to show that the F /B(VN )-measurable mappings YkN,M are well-defined by (7.67), although VN in general may not be complete. The integral h N 0 Ss ds ∈ L(VN , V ) is understood as an L(VN , V )-valued Bochner integral, where L(VN , V ) is the real Banach space of all bounded linear operators from VN to V . It follows from Proposition A.2.2 in [104] that PN 0
h
SsN ds =
h 0
PN SsN ds =
h 0
SsN ds
h
for all N , M ∈ N, which shows that 0 SsN ds is in fact in L(VN ) = L(VN , VN ) for each N ∈ N. Hence the iterates of the numerical approximation (7.67) are well-defined. The numerical scheme (7.67) is the simplest nontrivial Taylor scheme for SPDEs with additive noise obtained via the Taylor expansions considered here. It is in fact the additive noise version of the exponential Euler scheme (6.12) introduced in Chapter 6. In a similar way, the Taylor approximation (7.35) leads to the numerical scheme N,M Yk+1 = ShN YkN,M +
+
(k+1)h kh
0
h
N N SsN ds FN YkN,M + O(k+1)h − ShN Okh
N N S(k+1)h−s FN YkN,M OsN − Okh ds
(7.68)
for k = 0,1, . . . , M − 1 with fixed N and M in N. The well-definedness of the terms here can be shown in the same way as above.
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124
Chapter 7. Taylor Approximations for SPDEs with Additive Noise
7.7.2 A Runge–Kutta scheme for SPDEs In principle, the next Taylor approximations can be used to derive numerical schemes of higher order. These schemes, however, would be of limited practical use due to cost and difficulty of computing the higher iterated integrals as well as the higher order derivatives in the Taylor approximations. This situation is already well known for ODEs, where derivative-free Runge–Kutta schemes are often used, with the Taylor schemes being used for theoretical purposes such as for determining the convergence order. Similar, derivative-free numerical schemes with simple integrals can also be derived for SPDEs and will also be called Runge–Kutta schemes here. To show how a Runge–Kutta scheme can be derived for SPDEs consider the Taylor approximation (7.35) on the time subinterval [kh, (k + 1)h], namely, X(k+1)h ≈ Sh Xkh + +
0 (k+1)h
h
Ss ds F (Xkh ) + O(k+1)h − Sh Okh
S(k+1)h−s F (Xkh ) (Os − Okh ) ds
kh
≈ Sh Xkh + h F (Xkh ) + + O(k+1)h − Sh Okh
(k+1)h
F (Xkh ) (Os − Okh ) ds
kh
1 (k+1)h (Os − Okh ) ds = Sh Xkh + h F (Xkh ) + h F (Xkh ) h kh + O(k+1)h − Sh Okh for k = 0,1, . . . , M − 1 for a fixed M ∈ N. The expression in the second to last line is the classical Taylor approximation
1 F Xkh + h
(k+1)h
(Os − Okh ) ds
kh
1 (k+1)h (Os − Okh ) ds . ≈ F (Xkh ) + F (Xkh ) h kh
Using this in the approximation above gives
X(k+1)h ≈ Sh Xkh + h F Xkh +
1 h
(k+1)h
(Os − Okh ) ds
kh
+ O(k+1)h − Sh Okh ,
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7.7. Numerical schemes from Taylor expansions and hence the one-step numerical scheme N,M = ShN YkN,M + h FN YkN,M + Yk+1
+
N N O(k+1)h − ShN Okh
1 h
125
(k+1)h
kh
N OsN − Okh ds (7.69)
for k = 0, 1, . . . , M − 1 and fixed N , M ∈ N, which is also well-defined. This numerical scheme was originally introduced and analyzed by Jentzen [63, 65]. Since it is based on a higher order Taylor approximation it has a higher order convergence rate than the usual numerical schemes that have been used for SPDEs. Moreover, like the exponential Euler scheme, it is very easy to simulate.
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Chapter 8
Taylor Approximations for SPDEs with Multiplicative Noise
The key idea in deriving a higher order Taylor expansion of the solution of an SPDE with nonadditive noise is essentially the same as for an SPDE with additive noise in Chapter 7. Simply said, one uses classical Taylor expansions of the drift and the diffusion coefficients of the SPDE in the mild form and then recursively inserts lower order Taylor expansions of the solution of this SPDE to obtain a closed form with remainder. Iterating this procedure yields Taylor expansions of the solution of an SPDE of arbitrarily high orders. These Taylor expansions contain multiple stochastic integrals involving the infinite dimensional stochastic process and the semigroup, which provide more information about the SPDE solution and the noise and, hence, allow an approximation of higher order to be obtained. This method for deriving Taylor expansions depends strongly on the semigroup approach, i.e., interpreting the SPDE as a mild integral equation. The multiplicative noise case considered here is technically more demanding than the additive noise case in Chapter 7 and is presented for analytic semigroups with a diagonal generator rather than for general ones as well as with the noise restricted to Wiener processes. The convergence obtained is strong convergence, not pathwise convergence. In addition, a combinatorial notation involving stochastic trees and woods is introduced to allow a compact representation of Taylor approximations and numerical schemes of arbitrary order. This chapter is a slightly modified version of the article [65]. Further results on Taylor expansions for SPDEs can be found in Bayer & Teichmann [5], Buckdahn, Bulla & Ma [11], Buckdahn & Ma [10], and Conus [21].
8.1
Heuristic derivation of Taylor expansions
The underlying idea for deriving Taylor approximations for SPDEs will be sketched briefly again here, so this chapter can be read independently of Chapter 7. For 127
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128
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise
simplicity the remainder terms are omitted and restrictive assumptions on the coefficients are used, which will be relaxed later. Let U , H be two separable real Hilbert spaces and let A : D(A) ⊂ H → H be a linear operator which generates an analytic semigroup on H , i.e., eAt : H → H for t ≥ 0. In addition, let F : H → H and B : H → H S(U , H ) be two infinitely often Fréchet differentiable mappings with globally bounded derivatives, where H S(U , H ) is the real Hilbert space of Hilbert–Schmidt operators from U to H . Moreover, let (, F , P) be a probability space with a normal filtration Ft and let Wt be a cylindrical I -Wiener process on U with respect to Ft . Finally, let T ∈ (0, ∞) be fixed. Consider the SPDE dXt = [AXt + F (Xt )] dt + B(Xt ) dWt for t ∈ [0, T ], which is understood in the mild form Xt = eAt X0 +
t
eA(t−s) F (Xs ) ds +
0
t
eA(t−s) B(Xs ) dWs ,
P-a.s.,
(8.1)
0
for every t ∈ [0, T ]. Here and below assume that all integrals are well-defined in an appropriate sense and that X : [0, T ] × → H is a predictable stochastic process, which satisfies (8.1). Let t0 ∈ [0, T ). Then the solution process X also satisfies Xt = eA(t−t0 ) Xt0 +
t
eA(t−s) F (Xs ) ds +
t0
t
eA(t−s) B(Xs ) dWs ,
P-a.s., (8.2)
t0
for every t ∈ [t0 , T ]. The aim is to derive Taylor approximations of X about the pivot time t0 ∈ [0, T ). A first simple Taylor approximation of X can be obtained by omitting the second term in (8.2) and by using the classical Taylor approximation B(Xs ) ≈ B(Xt0 ) for s ∈ [t0 , t], which yields t A(t−t0 ) Xt ≈ e Xt0 + eA(t−s) B(Xt0 ) dWs (8.3) t0
for t ∈ [t0 , T ]. The Taylor approximation (8.3) is obviously an approximation of low order. To derive a Taylor approximation of higher order consider the classical Taylor approximations (8.4) F (Xs ) ≈ F (Xt0 ) and
B(Xs ) ≈ B(Xt0 ) + B (Xt0 ) Xs − Xt0
(8.5)
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8.1. Heuristic derivation of Taylor expansions
129
for s ∈ [t0 , t], where B denotes the Fréchet derivative of B. Inserting (8.4) and (8.5) into (8.2) yields t A(t−t0 ) Xt ≈ e Xt0 + eA(t−s) F (Xt0 ) ds t0
+
t t0
eA(t−s) B(Xt0 ) + B (Xt0 ) Xs − Xt0 dWs
and therefore Xt ≈ e
A(t−t0 )
+
Xt0 +
t
t0
t
e
A(t−s)
t0
F (Xt0 ) ds +
t
t0
eA(t−s) B(Xt0 ) dWs
eA(t−s) B (Xt0 ) Xs − Xt0 dWs
(8.6)
for t ∈ [t0 , t]. The right-hand side of (8.6) is not an appropriate Taylor approximation since the integral t eA(t−s) B (Xt0 ) Xs − Xt0 dWs , t ∈ [t0 , T ], t0
contains the unknown solution Xs for s ∈ (t0 , t], which is what is to be approximated. The trick is to replace Xs for s ∈ [t0 , t] in the integral in (8.6) by the lower order Taylor approximation (8.3). This yields t t A(t−s) A(t−t0 ) Xt ≈ e e F (Xt0 ) ds + eA(t−s) B(Xt0 ) dWs Xt0 + t0
+
t
t0
t0
s eA(t−s) B (Xt0 ) eA(s−t0 ) Xt0 + eA(s−u) B(Xt0 ) dWu − Xt0 dWs t0
and, finally, after rearranging t t Xt ≈ eA(t−t0 ) Xt0 + eA(t−s) F (Xt0 ) ds + eA(t−s) B(Xt0 ) dWs t0
+
t
t0
+
t
t0
eA(t−s) B (Xt0 ) eA(t−s) B (Xt0 )
(8.7)
t0
eA(s−t0 ) − I Xt0 dWs
s
t0
eA(s−u) B(Xt0 ) dWu dWs
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130
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise
for t ∈ [t0 , T ]. The right-hand side of (8.7) now contains the solution X at time t0 only and is therefore an appropriate Taylor approximation. It will be seen later that this Taylor approximation is indeed of higher order in the mean-square sense. The method for deriving the higher order Taylor approximation (8.7) has two steps: 1. First, use classical Taylor approximations for the coefficients F and B in the mild integral equation of the SPDE; see (8.6). 2. Second, insert recursively a lower order Taylor approximation; see (8.3). The essential ingredients of such Taylor approximations are iterated derivatives of the coefficients and iterated stochastic integrals involving the semigroup generated by the dominant linear operator of the SPDE, such as t s A(t−s) eA(s−u) B(Xt0 ) dWu dWs e B (Xt0 ) t0
t0
in the Taylor approximation (8.7). More Taylor approximations and Taylor expansions of the solution of an SPDE can be derived in a similar way. A mechanism using appropriate integral operators and rooted trees to describe these Taylor expansions and Taylor approximations precisely will be presented in this chapter. In addition, the order of the Taylor approximations will be estimated, and it will be shown how Taylor expansions of arbitrarily high orders can be achieved.
8.2
Setting and assumptions
Let (H , ·, ·H , · H ) and (U , ·, ·U , · U ) be two separable real Hilbert spaces and let (D, · D ) be a separable real Banach space with H ⊂ D continuously. Recall that · L(n) (H ,H ) denotes the operator norm in the real Banach space (n) L (H , H ) of all n-multilinear bounded operators from H n := H × · · · × H to H for every n ∈ N and write · L(H ) instead of · L(1) (H ,H ) when n = 1. In addition, recall that (L(U , D), · L(U ,D) ) denotes the real Banach space of all bounded linear operators from U to D and that (H S(U , H ), ·, ·H S(U ,H ) , · H S(U ,H ) ) denotes the real Hilbert space of all Hilbert–Schmidt operators from U to H . Fix T > 0 and a probability space (, F , P) with a normal filtration Ft and let Wt be a cylindrical I -Wiener process on U with respect to Ft . Note that Wt does not take values in U but instead in a larger space into which U is embedded in the Hilbert–Schmidt sense (see Section 5.2.4). The following are the first two of four assumptions used in this chapter. Assumption 15 (linear operator A). Let I be a finite or countable set. In addition, let (λi )i∈I be a family of real numbers with inf i∈I λi > −∞ and let (ei )i∈I be an
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8.2. Setting and assumptions
131
orthonormal basis of H . The linear operator A : D(A) → H is given by −λi ei , vH ei Av = i∈I
for all v ∈ D(A) with D(A) = v ∈ H :
2 2 i∈I |λi | |ei , vH |
< ∞ ⊂ H.
The linear operator A : D(A) → H is closed and densely defined and generates an analytic semigroup eAt : H → H , t ≥ 0. See [110]. Assumption 16 (drift F ). Let + F : H+→ H be an infinitely often Fréchet differentiable mapping with supv∈H +F (n) (v)+L(n) (H ,H ) < ∞ for all n ∈ N. To formulate the other assumptions, appropriate intermediate spaces between H and the domain D(A) of the linear operator A : D(A) → H are required. To this end fix a real number κ ∈ [0, ∞) with κ > − inf i∈I λi (such a κ exists due to Assumption 15). Then the linear operator κ − A defined by (κ + λi ) ei , vH ei κ − A : D(κ − A) ⊂ H → H , v → κv − Av = i∈I
for every v ∈ D(κ − A) = D(A) has strictly positive real eigenvalues and, moreover, inf i∈I (κ + λi ) > 0. This allows linear operators (κ − A)r : D((κ − A)r ) → H to be defined by (κ − A)r v := (κ + λi )r ei , vH ei i∈I
for every v ∈ D((κ − A)r ) and every r ∈ [0, ∞), where 5 6 (κ + λi )2r |ei , vH |2 < ∞ ⊂ H . D((κ − A)r ) := v ∈ H :
(8.8)
i∈I
The space D((κ − A)r ) equipped with the norm + + v D((κ−A)r ) = +(κ − A)r v +H ,
v ∈ D((κ − A)r ),
is a real Banach space and, in fact, even a real Hilbert space for every r ∈ [0, ∞). By definition, D((κ − A)r ) ⊂ H continuously for every r ∈ [0, ∞). In particular, D((κ − A)1 ) = D(A) and D((κ − A)0 ) = H . In addition, define the space D((κ − A)−r ) := D((κ − A)r ) as the topological dual space of D((κ − A)r ) for every r ∈ (0, ∞), where the norm in D((κ − A)−r ) is the operator norm given by µ D((κ−A)−r ) =
sup v∈D((κ−A)r ) |(κ−A)r v |H ≤1
|µ(v)|
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132
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise
for every µ ∈ D((κ − A)−r ) and every r ∈ (0, ∞). By definition, D((κ − A)−r1 ) ⊂ D((κ − A)−r2 ) continuously for every r1 , r2 ∈ (0, ∞) with r1 ≤ r2 . By the embedding H → D((κ − A)−r ),
v → v, ·H ,
v∈H
for every r ∈ (0, ∞) it follows that D((κ − A)r2 ) ⊂ D((κ − A)r1 ) continuously for every r1 , r2 ∈ R with r1 ≤ r2 . Although the particular norms in these spaces depend on κ, which is held fixed, they essentially depend on A only. See Example 37.1 in [110] for more details about this scaling of spaces. Finally, the definition of the semigroup eAt : H → H , t ≥ 0, can be extended to eAt : D((κ − A)−r ) → D((κ − A)−r ) by (eAt µ)(v) := µ(eAt v) for every v ∈ D((κ − A)r ), µ ∈ D((κ − A)−r ), r ∈ (0, ∞), and t ∈ [0, ∞). In fact, eAt (D((κ − A)−r )) ⊂ H for every r, t ∈ (0, ∞). Assumption 17 (diffusion B). The embedding D ⊂ D((κ − A)−r ) is continuous for a given r ∈ [0, ∞), the operator B : H → L(U , D) is infinitely often Fréchet differentiable, and the operators eAt B (n) (v)(w1 , . . . , wn ) and (κ − A)γ eAt B(v) are Hilbert–Schmidt operators in H S(U , H ). Moreover, there exist a family of real positive numbers (Ln )n∈N and real numbers θ, ρ ∈ (0, 12 ], and γ ∈ (0, 1) such that + + 1 + At (n) + ≤ Ln · (1 + v H ) · w1 H · . . . · wn H · t θ− 2 , +e B (v)(w1 , . . . , wn )+ H S(U ,H )
+ + + + At +e (B(v) − B(w))+
≤ L0 · v − w H · t ρ− 2 ,
+ + + + +(κ − A)γ eAt B(v)+
≤ L0 · (1 + v H ) · t ρ− 2
H S(U ,H )
1
1
H S(U ,H )
for all v, w, w1 , . . . , wn ∈ H , n ∈ {0, 1, 2, . . .}, and all t ∈ (0, T ]. γ γ Assumption 18 (initial value). +pD((κ − A) ) be an F0 / B (D((κ − A) ))+ Let x0 :γ → measurable mapping with E +(κ − A) x0 +H < ∞ for some p ∈ [1, ∞), where γ ∈ (0, 1) is given in Assumption 17.
Now consider the SPDE # $ dXt = AXt + F (Xt ) dt + B(Xt ) dWt ,
X0 = x 0 ,
(8.9)
for t ∈ [0, T ]. Under Assumptions 15–18 the SPDE (8.9) has a unique mild solution.
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8.2. Setting and assumptions
133
Proposition 7. Let Assumptions 15–18 be satisfied and let γ ∈ (0, 1) be given by Assumption 17. Then there is an up-to-modifications unique +p stochastic + predictable process X : [0, T ] × → D((κ − A)γ ) with sup0≤t≤T E +(κ − A)γ Xt +H < ∞ for p ∈ [1, ∞) and with t t (8.10) eA(t−s) F (Xs ) ds + eA(t−s) B(Xs ) dWs = 1 P Xt = eAt x0 + 0
0
for all t ∈ [0, T ]. Moreover, X is the unique mild solution of the SPDE (8.9) in the sense of equation (8.10). The proof of Proposition 7 can be found in [65]. The integrals in (8.10) are well-defined predictable stochastic processes under the assumptions above. The solution process X : [0, T ] × → D((κ − A)γ ) is up-to-modifications, in fact, unique in a much larger class of stochastic processes (see Theorem 7.4 in [24]). Remark 16. Taylor expansions need smooth coefficients, but Assumptions 16 and 17 are somewhat restrictive. However, on the one hand it would suffice to postulate that only the first k-derivatives of F and B exist and are locally bounded in some sense in appropriate spaces with k ∈ N sufficiently large. Assumptions 16 and 17 are used here just for simplicity. On the other hand, the stochastic heat equation on the one-dimensional domain (0, 1) with multiplicative space–time white noise is a nontrivial example satisfying the assumptions above (see Section 8.2.1), where nontrivial means that there is no explicit representation of the solution of the stochastic heat equation with multiplicative noise and, hence, numerical approximations of the solution process are needed.
8.2.1
Stochastic heat equation
The stochastic heat equation with multiplicative space–time white noise on a onedimensional spatial domain provides a simple example of an SPDE which satisfies Assumptions 15–18. It is often used to test numerical schemes for SPDEs with nonadditive noise (see, e.g., Davie & Gaines [27] and Müller-Gronbach & Ritter [99] and the references therein). Let H = U = L2 ((0, 1), R) be the real Hilbert space of equivalence classes of B ((0, 1)) /B (R)-measurable and square integrable functions from (0, 1) to R. The scalar product and the norm in H and U are given by v, wH = v, wU =
1
v(x) w(x) dx, 0
v H = v U =
1
1 2
|v(x)|2 dx
0
for every v, w ∈ H = U . In addition, let D = L1 ((0, 1), R) be the real Banach space of all equivalence classes of B ((0, 1)) /B (R)-measurable and absolute integrable
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134
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise
functions from (0, 1) to R equipped with the norm 1 v D = |v(x)| dx 0
for every v ∈ D. (Here and below the distinction between integrable functions and their corresponding equivalence classes in D and H = U will not be emphasized.) ∂2 In addition, let A = ϑ ∂x 2 be the Laplacian times a constant ϑ > 0 with Dirichlet boundary conditions on the one-dimensional domain (0, 1), so I = N and √ λi = ϑπ 2 i 2 , ei (x) = 2 sin(iπ x), for all x ∈ (0, 1) and all i ∈ N. The operator A thus reduces to Av =
∞
−ϑπ 2 i 2 ei , vH ei ,
v ∈ D(A),
(8.11)
i=1
with D(A) = w ∈ H : i∈I i 4 |ei , wH |2 < ∞ . Thus Assumption 15 is fulfilled. In addition, let the drift F : H → H be given by F (v) ≡ 0 for all v ∈ H . Hence Assumption 16 is also fulfilled. Furthermore, choose κ = 0, let σ ∈ R be a real number, and let B be given by B : H → L(H , D),
(B(v)(w)) (x) = σ · v(x) · w(x)
(8.12)
for every x ∈ (0, 1) and v, w ∈ H . Then B is indeed well-defined since 1 B(v)(w) D = |σ | |v(x) · w(x)| dx 0
≤ |σ | 0
1
1 2
|v(x)| dx 2
1
1 2
|w(x)| dx 2
0
= |σ | · v H · w H
holds for all v, w ∈ H by the Cauchy–Schwarz inequality. Thus B(v) is a bounded linear operator from H to D with B(v) L(H ,D) ≤ |σ | v H for all v ∈ H . Moreover, B is a bounded linear operator from H to L(H , D). In particular, B is infinitely often Fréchet differentiable with the derivatives B (v) = B and B (i) (v) = 0 for all i = 2, 3, . . . and v ∈ H . The following lemma (see Lemma 27 in [62]) provides an embedding for D into D((−A)−r ) for a particular r ∈ [0, ∞), namely, r = 12 , which is needed for Assumption 17. Lemma 8.1. Let H = L2 ((0, 1), R), D = L1 ((0, 1), R) and let the linear operator A : D(A) ⊂ H → H be given by (8.11). Then the linear operator χ given by 1 1 (χ (µ)) (v) = µ(x) v(x) dx χ : D → D (−A)− 2 , 0
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8.3. Taylor expansions for SPDEs
135
1
for every v ∈ D((−A) 2 ) and every µ ∈ D is bounded. In particular, the embedding 1 D ⊂ D((−A)− 2 ) is continuous. The mapping χ in Lemma 8.1 is the usual embedding of functions into distributions. Hence, Lemma 8.1 yields that L1 ((0, 1) , R) is continuously embedded into 1 D((−A)− 2 ). Finally, Assumption 17 is satisfied with the parameters γ = 14 − ε and θ = 14 for every arbitrarily small ε ∈ (0, 14 ) due to the following lemma (see Lemma 28 in [62]). Lemma 8.2. Let A : D(A) ⊂ H → H and B : H → L(H , D) be given by (8.11) and (8.12). Then (−A)γ eAt B(v) is a Hilbert–Schmidt operator from H to H with + + 1 4 |σ | (T + 1) + + v H t − 4 +γ ≤ +(−A)γ eAt B(v)+ H S(H ) min(ϑ, 1) for every v ∈ H , t ∈ (0, T ], and γ ∈ [0, 14 ). 1 For the initial value the choice x0 = ∞ i=1 i ei will be used, so Assumption 18 is also fulfilled. Finally, the stochastic heat equation, precisely described above, can in a short form also be written as ∂2 (8.13) dXt = ϑ 2 Xt dt + (σ Xt ) dWt ∂x with Xt (0) = Xt (1) = 0,
X 0 = x0 ,
for t ∈ [0, T ] on the spatial domain (0, 1), where Wt is a cylindrical I -Wiener process on L2 ((0, 1) , R).
8.3 Taylor expansions for SPDEs The basic notation for the systematic derivation of Taylor expansions of the solution of the SPDE (8.9) is presented in this section. Henceforth, fix t0 ∈ [0, T ) and let denote the real vector space of all equivalence classes of predictable stochastic processes Y : [t0 , T ] × → H with sup Yt Lp (;H ) < ∞
t0 ≤t≤T
(8.14)
for p ∈ [1, ∞), where two processes are in an equivalence class if and only if they are modifications of each other. As usual, no distinction will be made between predictable stochastic processes satisfying (8.14) and their equivalence class in .
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136
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise Write Xt := Xt − Xt0 ,
t := t − t0 ,
for t ∈ [t0 , T ] ⊂ [0, T ]. Here and below X is always the unique (up-to-modifications) solution process of the SPDE (8.9), which exists by Proposition 7. Moreover, X ∈ by Proposition 7.
8.3.1
Integral operators
Let j ∈ {0, 1, 1∗ , 2, 2∗ }, where the indices 0, 1, and 2 label expressions containing only a constant value of the SPDE solution, while 1∗ and 2∗ label certain integrals with time-dependent values of the solution in the integrand. Specifically, define the stochastic processes Ij0 ∈ by At − I Xt0 , e t A(t−s) F (Xt0 ) ds, t0 e t Ij0 (t) := t0 eA(t−s) F (Xs ) ds, t A(t−s) B(Xt0 ) dWs , t e 0 t A(t−s) B(Xs ) dWs , t0 e
j = 0, j = 1, j = 1∗ ,
P-a.s.,
j = 2, j = 2∗ ,
for each t ∈ [t0 , T ]. Let i ∈ N and j ∈ {1, 1∗ , 2, 2∗ }. Then, for all t ∈ [t0 , T ] and all g1 , . . . , gi ∈ , · · × → by define the i-multilinear symmetric mappings Iji : i := × · i-times
Iji [g1 , . . . , gi ](t) :=
1 i!
t
t0
eA(t−s) F (i) (Xt0 ) (g1 (s), . . . , gi (s) ) ds,
P-a.s.,
when j = 1, by Iji [g1 , . . . , gi ](t)
t
:=
e
A(t−s)
t0
0
1
F (i) (Xt0 + rXs ) (g1 (s), . . . , gi (s))
when j = 1∗ , by Iji [g1 , . . . , gi ](t) :=
1 i!
t
t0
(1 − r)i−1 dr ds, (i − 1)!
eA(t−s) B (i) (Xt0 ) (g1 (s), . . . , gi (s)) dWs ,
P-a.s.,
P-a.s.,
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8.3. Taylor expansions for SPDEs
137
when j = 2, and by Iji [g1 , . . . , gi ](t)
t
:= t0
eA(t−s) 0
1
B (i) (Xt0 + rXs ) (g1 (s), . . . , gi (s))
(1 − r)i−1 dr dWs , (i − 1)!
P-a.s.,
when j = 2∗ . It follows from Assumptions 15–18 that the stochastic processes Ij0 ∈ , j ∈ {0, 1, 1∗ , 2, 2∗ }, and the mappings Iji : i → , i ∈ N, j ∈ {1, 1∗ , 2, 2∗ }, are welldefined. The proof that the processes Iji [g1 , . . . , gi ] for j ∈ {1, 1∗ , 2, 2∗ }, g1 , . . . , gi ∈ , and i ∈ N are predictable (and in ) is a little tricky: one shows first that these processes are mean-square continuous by using Assumptions 16 and 17, and this yields that they have a predictable version (see Proposition 3.6(ii) in [24]). The increment Xt of the solution X of the SPDE (8.9) obviously satisfies t t Xt = eAt − I Xt0 + eA(t−s) F (Xs ) ds + eA(t−s) B(Xs ) dWs , t0
t0
P-a.s., (8.15)
or, in terms of the above integral operators, Xt = I00 (t) + I10∗ (t) + I20∗ (t),
P-a.s.,
for all t ∈ [t0 , T ], which can be written symbolically in the space as X = I00 + I10∗ + I20∗ .
(8.16)
The stochastic processes I00 , I1i [g1 , . . . , gi ], and I2i [g1 , . . . , gi ] for g1 , . . . , gi ∈ and i ∈ N depend on the solution only at time t = t0 and are therefore useful approximations for the solution Xt , t ∈ [t0 , T ]. On the other hand, the stochastic processes I1i∗ [g1 , . . . , gi ] and I2i∗ [g1 , . . . , gi ] for g1 , . . . , gi ∈ and i ∈ N depend on the whole solution process Xt for t ∈ [t0 , T ], which is what is to be approximated. In this sense, as in Chapter 7, the subscript ∗ on the 1 and 2 denotes remainder terms that can be further expanded with the help of the following lemma. Lemma 8.3. Suppose that Assumptions 15–18 hold. Then I00 , g1 , . . . , gi I1i∗ [g1 , . . . , gi ] = I1i [g1 , . . . , gi ] + I1i+1 ∗ + I1i+1 I10∗ , g1 , . . . , gi + I1i+1 I20∗ , g1 , . . . , gi ∗ ∗
(8.17)
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138 and
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise I00 , g1 , . . . , gi I2i∗ [g1 , . . . , gi ] = I2i [g1 , . . . , gi ] + I2i+1 ∗ + I2i+1 I10∗ , g1 , . . . , gi + I2i+1 I20∗ , g1 , . . . , gi ∗ ∗
(8.18)
for every g1 , . . . , gi ∈ and every i ∈ N. These also hold for i = 0, in which case the g1 , . . . , gi are omitted.
8.3.2
Derivation of simple Taylor expansions
A further expansion of (8.16) can be obtained by applying formula (8.17) to the stochastic process I10∗ and formula (8.18) to the stochastic process I20∗ , i.e., I10∗ = I10 + I11∗ [I00 ] + I11∗ [I10∗ ] + I11∗ [I20∗ ], I20∗ = I20 + I21∗ [I00 ] + I21∗ [I10∗ ] + I21∗ [I20∗ ], and inserting these into (8.16) to obtain X = I00 + I10 + I11∗ [I00 ] + I11∗ [I10∗ ] + I11∗ [I20∗ ] + I20 + I21∗ [I00 ] + I21∗ [I10∗ ] + I21∗ [I20∗ ] . This can also be written as X = I00 + I10 + I20 + R
(8.19)
with the remainder R ∈ given by R = I11∗ [I00 ] + I11∗ [I10∗ ] + I11∗ [I20∗ ] + I21∗ [I00 ] + I21∗ [I10∗ ] + I21∗ [I20∗ ]. If the double integral terms I11∗ [Ij0 ], I21∗ [Ij0 ] for j ∈ {0, 1∗ , 2∗ } in R are sufficiently small (which they are, as will be shown below), then the expansion (8.19) gives the approximation (8.20) X ≈ I00 + I10 + I20 . By the definition of the stochastic processes I00 , I10 , and I20 this gives t t Xt ≈ eAt − I Xt0 + eA(t−s) F (Xt0 ) ds + eA(t−s) B(Xt0 ) dWs t0
for t ∈ [t0 , T ]. Hence Xt ≈ e
At
Xt0 +
t
e 0
t0
As
ds F (Xt0 ) +
t
t0
eA(t−s) B(Xt0 ) dWs
(8.21)
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8.3. Taylor expansions for SPDEs
139
for t ∈ [t0 , T ] is an approximation for the solution of SPDE (8.9). Since the righthand side of (8.21) depends on the solution only at time t = t0 , it is the first nontrivial Taylor approximation of the solution of the SPDE (8.9). p
1
Define Y Lp (;H ) := (E Y H ) p for an F /B(H )-measurable mapping Y : → H and a real number p ∈ [1, ∞). It will be shown in Theorem 8.4 that the remainder term R ∈ of the Taylor approximation (8.21) can be estimated by R(t) Lp (;H ) ≤ Cp (t)θ+min(γ ,θ)
(8.22)
for every t ∈ [t0 , T ] with constant Cp ≥ 0. Henceforth, the notation Yt = Zt + O (t)r will be used for two stochastic processes Y , Z ∈ and a real number r > 0 if Yt − Zt Lp (;H ) < ∞. sup (t)r t∈(t0 ,T ] Then it follows from (8.21) and (8.22) that t t Xt = eAt Xt0 + eAs ds F (Xt0 ) + eA(t−s) B(Xt0 ) dWs 0
t0
+ O (t)θ+min(γ ,θ) .
(8.23)
The approximation (8.23) thus has order θ + min(γ , θ ) in the above strong sense. It will be called the exponential Euler approximation, since the corresponding numerical one-step scheme is the exponential Euler scheme introduced in Chapter 6. See also Section 8.6.
8.3.3
Higher order Taylor expansions
Further expansions of the remainder terms in a Taylor expansion give a Taylor expansion of higher order. To illustrate this the terms I21∗ [I00 ] and I21∗ [I20∗ ] in the remainder R in (8.19) will be expanded. The expansion formulas (8.17) and (8.18) yield I21∗ [I00 ] = I21 [I00 ] + I22∗ [I00 , I00 ] + I22∗ [I10∗ , I00 ] + I22∗ [I20∗ , I00 ] and I21∗ [I20∗ ] = I21 [I20∗ ] + I22∗ [I00 , I20∗ ] + I22∗ [I10∗ , I20∗ ] + I22∗ [I20∗ , I20∗ ] = I21 [I20 ] + I21 [I21∗ [I00 ]] + I21 [I21∗ [I10∗ ]] + I21 [I21∗ [I20∗ ]] + I22∗ [I00 , I20∗ ] + I22∗ [I10∗ , I20∗ ] + I22∗ [I20∗ , I20∗ ], which are inserted into (8.19) to give X = I00 + I10 + I20 + I21 [I00 ] + I21 [I20 ] + R,
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140
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise
where the remainder term R ∈ here is given by R = I11∗ [I00 ] + I11∗ [I10∗ ] + I11∗ [I20∗ ] + I21∗ [I10∗ ] + I22∗ [I00 , I00 ] + I22∗ [I10∗ , I00 ] + I22∗ [I20∗ , I00 ] + I21 [I21∗ [I00 ]] + I21 [I21∗ [I10∗ ]] + I21 [I21∗ [I20∗ ]] + I22∗ [I00 , I20∗ ] + I22∗ [I10∗ , I20∗ ] + I22∗ [I20∗ , I20∗ ]. By Theorem 8.4 this remainder term can be estimated by R(t) = O (t)θ+2 min(γ ,θ ) . Thus Xt = I00 (t) + I10 (t) + I20 (t) + I21 [I00 ](t) + I21 [I20 ](t) + O (t)θ+2 min(γ ,θ) , which can also be written as t t eAs ds F (Xt0 ) + eA(t−s) B(Xt0 ) dWs Xt = eAt Xt0 + 0
+
t
t0
+
eA(t−s) B (Xt0 ) eAs − I Xt0 dWs
t
e t0
t0
A(t−s)
B (Xt0 )
s
t0
eA(s−r) B(Xt0 ) dWr dWs + O (t)θ+2 min(γ ,θ ) .
This approximation is of order θ + 2 min(γ , θ ) and was sketched in Section 8.1. The double integral dWr dWs indicates that this Taylor approximation is in a way an infinite dimensional analogue of the Milstein approximation for SODEs (see Section 8.5.3).
8.4
Stochastic trees and woods
The construction of further Taylor approximations of arbitrarily high order requires a succinct way to determine and label systematically the integral operators that need to be used. This is provided by the combinatorial concepts of rooted trees and woods.
8.4.1
Stochastic trees and woods
Let N ∈ N be a natural number and let t : {2, . . . , N } → {1, . . . , N − 1} ,
t : {1, . . . , N } → 0, 1, 2, 1∗ , 2∗
be two mappingswith the property that t (j ) < j for all j ∈ {2, . . . , N }. Then the pair of mappings t = t , t is called an S-tree (stochastic tree) with l(t) := N nodes. Let ST denote the set of all stochastic trees.
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8.4. Stochastic trees and woods 3
141
5
6
4⊗
2
2⊗
3
7
4
5
1⊗
1
Figure 8.1. Two examples of stochastic trees [65]. Every S-tree can be represented as a graph whose nodes are given by the set nd(t) := {1, . . . , N } and whose arcs are described by the mapping t in the sense that there is an edge from j to t (j ) for every node j ∈ {2, . . . , N }. The mapping t is an additional labeling of the nodes with t (j ) ∈ {0, 1, 2, 1∗ , 2∗ } indicating the type of node j for each j ∈ nd(t). The left picture in Figure 8.1 corresponds to the tree t1 = (t1 , t1 ) with nd(t1 ) = {1, 2, 3, 4, 5} given by t1 (5) = 2,
t1 (4) = 1,
t1 (3) = 2,
t1 (2) = 1
and t1 (1) = 1,
t1 (2) = 1∗ ,
t1 (3) = 2,
t1 (4) = 0,
t1 (5) = 2∗ .
The root is always represented by the lowest node. The number on the left of a node in Figure 8.1 is the number of the node of the corresponding tree, while the type of the node in Figure 8.1 depends on the additional labeling of the nodes given by t2 . More precisely, a node j ∈ nd(t1 ) is represented by ⊗ if t1 (j ) = 0, by if t1 (j ) = 1, by if t1 (j ) = 2, by if t1 (j ) = 1∗ , and, finally, by if t1 (j ) = 2∗ . The right picture in Figure 8.1 corresponds to the tree t2 = (t2 , t2 ) with nd(t2 ) = {1, . . . , 7} given by
t2 (7) = 4, t2 (6) = 4, t2 (5) = 1, t2 (4) = 1, t2 (3) = 1, t2 (2) = 1 and t2 (1) = 0, t2 (2) = 0, t2 (3) = 2, t2 (4) = 1, t2 (5) = 1∗ , t2 (6) = 1, t2 (7) = 2∗ . The set of S-woods (stochastic woods) is now defined by SW :=
∞ ;
(ST)n .
n=1
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142
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise 1⊗
1
1
Figure 8.2. The stochastic wood w0 in SW [65]. Of course, the embedding ST ⊂ SW holds. A simple example of an S-wood required later is w0 = (t1 , t2 , t3 ) ∈ SW with t1 , t2 , and t3 in ST given by l(t1 ) = l(t2 ) = l(t3 ) = 1 and t1 (1) = 0, t2 (1) = 1∗ , t3 (1) = 2∗ . This is illustrated in Figure 8.2, where the left tree corresponds to t1 , the middle one to t2 , and the right one to t3 .
8.4.2
Construction of stochastic trees and woods
Certain stochastic woods in SW are used to represent Taylor expansions of the solution X of the SPDE (8.9). An operator on the set SW that enables the step-by-step construction of an appropriate stochastic woods will now be defined. Let w = (t1 , . . . , tn ) ∈ SW with n ∈ N be an S-wood with ti = (ti , ti ) ∈ ST for i ∈ {1, . . . , n}. In addition, let i ∈ {1, . . . , n} and j ∈ {1, . . . , l(ti )} be given and suppose that either ti (j ) = 1∗ or ti (j ) = 2∗ , in which case the pair (i, j ) will be called an active node of w. Denote the set of all active nodes of w by acn(w) ⊂ N2 . In figures of woods and trees, e.g., in Figure 8.1, active nodes are represented by a filled square for 1∗ and a simple square for 2∗ . Now introduce the trees tn+1 = (tn+1 , tn+1 ), tn+2 = (tn+2 , tn+2 ), and tn+3 = (tn+3 , tn+3 ) ∈ ST by nd(tn+m ) = {1, . . . , l(ti ), l(ti ) + 1} and
tn+m (k) = ti (k), k = 2, . . . , l(ti ), tn+m (l(ti ) + 1) = j ,
tn+m (k) = ti (k), k = 1, . . . , l(ti ), 0, m = 1, tn+m (l(ti ) + 1) = 1∗ , m = 2, 2∗ , m = 3,
for m ∈ {1, 2, 3}. Further, consider the S-tree ˜t = (˜t , ˜t ) ∈ ST given by ˜t = ti , but with ˜t : {1, . . . , l(ti )} → {0, 1, 2, 1∗ , 2∗ } given by ˜t (k) = ti (k) for k ∈ nd(ti )/{j } and by ˜t (j ) = 1 if ti (j ) = 1∗ and ˜t (j ) = 2 if ti (j ) = 2∗ . Then define E(i,j ) (w) = E(i,j ) ((t1 , . . . , tn )) := t1 , . . . , ti−1 , ˜t, ti+1 , . . . tn+3 ∈ SW
and consider the set of all woods that can be constructed iteratively by applying the E(i,j ) operations, i.e., define ∃ n ∈ N, i1 , . . ., in , j1 , . . . , jn ∈ N : ∀ k ∈ {1, . . . , n} ; SW’ := {w0 } w ∈ SW : (ik , jk ) ∈ acn E(ik−1 ,jk−1 ) . . . E(i1 ,j1 ) w0 , w = E(in ,jn ) . . . E(i1 ,j1 ) w0 for the stochastic wood w0 introduced in Figure 8.2.
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8.4. Stochastic trees and woods
1⊗
1
143
1
2⊗
2
2
1
1
1
Figure 8.3. The stochastic wood w1 in SW [65].
1⊗ 1
1
2⊗ 2
2
2⊗ 2
2
1
1
1
1
1
1
Figure 8.4. The stochastic wood w2 in SW [65]. The following examples using the initial stochastic wood w0 illustrate these definitions. First, the active nodes of w0 are acn(w0 ) = {(2, 1), (3, 1)}, since the first node in the second tree and the first node in the third tree are represented by squares. Hence, E(3,1) w0 is well-defined, and the resulting stochastic wood w1 = E(3,1) w0 , which contains six trees, is presented in Figure 8.3. Writing w1 = (t1 , . . . , t6 ), the left tree in Figure 8.3 corresponds to t1 , the second tree to t2 , and so on. Moreover, the active nodes of w1 are acn(w1 ) = {(2, 1), (4, 1), (5, 1), (5, 2), (6, 1), (6, 2)} ,
(8.24)
so w2 = E(2,1) w1 is also well-defined. It is shown in Figure 8.4. Figure 8.5 shows the stochastic wood w3 = E(4,1) w2 , which is well-defined since acn(w2 ) = {(4, 1), (5, 1), (5, 2), (6, 1), (6, 2), (7, 1), (8, 1), (8, 2), (9, 1), (9, 2)} . (8.25) For the S-wood w3 acn(w3 ) =
(5, 1), (5, 2), (6, 1), (6, 2), (7, 1), (8, 1), (8, 2), (9, 1), (9, 2), (10, 1), (11, 1), (11, 3), (12, 1), (12, 3)
Since (6, 1) ∈ acn(w3 ), the stochastic wood w4 = E(6,1) w3 , given in also well-defined. Its active nodes are (5, 1), (5, 2), (6, 2), (7, 1), (8, 1), (8, 2), (9, 1), (9, 2), (10, 1), (11, 1), (11, 3), (12, 1), (12, 3), (13, 1), acn(w4 ) = (13, 2), (14, 1), (14, 2), (14, 3), (15, 1), (15, 2), (15, 3)
.
(8.26)
Figure 8.6, is
.
(8.27)
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144
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise
1⊗ 1
1 2⊗
2⊗
2
2
2⊗
2
2
1
1
1
1
1
1
3⊗ 2⊗
1
2⊗
3
1
3
1
Figure 8.5. The stochastic wood w3 in SW [65].
2⊗ 2
2
1
1
1
1⊗ 1 2⊗
1
2⊗ 2
2
1
1
3⊗ 2⊗
1 2
3
1
1
2⊗
3
1 2
3
1
3⊗
2
1
3
1
Figure 8.6. The stochastic wood w4 in SW [65]. Finally, the stochastic wood w5 = E(6,2) w4 with (5, 1), (5, 2), (7, 1), (8, 1), (8, 2), (9, 1), (9, 2), (10, 1), (11, 1), acn(w5 ) = (11, 3), (12, 1), (12, 3), (13, 1), (13, 2), (14, 1), (14, 2), (14, 3), (8.28) (15, 1), (15, 2), (15, 3), (16, 2), (17, 2), (17, 3), (18, 2), (18, 3)
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8.4. Stochastic trees and woods
1⊗ 1 1 2⊗ 3⊗ 2⊗
1
145
2⊗ 2
2
2⊗ 2
2
1
1
1
1
1 2⊗
3
1
3
3⊗ 2
2
1
2
3
1
1
1 3⊗ 3
3
2
2
2
1
1
1
3
1
Figure 8.7. The stochastic wood w5 in SW [65].
is show in Figure 8.7. By definition the S-woods w0 , w1 , . . . , w5 are in SW’ but the stochastic wood given by Figure 8.1 is not in SW’.
8.4.3
Subtrees
Let t = (t , t ) ∈ SW be a given S-tree with cardinality |nd(t)| ≥ 2. For two nodes k, l ∈ nd(t) with k ≤ l the node l is said to be a grandchild of k if there exists a sequence k1 = k < k2 < · · · < kn = l of nodes with n ∈ N such that t (kv+1 ) = kv for every v ∈ {1, . . . , n − 1}. Suppose now that j1 < · · · < jn with n ∈ N are the nodes of t such that t (ji ) = 1 for i ∈ {1, . . . , n}. For a given i ∈ {1, . . . , n} suppose that ji,1 , . . . , ji,li ∈ nd(t) with li ∈ N and ji = ji,1 < ji,2 < · · · < ji,li ≤ l(t) are the grandchildren of ji ∈ nd(t). Then define the trees ti = ti , ti ∈ ST with l(ti ) := li by ji,ti (k) = t (ji,k ), ti (k) = t (ji,k ) for k ∈ {2, . . . , li } and by ti (1) = t (ji ) for every i ∈ {1, . . . , n}. The trees t1 , . . . , tn ∈ ST
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146
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise 2
1⊗
1
3
1
1
Figure 8.8. Subtrees of the right tree in Figure 8.1 [65]. defined in this way are called subtrees of t. For example, the subtrees of the right tree in Figure 8.1 are given in Figure 8.8. Moreover, a node is called a leaf if it has no grandchildren except itself. The following subset of stochastic trees will be used, too: ST’ := t = t , t ∈ ST : t (j ) = 0 ⇒ j is a leaf ∀ j ∈ {1, 2, . . . , l(t)} . Of course, if w = (t1 , . . . , tn ) ∈ SW’ is a stochastic wood in SW’, then t1 , . . . , tn ∈ ST’.
8.4.4
Order of stochastic trees and woods
The characterization of Taylor approximations in terms of stochastic trees and woods requires concepts of the order of a stochastic tree in ST’ and the order of a stochastic wood in SW’. The order of an S-tree t ∈ ST’ is given by the function ordt : ST’ → [0, ∞) defined by ordt(t) := j ∈ nd(t) : t (j ) = 1 or t (j ) = 1∗ + γ j ∈ nd(t) : t (j ) = 0 + θ j ∈ nd(t) : t (j ) = 2 or t (j ) = 2∗ for every S-tree t = (t , t ) ∈ ST’. This definition of ordt is motivated by Lemma 8.7 below. For example, the order of the left tree in Figure 8.1 is 2 + γ + 2θ since the left tree has one node of type 0, two nodes of type 1 and type 1∗ , respectively, and also two nodes of type 2 and type 2∗ , respectively. The order of the right tree in Figure 8.1 is not defined, since the right tree in Figure 8.1 is not in ST’. A tree t = (t , t ) ∈ ST’ is said to be active if there is a j ∈ nd(t) such that t (j ) = 1∗ or t (j ) = 2∗ , i.e., an S-tree is active if it has an active node. The order of an S-wood w ∈ SW’ is given by the function ord : SW’ → [0, ∞) defined by ord(w) := min { ordt(ti ) ∈ [0, ∞) : i ∈ {1, . . . , n} such that ti is active } for every S-wood w = (t1 , . . . , tn ) ∈ SW’ with n ∈ N.
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8.4. Stochastic trees and woods
147
For example, the stochastic wood in Figure 8.2 has order θ since the middle tree and the last (i.e., third) tree in Figure 8.2 are active. More precisely, the nodes (2, 1) and (3, 1) of the S-wood w0 are active nodes and, therefore, the second and the third tree are active. The second tree in Figure 8.2 has order 1, since it consists of only one node of type 1∗ , and the third three in Figure 8.2 has order θ . Hence, since θ ≤ 1, the S-wood w0 has order θ . Similarly, the second tree and the last three trees are active in the stochastic wood w1 in Figure 8.3 (see (8.24) for the active nodes of w1 ), and the stochastic wood in Figure 8.3 has order θ + min(γ , θ ). The second tree in w1 has order 1 and the last three trees in the S-wood w1 have order θ + γ , θ + 1 and 2θ , respectively. As a third example, consider the S-wood w2 in Figure 8.4. The active nodes of w2 are shown in (8.25); i.e, the last six S-trees are active and have the orders θ + γ , θ + 1, 2θ , 1 + γ , 2, and 1 + θ , respectively. The minimum of these six real numbers is θ + min(θ , γ ), so the order of the S-wood w2 in Figure 8.4 is also θ + min(θ, γ ). A similar calculation shows that the order of the stochastic wood w3 in Figure 8.5 is θ + min(θ , 2γ ) and that the order of the stochastic wood w4 in Figure 8.6 is also θ + min(θ , 2γ ). Finally, the stochastic wood w5 in Figure 8.7 is of order θ + 2 min(θ , γ ).
8.4.5
Stochastic woods and Taylor expansions
Each stochastic wood in w ∈ SW’ characterizes a Taylor expansion of the solution X of the SPDE (8.9). In particular, every stochastic tree ti ∈ ST’ for i ∈ {1, . . . , n} of a stochastic wood w = (t1 , . . . , tn ) ∈ SW’ with n ∈ N represents a certain summand in the Taylor expansion. To show this consider the mappings φ, ψ : ST’ → , which are defined recursively as follows. For a given S-tree t = (t , t ) ∈ ST’ define φ(t) := It0 (1) when l(t) = 1 and define φ(t) := Itn (1) [φ(t1 ), . . . , φ(tn )] when l(t) ≥ 2, where t1 , . . . , tn ∈ ST’ with n ∈ N are the subtrees of t. Note that if l(t) ≥ 2 for a tree t = (t , t ) ∈ ST’, then t (1) = 0. In addition, for an arbitrary t ∈ ST’ define ψ(t) := 0 if t is an active tree with ψ(t) = φ(t) otherwise. Finally, define mappings , : SW’ → by (w) = φ(t1 ) + · · · + φ(tn ),
(w) = ψ(t1 ) + · · · + ψ(tn )
for every S-wood w = (t1 , . . . , tn ) ∈ SW’ with n ∈ N. For example, for the initial stochastic wood w0 (see Figure 8.2) (w0 ) = I00 + I10∗ + I20∗ ,
(w0 ) = I00 .
(8.29)
Hence, (w0 ) = X
(8.30)
from (8.29) and (8.16).
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148
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise Furthermore, (w1 ) = I00 + I10∗ + I20 + I21∗ [I00 ] + I21∗ [I10∗ ] + I21∗ [I20∗ ]
(8.31)
(w1 ) = I00 + I20
(8.32)
and
for the S-wood w1 = E(3,1) w0 shown in Figure 8.3. This notation will now be used to formulate the main theorem of this chapter, Theorem 8.4, which shows that every stochastic wood w ∈ SW’ represents a Taylor expansion Xt0 + (w)(t) and the corresponding Taylor approximation Xt0 + (w)(t) for t ∈ [t0 , T ] of the solution process X of the SPDE (8.9). In this theorem the real numbers + + 1 + −1 + R := max 1, κ, +(κ − A) + (8.33) , , T , F (0) H , L0 L(H ) θ and
+ + + + Rn := sup +F (n) (v)+ v∈H
L(n) (H ,H )
+ sup 0≤t≤T
+ + +(κ − A)γ Xt +
Ln (;H )
+ Xt Ln (;H ) + Ln , (8.34)
n ∈ N, are used. The finiteness of R and (Rn )n∈N follows from Assumptions 15–18 and Proposition 7. Theorem 8.4. Let Assumptions 15–18 be fulfilled, and let w ∈ SW’. Then there exists a constant Cp > 0 such that P Xt = Xt0 + (w) (t) = 1 and 1 p + +p E +Xt − Xt0 − (w)(t)+H ≤ Cp (t − t0 )ord(w)
(8.35)
holds for every t ∈ [t0 , T ], where X is the up-to-modifications unique solution of the SPDE (8.9). The constant Cp here depends only on the S-wood w, p ≥ 1, R > 0 given in (8.33) and on (Rn )n∈N given in (8.34). The proof of Theorem 8.4 is postponed to the final section. The Taylor approximation in Theorem 8.4 can also be written as Xt = Xt0 + (w) + O (t)ord(w) (8.36)
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8.5. Examples of Taylor approximations
149
for every stochastic wood w ∈ SW’ in view of the definition of the O(·)-notation in Section 8.3.2. Theorem 8.4 is in a sense the analogue for SPDEs of Theorem 2.6 in [46] for ODEs and Theorem 5.5.1 in [82] for SODEs. The following lemma shows that there are woods in SW’ with arbitrarily high orders. Lemma 8.5. Let Assumptions 15–18 be fulfilled. Then sup ord (w) = ∞. w∈SW’
Hence Taylor approximations of arbitrarily high orders can be constructed by successively applying the E(i,j ) operator.
8.5
Examples of Taylor approximations
Various examples of Taylor approximations Xt0 + (w)(t) in Theorem 8.4 and (8.36) are illustrated in this section. These are based on the stochastic woods w ∈ {w0 , w1 , . . . , w5 } in the figures above.
8.5.1 Abstract examples of Taylor approximations The Taylor approximations Xt0 + (w)(t) and their orders ord(w) will be calculated here for different stochastic woods w ∈ SW’. Theorem 8.4 then yields 1 p +p + + + ≤ Cp (t)ord(w) E Xt − Xt0 − (w)(t) H for every t ∈ [t0 , T ] with an appropriate constant Cp > 0, where p ∈ [1, ∞). The orders of the Taylor approximations here depend on the two parameters θ ∈ (0, 12 ] and γ ∈ (0, 1) in Assumption 17. Taylor approximation of order θ The first Taylor expansion of the solution is given by the initial stochastic wood w0 (see Figure 8.2); i.e., (w0 ) = X is approximated by (w0 ) with order ord(w0 ). Specifically, by (8.29), (w0 )(t) = (eAt − I )Xt0 ,
P-a.s.,
and (w0 )(t) = (eAt − I )Xt0 +
t
t0
eA(t−s) F (Xs ) ds +
t
eA(t−s) B(Xs ) dWs , P-a.s.,
t0
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150
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise
for every t ∈ [t0 , T ]. Since ord(w0 ) = θ, the corresponding Taylor approximation (8.37) Xt = eAt Xt0 + O (t)θ has order θ. Taylor approximation of order θ + min(γ , θ) The Taylor approximation given by the S-wood w1 (see Figure 8.3) is based on (w1 ) and (w1 ), which were given in (8.31) and (8.32). Since ord(w1 ) = θ + min(γ , θ), the Taylor approximation t (8.38) eA(t−s) B(Xt0 ) dWs + O (t)θ+min(γ ,θ ) Xt = eAt Xt0 + t0
has order θ + min(γ , θ). Another Taylor approximation of order θ + min(γ , θ ) The stochastic wood w2 (see Figure 8.4) has order θ + min(γ , θ ) and the Taylor approximation (w2 ) of (w2 ) = X is given by (w2 ) = I00 + I10 + I20 , i.e., Xt = e
At
Xt0 +
t
e 0
As
ds F (Xt0 ) +
t
t0
eA(t−s) B(Xt0 ) dWs
(8.39)
+ O (t)θ+min(γ ,θ ) , which is another Taylor approximation of order θ + min(γ , θ). It is the exponential Euler approximation that was already considered in Section 8.3.2. A natural question to ask is why one should use this Taylor approximation if it is of the same order as the Taylor approximation (8.38). The reason is the following: Although both of these Taylor approximations have the same local approximation order, for numerical schemes it is the global approximation order and behavior that are of primary importance, as will be discussed in Section 8.6.4. It turns out that the one-step numerical scheme induced by the Taylor approximation (8.39) has very good global approximation properties, which was seen in Chapter 7 in the case of additive noise and will be seen in Section 8.6 in the case of multiplicative noise. Taylor approximation of order θ + min(2γ , θ ) The stochastic wood w3 (see Figure 8.5) has order θ + min(2γ , θ) and (w3 ) = I00 + I10 + I20 + I21 [I00 ].
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8.5. Examples of Taylor approximations
151
The corresponding Taylor approximation t t eAs ds F (Xt0 ) + eA(t−s) B(Xt0 ) dWs Xt = eAt Xt0 + 0
+
t
t0
eA(t−s) B (Xt0 )
(8.40)
t0
eAs − I Xt0 dWs + O (t)θ+min(2γ ,θ)
thus has order θ + min(2γ , θ). It is left as an exercise for the reader to show that the stochastic wood w4 gives the same Taylor approximation as w3 , i.e., (w4 ) = (w3 ). Taylor approximation of order θ + 2 min(γ , θ ) The Taylor approximation corresponding to the S-wood w5 (see Figure 8.7) is t t At As Xt = e Xt0 + e ds F (Xt0 ) + eA(t−s) B(Xt0 ) dWs 0
+
t
e t0
+
t
t0
A(t−s)
t0
B (Xt0 )
eA(t−s) B (Xt0 )
s
t0
eA(s−r) B(Xt0 ) dWr dWs
(8.41)
eAs − I Xt0 dWs + O (t)θ+2 min(γ ,θ)
and thus has order θ + 2 min(γ , θ ). It corresponds to the Taylor approximation introduced in Section 8.3.3.
8.5.2 Application to the stochastic heat equation The abstract Taylor approximations above are applied here to the stochastic heat equation (8.13) introduced in Section 8.2.1. The stochastic heat equation satisfies Assumptions 15–18 with the parameters θ = 14 and γ = 14 − ε for every arbitrarily small ε ∈ (0, 14 ). First, the Taylor approximation (8.37) here is 1 (8.42) Xt = eAt Xt0 + O (t) 4 , while the Taylor approximation (8.38) (see also (8.39)) gives t 1 eA(t−s) B(Xt0 ) dWs + O (t) 2 −ε . Xt = eAt Xt0 +
(8.43)
t0
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152
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise
In addition, Xt = e
Xt0 +
At
t0
t0
=e
At
eA(t−s) B(Xt0 ) dWs
1 eA(t−s) B((eAs − I )Xt0 ) dWs + O (t) 2
t
+
t
Xt0 +
t
t0
1 eA(t−s) B(eAs Xt0 ) dWs + O (t) 2
(8.44)
comes from the Taylor approximation (8.40). The Taylor approximations (8.43), (8.44), and that below exploit the special structure of the stochastic heat equation (8.13); i.e., its drift term vanishes and its diffusion term is linear. In particular, the Taylor approximation (8.41) reduces to t Xt = eAt Xt0 + eA(t−s) B(eAs Xt0 ) dWs +
t0
t
e
A(t−s)
B
e
=e
A(s−r)
t0
t0 t
B(Xt0 ) dWr
Xt0 + e t0 3 + O (t) 4 −ε . At
s
A(t−s)
B e
As
Xt0 +
s
e t0
3 dWs + O (t) 4 −ε
A(s−r)
B(Xt0 ) dWr
dWs (8.45)
For later comparison, the linear-implicit Crank–Nicolson approximation for the SPDE (8.13) is −1 −1 t t t t Xt ≈ I + I− Xt0 + B(Xt0 ) dWs . A I− A A 2 2 2 t0
8.5.3
(8.46)
Finite dimensional SODEs
The abstract setting for SPDE here includes finite dimensional SODEs. It is interesting to see how the Taylor approximations and numerical schemes derived here when applied to SODE compare with those in the monographs of Kloeden & Platen [82] and Milstein [95]; see also Chapter 3. This will be shown for simplicity for a scalar SODE, i.e., an SODE with a one-dimensional state space driven by a onedimensional standard Wiener process. In this context, H = R with the scalar productv, wH = v · w for real numbers v, w ∈ H = R. In addition, let U = H = R and let Wt be a one-dimensional standard Wiener process.
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8.5. Examples of Taylor approximations
153
In addition, suppose that the eigenfunctions and the eigenvalues of the linear operator −A in Assumption 15 are given by e1 = 1 ∈ H and λ1 = 0 with the index set I = {1}. Thus A is the boring bounded linear operator with Av = 0 for every real number v ∈ D(A) = H = R. Furthermore, D((κ − A)r ) = H = R for every r ∈ R. Define D = H = R, so L(U , D) = H S(U , D) = R. In view of Assumptions 16 and 17 the drift term F : R → R and the diffusion term B : R → R are smooth functions with globally bounded derivatives (as assumed in [82] and Chapter 3). Finally, the initial value x0 : → R is then simply an F /B(R)-measurable mapping, i.e., a random variable, which satisfies E |x0 |p < ∞ for some p ∈ [1, ∞). The SPDE (8.9) in this setup is thus the SODE dXt = F (Xt ) dt + B(Xt ) dWt
(8.47)
for t ∈ [0, T ] with initial value X0 = x0 . To apply the abstract Taylor expansions introduced above to this simple example note that the appropriate parameters in Assumption 17 here are γ = 1 − ε and θ = 12 for every arbitrarily small ε ∈ (0, 1). First, the simple approximation (8.37) reads 1 Xt = Xt0 + O (t) 2 , which simply states the Hölder smoothness of the solution of the SODE (8.47). Moreover, the Taylor approximation (8.38) reduces to t Xt = Xt0 + B(Xt0 ) dWs + O(t) = Xt0 + B(Xt0 ) · Wt − Wt0 + O(t). t0
It corresponds to the stochastic Taylor expansion for SODEs with the hierarchical set A = {∅, (1)} in Chapter 3 (see also Theorem 5.5.1 in [82]). The exponential Euler approximation (8.39) here becomes Xt = Xt0 + F (Xt0 ) · t + B(Xt0 ) · Wt − Wt0 + O(t). This is nothing other than the corresponding one-step approximation of the classical Euler–Maruyama scheme corresponding to the hierarchical set A = {∅, (0), (1)}. In view of this the name “exponential Euler scheme” is indeed justified. Finally, the Taylor approximation (8.41) reduces to Xt = Xt0 + F (Xt0 ) · t + B(Xt0 ) · Wt − Wt0 t s 3 + B (Xt0 )B(Xt0 ) dWr dWs + O (t) 2 . t0
t0
This is just the one-step approximation of the Milstein scheme and corresponds to the hierarchical set A = {∅, (0), (1), (1, 1)} in Chapter 3.
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154
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise
Exercise 1. Derive Taylor approximations for the SODE (8.47) by first writing it in the form dXt = (Xt + [F (Xt ) − Xt ]) dt + B(Xt ) dWt and using the operator Av = v for all v ∈ R.
8.6
Numerical schemes for SPDEs
Numerical approximations of the SPDE (8.9) require the discretization of both the infinite dimensional space H and the time interval [0, T ]. A spectral Galerkin approximation based on the eigenfunctions of the infinite dimensional operator A will be used as the spatial discretization and the above Taylor expansions of the SPDE (8.9) for the temporal discretization. First, recall that I is the set of indices of the eigenfunctions (ei )i∈I of the linear operator A given in Assumption 15. Let J be a further finite or countable set and let fj j ∈J be an orthonormal basis of the real Hilbert space U . In addition, let (IN )N∈N be an increasing sequence of finite nonempty subsets of I and let (JN )N∈N be an increasing sequence of finite nonempty subsets of J, i.e., ∅ = IM ⊂ IN ⊂ I,
∅ = J M ⊂ JN ⊂ J
(8.48)
for all M, N ∈ N with M ≤ N . In the next step consider the finite dimensional subspaces HN = span ei : i ∈ IN , of H and UN = span fj : j ∈ JN of U and the projections PN : D((−A)−r ) → HN and P˜N : U → UN given by , PN v = f j , u U fj P˜N u = µ(ei ) ei , j ∈JN
i∈IN
for every µ ∈ D((−A)−r ), u ∈ U , and N ∈ N, where r ∈ [0, ∞) is given in Assumption 17. Then truncate the initial random variable x0 , the linear operator A, the drift term F , and the diffusion term B by x0N : → HN , AN : HN → HN , FN : HN → HN , BN,L : HN → L(U , HN ), which are defined by x0N (ω) := PN (x0 (ω)), and FN (v) := PN (F (v)) ,
AN v := Av,
BN,L (v) = PN B(v)(P˜L u)
for all v ∈ HN , u ∈ U , ω ∈ and each N , L ∈ N.
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8.6. Numerical schemes for SPDEs
155
In the following subsections, one-step numerical schemes based on the Taylor expansions in the previous sections to approximate the solution process Xt of the SPDE (8.9) are presented. These schemes will always be denoted by F /B(HN )measurable mappings YkN,M,L : → HN
with
Y0N,M,L (ω) := x0N (ω)
for k ∈ {0, 1, . . . , M −1}, N, M, L ∈ N, and ω ∈ . Here N ∈ N provides an indication of the accuracy of the spatial discretization of the space H , while L ∈ N indicates the accuracy of the spatial discretization of the space U and M ∈ N is the number of times steps on the interval [0, T ] with constant time step size h = T /M. (In principle an arbitrary nonequidistant time discretization could also be considered.) Thus Xkh ≈ YkN,M,L for k ∈ {0, 1, . . . , M − 1} and N , M, L ∈ N.
8.6.1 The exponential Euler scheme The global convergence orders of the Taylor approximations (8.37) and (8.38) are too low to be consistent, so the Taylor approximation (8.39) will be used here to define the numerical scheme h N,M,L AN h N,M,L AN s Yk+1 = e (8.49) Yk + e ds FN YkN,M,L 0
+
(k+1)h
kh
eAN ((k+1)h−s) BN,L YkN,M,L dWs ,
P-a.s.,
for k ∈ {0, 1, . . . , M − 1} and N , M, L ∈ N. The conditional distribution with respect to Fkh of the Itô integrals (k+1)h eAN ((k+1)h−s) BN,L YkN,M,L dWs kh
in this numerical scheme is the normal distribution. Similarly, the Taylor approximation (8.40) gives the numerical scheme h N,M,L AN h N,M,L AN s Yk+1 = e Yk + e ds FN YkN,M,L 0
+
(k+1)h
kh
+
(k+1)h kh
eAN ((k+1)h−s) BN,L YkN,M,L dWs
(8.50)
eAN ((k+1)h−s) BN,L YkN,M,L eAN (s−kh) − I YkN,M,L dWs ,
P-a.s., for k ∈ {0, 1, . . . , M − 1} and N , M, L ∈ N.
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156
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise
Here and below BN ,L : HN → L (HN , L (U , HN )) is the Fréchet derivative of the mapping BN,L : HN → L(U , HN ) defined above. Moreover, the conditional distributions with respect to Fkh of the integrals
(k+1)h
kh
(k+1)h
kh
eAN ((k+1)h−s) BN,L YkN,M,L dWs ,
eAN ((k+1)h−s) BN,L YkN,M,L eAN (s−kh) − I YkN,M,L dWs
are also the normal distribution. However, it seems to be much more complicated to compute the covariance matrix of these normal distributed random variables instead of the normal distributed random variables used in the exponential Euler scheme.
8.6.2 An infinite dimensional analogue of Milstein’s scheme The Taylor approximation (8.41) gives the one-step numerical scheme N,M,L Yk+1 = eAN h YkN,M,L +
+
(k+1)h
kh
+
(k+1)h
kh
+
(k+1)h
kh
0
h
eAN s ds FN YkN,M,L
eAN ((k+1)h−s) BN,L YkN,M,L dWs eAN ((k+1)h−s) BN,L YkN,M,L eAN (s−kh) − I YkN,M,L dWs s eAN ((k+1)h−s) BN,L eAN (s−r) BN,L YkN,M,L dWr dWs , YkN,M,L kh
P-a.s., for k ∈ {0, 1, . . . , M − 1} and N , M, L ∈ N. This is, in a sense, an infinite dimensional analogue of the Milstein scheme for SODEs in Chapter 3. The Milstein scheme for multidimensional SODEs is not always easy to use because of difficulty and additional cost in computing the iterated stochastic integrals involving independent Wiener processes. However, Wiktorsson [124] has shown that it is possible to approximate the distribution of such iterated integrals efficiently to obtain numerical approximations with a higher convergence order with respect to the computational effort than the classical Euler–Maruyama scheme. In addition, it is demonstrated in [73] that an appropriate infinite dimensional analogue of Milstein’s scheme (similar to the above one) can be simulated efficiently for a certain class of SPDEs. Further results on Milstein-type schemes for SPDEs can, e.g., be found in [98, 89, 4, 73] and in the references therein.
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8.6. Numerical schemes for SPDEs
8.6.3
157
Linear-implicit Euler and Crank–Nicolson schemes
Two representative numerical schemes used in the literature for the SPDE (8.9) are the linear-implicit Euler scheme N,M,L (8.51) = (I − hAN )−1 YkN,M,L + h FN YkN,M,L Yk+1 + (I − hAN )−1 BN,L YkN,M,L Wk , P-a.s., and the linear-implicit Crank–Nicolson scheme
N,M,L Yk+1
h = I − AN 2
−1 h N,M,L N,M,L (8.52) I + A N Yk + h FN Yk 2
−1 h BN,L YkN,M,L Wk , + I − AN 2
P-a.s.,
for k ∈ {0, 1, . . . , M − 1} and N , M, L ∈ N . Here it is necessary to assume that λi ≥ 0 for all i ∈ I in Assumption 15 in order to ensure that (I − hA) is invertible for every h ≥ 0.
8.6.4
Global and local convergence orders
Taylor expansions provide a fundamental instrument to understand local approximation properties of the solution of a differential equation. In the case of deterministic ODEs it is a general result (under suitable assumptions) that a numerical scheme for an ODE has global convergence order r − 1 if it has local order r with r > 1. The reason is that the local errors for the individual steps accumulate to give a larger global error. For SODEs the situation is different. There is a general result of Milstein, Theorem 1.1 in [95] (see also Beyn & Kruse [7]), which says that a numerical scheme for an SODE with local order r > 12 has global convergence order r − 12 . The reason is that the errors in every step do not accumulate so rapidly since they are centered, independent random variables and one therefore usually loses a half order only (see also Lemma 10.8.1 in Kloeden & Platen [82] for details). In the case of SPDEs the situation is different again. For example, the linearimplicit Euler scheme converges temporally to the exact solution of the SPDE (8.13) with local order 14 − ε and also with global order 14 − ε (compare Figure 8.9 for the local error with Figure 8.10 for the global error). Hence, the temporal discretization error does not accumulate at all! See Remark 2 in Davie & Gaines [27]. A similar situation (but with a higher convergence order) holds for many Taylor schemes introduced above. For example, the exponential Euler scheme (8.49) converges temporally to the exact solution of the SPDE (8.13) with local order 12 − ε
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158
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise
(see Figure 8.9) and it also seems to converge temporally with global order 12 − ε (see Figure 8.10). The reason seems to be a combination of independent errors and the parabolic regularization effect of the semigroup. However, such global phenomena have been barely investigated and a general result such as Milstein’s Theorem 1.1 in [95] for SODEs is missing in infinite dimensions for SPDEs.
8.6.5
Numerical simulations
In this subsection the exponential Euler scheme, the linear-implicit Euler scheme, and the linear-implicit Crank–Nicolson scheme are applied to the above stochastic heat 1 equation. To have a concrete example set ϑ = 10 , σ = 15 , and T = 1 for the parameters in the stochastic heat equation (8.13) in Section 8.2.1, which thus reduces to 1 1 ∂2 Xt dt + (8.53) Xt dWt dXt = 10 ∂x 2 5 for t ∈ [0, 1] and x ∈ (0, 1) with boundary conditions and initial value Xt (0) = Xt (1) = 0,
X0 =
∞ 1 i=1
i
ei .
Here Wt is a cylindrical I -Wiener process on L2 ((0, 1)√, R). Also choose J := I = N and fi (x) := ei (x) = 2 sin(iπ x) for all x ∈ (0, 1) and each i ∈ N as well as IN = JN = {1, 2, . . . , N } ⊂ I = J = N for every N ∈ N in (8.48) for the Galerkin projections. In this setup, the exponential Euler scheme reduces to (k+1)h N,M,L eAN ((k+1)h−s) BN,L YkN,M,L dWs , = eAN h YkN,M,L + Yk+1 kh
P-a.s., (8.54)
the linear-implicit Euler scheme to
N,M,L Yk+1 = (I − hAN )−1 YkN,M,L +BN,L YkN,M,L dWs , BN,L YkN,M,L Wk , P-a.s., (8.55) and the linear-implicit Crank–Nicolson scheme to −1 h h N,M,L I + AN YkN,M,L = I − AN (8.56) Yk+1 2 2 −1 h BN,L YkN,M,L Wk , + I − AN 2
P-a.s.,
for k ∈ {0, 1, . . . , M − 1} and N , M, L ∈ N.
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8.6. Numerical schemes for SPDEs
159
Comparison of local convergence orders Let t0 = 12 . In Figure 8.9 the root-mean-square approximation error is plotted for the Taylor approximations (8.42) and (8.43) and the linear-implicit Crank–Nicolson approximation (8.46) versus time steps t = t − 12 for different t ∈ [ 12 , 1]. The infinite dimensional space is approximated by the finite dimensional span of the first 100 eigenfunctions of the Laplacian with Dirichlet boundary conditions, and the expectations are approximated by 50 independent random realizations. −1
10
Root mean square error
Taylor appr. order 1/4− Crank−Nicolson appr. Taylor appr. order 1/2− Orderlines 1/4, 1/2
−2
10
−3
10
−4
10
−3
−2
10
−1
10 Time step t − t0
10
Figure 8.9. Root-mean-square approximation error of (8.42), (8.43), and (8.46) vs. time steps t = t − t0 = t − 12 of the size t ∈ { 214 , 215 , . . . , 218 } [65]. Comparison of global convergence orders In Figure 8.10 the temporal discretization error * ) + + + 128,M,128 +2 E sup +X k − Yk + k∈{0,1,...,M}
M
H
1 2
(8.57)
of the exponential Euler scheme (8.54) and of the linear-implicit Euler scheme (8.55) is plotted against the number of time steps M for different M ∈ N using 50 independent random realizations to approximate the expectation. The two orderlines in Figure 8.10 correspond to the convergence orders 14 1 and 2 . Hence, Figure 8.10 indicates that the linear-implicit Euler scheme converges temporally with order 14 and that the exponential Euler scheme converges temporally with order 12 .
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160
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise
0
Root mean square discretization error
10
Linear implicit Euler scheme Exponential Euler scheme Orderlines 1/4, 1/2
−1
10
−2
10
0
1
10
2
10
10
3
10
Number of time steps
Figure 8.10. Root-mean-square discretization error (8.57) of the exponential Euler approximation Yk128,M,128 , k ∈ {0, 1, . . . , M}, given by (8.54), and of the linear-implicit Euler approximation Yk128,M,128 , k ∈ {0, 1, . . . , M}, given by (8.55), vs. M ∈ {2, 4, 8, 16} and M ∈ {4, 16, 64, 256}, respectively [65].
8.7 8.7.1
Proofs Proof of Lemma 8.3
First, F (Xs ) = F (Xt0 ) +
0
= F (Xt0 ) +
1
0
1
F (Xt0 + r(Xs − Xt0 )) Xs − Xt0 dr F (Xt0 + rXs ) (Xs ) dr
for every s ∈ [t0 , T ] by the Fundamental Theorem of Calculus. Therefore, t eA(t−s) F (Xs ) ds I10∗ (t) = t0
=
t
e t0
A(t−s)
F (Xt0 ) ds +
= I10 (t) + I11∗ [X](t),
t
e
A(t−s)
t0
0
1
F (Xt0 + rXs ) (Xs ) dr ds
P-a.s,
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8.7. Proofs
161
for all t ∈ [t0 , T ], which can also be written as I10∗ = I10 + I11∗ [X].
(8.58)
Moreover, by integration by parts
1
0
F (i) (Xt0 + rXs ) (g1 (s), . . . , gi (s))
(1 − r)i−1 dr (i − 1)!
)
(1 − r)i = −F (Xt0 + rXs ) (g1 (s), . . . , gi (s)) i!
*r=1
(i)
+
1
0
=
r=0
F (i+1) (Xt0 + rXs ) (Xs , g1 (s), . . . , gi (s))
(1 − r)i dr i!
1 (i) F (Xt0 ) (g1 (s), . . . , gi (s)) i! 1 (1 − r)i dr F (i+1) (Xt0 + rXs ) (Xs , g1 (s), . . . , gi (s)) + i! 0
for all s ∈ [t0 , T ] and g1 , . . . , gi ∈ , i ∈ N. Hence, I1i∗ [g1 , . . . , gi ] (t) t
=
eA(t−s) 0
t0
1 = i! +
t
t0 t
t0
1
F (i) (Xt0 + rXs ) (g1 (s), . . . , gi (s))
(1 − r)i−1 dr ds (i − 1)!
eA(t−s) F (i) (Xt0 ) (g1 (s), . . . , gi (s)) ds 1
eA(t−s) 0
F (i+1) (Xt0 + rXs ) (Xs , g1 (s), . . . , gi (s))
= I1i [g1 , . . . , gi ] (t) + I1i+1 ∗ [X, g1 , . . . , gi ] (t),
(1 − r)i dr ds i!
P-a.s.,
for every t ∈ [t0 , T ] and g1 , . . . , gi ∈ , i ∈ N, which can also be written as I1i∗ [g1 , . . . , gi ] = I1i [g1 , . . . , gi ] + I1i+1 ∗ [X, g1 , . . . , gi ]
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162
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise
for all g1 , . . . , gi ∈ and i ∈ N. Combining (8.58) and the equation above yields I1i∗ [g1 , . . . , gi ] = I1i [g1 , . . . , gi ] + I1i+1 ∗ [X, g1 , . . . , gi ] = I1i [g1 , . . . , gi ] + I1i+1 +I1i+1 I10∗ , g1 , . . . , gi I00 , g1 , . . . , gi ∗ ∗ + I1i+1 I20∗ , g1 , . . . , gi ∗ for all g1 , . . . , gi ∈ and i ∈ {0, 1, 2, . . .}, where (8.16) has been used. Therefore, it remains to show (8.18). Again the Fundamental Theorem of Calculus yields B(Xs ) = B(Xt0 ) +
0
= B(Xt0 ) +
1
0
1
B (Xt0 + r(Xs − Xt0 )) Xs − Xt0 dr B (Xt0 + rXs ) (Xs ) dr
for every s ∈ [t0 , T ]. Hence, t 0 I2∗ (t) = eA(t−s) B(Xs ) dWs t0 t
=
eA(t−s) B(Xt0 ) dWs
t0
+
t
t0
1
eA(t−s) 0
B (Xt0 + rXs ) (Xs ) dr dWs
= I20 (t) + I21∗ [X](t),
P-a.s,
for all t ∈ [t0 , T ], which can also be written as I20∗ = I20 + I21∗ [X].
(8.59)
Moreover,
(1 − r)i−1 dr (i − 1)! 0 *r=1 ) (1 − r)i (i) = −B (Xt0 + rXs ) (g1 (s), . . . , gi (s)) i! r=0 1 (1 − r)i + B (i+1) (Xt0 + rXs ) (Xs , g1 (s), . . . , gi (s)) dr i! 0 1
B (i) (Xt0 + rXs ) (g1 (s), . . . , gi (s))
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8.7. Proofs =
163 1 (i) B (Xt0 ) (g1 (s), . . . , gi (s)) i! 1 (1 − r)i dr B (i+1) (Xt0 + rXs ) (Xs , g1 (s), . . . , gi (s)) + i! 0
for all s ∈ [t0 , T ] and g1 , . . . , gi ∈ , i ∈ N. This implies that I2i∗ [g1 , . . . , gi ] t 1 (1 − r)i−1 A(t−s) = e B (i) (Xt0 + rXs ) (g1 (s), . . . , gi (s)) dr dWs (i − 1)! t0 0 =
1 i!
t0
+
t
eA(t−s) B (i) (Xt0 ) (g1 (s), . . . , gi (s)) dWs )
t
e t0
A(t−s)
1
B 0
(i+1)
* (1 − r)i dr dWs (Xt0 + rXs )(Xs , g1 (s), . . . , gi (s)) i!
= I2i [g1 , . . . , gi ] (t) + I2i+1 ∗ [X, g1 , . . . , gi ] (t),
P-a.s.,
for every t ∈ [t0 , T ] and g1 , . . . , gi ∈ , i ∈ N, which can also be written as I2i∗ [g1 , . . . , gi ] = I2i [g1 , . . . , gi ] + I2i+1 ∗ [X, g1 , . . . , gi ] for all g1 , . . . , gi ∈ , i ∈ N. Finally, due to (8.16) and (8.59), I2i∗ [g1 , . . . , gi ] = I2i [g1 , . . . , gi ] + I2i+1 ∗ [X, g1 , . . . , gi ] 0 = I2i [g1 , . . . , gi ] + I2i+1 , g , . . . , g I ∗ i 0 1 + I2i+1 I10∗ , g1 , . . . , gi + I2i+1 I20∗ , g1 , . . . , gi ∗ ∗ for every g1 , . . . , gi ∈ and every i ∈ {0, 1, 2, . . .}. This proves the assertion.
8.7.2
Proof of Lemma 8.5
First, note that acn(w) = ∅ for every stochastic wood w ∈ SW. Let ε := min(θ , γ ) ∈ (0, 12 ], define c := sup ord (w) ∈ [0, ∞], w∈SW’
and suppose that c < ∞ holds. Then, by definition of the supremum, there is a stochastic wood w ∈ SW’ with ord(w) > c − 2ε . Since acn(w) = ∅, let i1 , . . . , in ∈ N
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164
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise
and j1 , . . . , jn ∈ N with n ∈ N be such that acn(w) = {(i1 , j1 ) , . . . (in , jn )} . ˜ := E(i1 ,j1 ) . . . E(in ,jn ) w ∈ SW’ is a well-defined Therefore, the stochastic wood w stochastic wood in SW’, and by the definition of the operator E(·,·) it follows that ˜ ≥ ord(w) + ε. This yields ord(w) ε ε ˜ ≥ ord(w) + ε > c − ord(w) +ε = c+ , 2 2 ˜ > c. This, however, is a contradiction to the definition of c. which shows that ord(w) Hence c must be infinite.
8.7.3
Some additional lemmas
The proof of Theorem 8.4 requires the following lemmas. Lemma 8.6. Let Assumptions 15–18 be fulfilled and let w ∈ SW’ be an arbitrary stochastic wood in SW’. Then (w) = (E(i,j ) w)
(8.60)
for every active node (i, j ) ∈ acn(w) of the S-wood w. Proof. Let t1 = (t1 , t1 ), . . . , tn = (tn , tn ) ∈ ST’ with n ∈ N such that w = (t1 , . . . , tn ) ∈ ˜ := E(i,j ) w ∈ SW’ is given by SW’. Then i ∈ {1, . . . , n} and hence w ˜ = t1 , . . . , ti−1 , ˜t, ti+1 , . . . , tn , tn+1 , tn+2 , tn+3 w with ˜t = (˜t , ˜t ), tn+1 = (tn+1 , tn+1 ), tn+2 = (tn+2 , tn+2 ), and tn+3 = (tn+3 , tn+3 ) in ST’. More precisely, nd(tn+m ) = {1, . . . , l(ti ), l(ti ) + 1} and
tn+m (k) = ti (k), k = 2, . . . , l(ti ), tn+m (l(ti ) + 1) = j ,
tn+m (k) = ti (k), k = 1, . . . , l(ti ), 0, m = 1, tn+m (l (ti ) + 1) = 1∗ , m = 2, 2∗ , m = 3,
for m ∈ {1, 2, 3} due to the definition of the E(·,·) operator. Furthermore, the S-tree ˜t = (˜t , ˜t ) is given by ˜t = t but now with ˜t (k) = t (k) for every k ∈ nd(ti )\ {j } and i i 5 1, ti (j ) = 1∗ , t˜ (j ) = 2, ti (j ) = 2∗ .
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8.7. Proofs
165
˜ holds. To this end note that It is necessary to show that (w) = (w) (w) = φ(t1 ) + · · · + φ(tn ) and ˜ = φ(t1 ) + · · · + φ(ti−1 ) + φ(˜t) + φ(ti+1 ) + · · · + φ(tn+3 ) (w) hold. Hence, it is sufficient to show that φ(ti ) = φ ˜t + φ(tn+1 ) + φ(tn+2 ) + φ(tn+3 )
(8.61)
holds. Assume, for notational convenience, that j = 1. (The case j > 1 follows analogously, but with a more complicated notation.) Suppose therefore that ti,1 , . . . , ti,k ∈ ST’ with k ∈ {0, 1, 2, . . .} are the subtrees of ti . (Note that this includes the case k = 0.) Therefore, the subtrees of ˜t are also ti,1 , . . . , ti,k and the subtrees of tn+m are ti,1 , . . . , ti,k , tn+m,k+1 ∈ ST’ with nd(tn+m,k+1 ) = 1, tn+m,k+1 = (tn+m,k+1 , tn+m,k+1 ), and 0, m = 1, tn+m,k+1 (1) = 1∗ , m = 2, 2∗ , m = 3. Furthermore, since (i, 1) is an active node of the S-wood w, it follows that ti (1) ∈ {1∗ , 2∗ } and ˜t (1) ∈ {1, 2}. This yields # $ φ(ti ) = Itk (1) [φ(ti,1 ), . . . , φ(ti,k )], φ ˜t = I˜k φ(ti,1 ), . . . , φ(ti,k ) , t (1)
i
and # $ φ(tn+m ) = Itk+1 (1) φ(ti,1 ), . . . , φ(ti,k ), φ tn+m,k+1 i
for m ∈ {1, 2, 3}. Moreover, since φ tn+2,k+1 = I10∗ , φ tn+1,k+1 = I00 ,
φ tn+3,k+1 = I20∗
hold, it follows that k+1 0 0 φ(tn+1 ) = Itk+1 (1) φ(ti,1 ), . . . , φ(ti,k ), I0 = It (1) I0 , φ(ti,1 ), . . . , φ(ti,k ) , i
i
k+1 0 0 φ(tn+2 ) = Itk+1 (1) φ(ti,1 ), . . . , φ(ti,k ), I1∗ = It (1) I1∗ , φ(ti,1 ), . . . , φ(ti,k ) ,
i
i
k+1 0 0 φ(tn+3 ) = Itk+1 (1) φ(ti,1 ), . . . , φ(ti,k ), I2∗ = It (1) I2∗ , φ(ti,1 ), . . . , φ(ti,k ) i
i
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166
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise
due to the symmetry of the integral operators. Therefore, the application of Lemma 8.3 to φ(ti ) = Itk (1) [φ(ti,1 ), . . . , φ(ti,k )] yields i
$ # φ(ti ) = Itk (1) φ(ti,1 ), . . . , φ(ti,k ) i
$ 0 φ(ti,1 ), . . . , φ(ti,k ) + Itk+1 (1) I0 , φ(ti,1 ), . . . , φ(ti,k ) t (1) i k+1 0 0 + It (1) I1∗ , φ(ti,1 ), . . . , φ(ti,k ) + Itk+1 (1) I2∗ , φ(ti,1 ), . . . , φ(ti,k ) #
= I˜k
i
i
= φ(˜t) + φ(tn+1 ) + φ(tn+2 ) + φ(tn+3 ), which shows that (8.61) holds. Lemma 8.7. Let Assumptions 15–18 be fulfilled, let t = (t , t ) ∈ ST’. Then there exists a constant Cp > 0 such that 1 p p E (t) (t) H ≤ Cp · (t − t0 )ordt(t) holds for all t ∈ [t0 , T ], where Cp > 0 depends only on the S-tree t, on p ≥ 1, on R in (8.33), and on the sequence (Rn )n∈N in (8.34). Proof. Let t = (t , t ) ∈ ST’ and suppose that p ∈ [2, ∞) is arbitrary. (There is no loss of generality due to Jensen’s inequality.) The assertion will be shown by induction on the number of nodes l(t) ∈ N. Throughout the proof, C > 0 is a constant, which changes from line to line but depends on t, p, R, and (Rn )n∈N only. First, consider the case l(t) = 1. Then 0 t (1) = 0, I0 (t) = eAt − I Xt0 , t t (1) = 1, I10 (t) = t0 eA(t−s) F (Xt0 ) ds, t φ(t)(t) = I10∗ (t) = t0 eA(t−s) F (Xs ) ds, t (1) = 1∗ , t I20 (t) = t0 eA(t−s) B(Xt0 ) dWs , t (1) = 2, t 0 I2∗ (t) = t0 eA(t−s) B(Xs ) dWs , t (1) = 2∗ ,
P-a.s.,
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8.7. Proofs
167
for every t ∈ [t0 , T ] and γ , t (1) = 0, ord(t) = 1, t (1) = 1 or 1∗ , θ, t (1) = 2 or 2∗ . In the case t (1) = 0 it follows that + + + + φ(t)(t) Lp (;H ) = + eAt − I Xt0 +
Lp (;H )
+ + + + ≤ +(κ − A)−γ eAt − I +
L(H )
+ + · +(κ − A)γ Xt0 +Lp (;H ) ≤ C (t)γ
for every t ∈ [t0 , T ]. Moreover, by Lemma 18 in [64], + t + + + A(t−s) + φ(t)(t) Lp (;H ) = + e F (Xt0 ) ds + + t0
Lp (;H )
t+ + + A(t−s) + ≤ F (Xt0 )+ +e
Lp (;H )
t0
≤e
ds
t
κT
+ + F (0) H + R1 +Xt0 +Lp (;H ) ds
t0
≤ eκT F (0) H + R1 Rp (t) ≤ C (t) for every t ∈ [t0 , T ] in the case t (1) = 1 and + t + + + A(t−s) + φ(t)(t) Lp (;H ) ≤ + e F (Xs ) ds + + t0
≤
t+ + + A(t−s) + F (Xs )+ +e
Lp (;H )
t0
≤ eκT
Lp (;H )
t
t0
ds
F (0) H + R1 Xs Lp (;H ) ds
≤ eκT F (0) H + R1 Rp (t) ≤ C (t)
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168
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise
for every t ∈ [t0 , T ] in the case t (1) = 1∗ . Finally, Corollary A.2 yields + + t + + A(t−s) + φ(t)(t) Lp (;H ) = + e B(Xt0 ) dWs + + t0
Lp (;H )
12
t + +2 + + A(t−s) B(Xt0 )+ p ≤p +e
≤ L0 p
ds
L (;H S(U ,H ))
t0
t
2
+ + 1 + +Xt +
Lp (;H )
0
t0
≤ 1 + Rp L 0 p
t
(t − s)
2θ−1
(t − s)
2θ−1
12 ds
12 ds
t0
t
= 1 + Rp L 0 p
s
2θ−1
12 ds
0
=
(1 + Rp )L0 p (t)θ ≤ C (t)θ √ 2θ
for every t ∈ [t0 , T ] in the case t (1) = 2 and + + t + + A(t−s) + φ(t)(t) Lp (;H ) = + e B(Xs ) dWs + + t0
≤p
Lp (;H )
12
t + +2 + A(t−s) + B(Xs )+ p +e
L (;H S(U ,H ))
t0
≤ L0 p
t
t0
2
1 + Xs Lp (;H ) (t − s)
≤ 1 + Rp L 0 p
t
(t − s)
2θ−1
ds
2θ−1
12 ds
12 ds
t0
= (1 + Rp )L0 p
t
s
2θ−1
12 ds
0
=
(1 + Rp )L0 p (t)θ ≤ C (t)θ √ 2θ
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8.7. Proofs
169
for every t ∈ [t0 , T ] in the case t (1) = 2∗ . This shows the assertion in the case l(t) = 1. Now suppose that l(t) ∈ {2, 3, . . .}. Then let t1 , . . . , tn ∈ ST, n ∈ N, be the subtrees of t. Hence, φ(t)(t) = Itn (1) [φ(t1 ), . . . , φ(tn )](t),
P-a.s.,
for every t ∈ [t0 , T ] by the definition of φ. In the case t (1) = 1∗ , φ(t)(t) =
t
1
eA(t−s) 0
t0
F (n) (Xt0 + rXs )( (t1 )(s), . . . , (tn )(s))
(1 − r)n−1 dr ds, (n − 1)!
×
P-a.s.,
for every t ∈ [t0 , T ] and therefore (t)(t) H ≤ e
κT
t
(t1 )(s) H · · · (tn )(s) H ds,
Rn t0
P-a.s.,
for every t ∈ [t0 , T ]. Hence, Hölder’s inequality implies (t)(t) Lp (;H ) ≤ e
κT
≤e
κT
+ t+ + + + + Rn + (t1 )(s) H · · · (tn )(s) H + t0
t
Rn t0
ds Lp (;R)
(t1 )(s) Lpn (;H ) · · · (tn )(s) Lpn (;H ) ds
≤ eκT Rn C (t)1+ordt(t1 )+···+ordt(tn ) = C (t)ordt(t) for every t ∈ [t0 , T ], since l(t1 ), . . . , l(tn ) ≤ l(t) − 1 and the induction hypothesis can be applied to subtrees. A similar calculation shows the result when t (1) = 1. In the case t (1) = 2∗ , φm (t)(t) =
t t0
1
eA(t−s) 0
×
B (n) (Xt0 + rXs ) ( (t1 )(s), . . . , (tn )(s))
(1 − r)n−1 dr dWs , (n − 1)!
P-a.s.,
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170
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise
for every t ∈ [t0 , T ]. Thus φm (t)(t) Lp (;H ) ≤ p
t t0
+ 1 + + eA(t−s) B (n) (Xt0 + rXs ) + 0
+2 1 2 (1 − r)n−1 + + ds dr + × ( (t1 )(s), . . . , (tn )(s)) (n − 1)! Lp (;H S(U ,H )) for every t ∈ [t0 , T ] by Corollary A.2. This yields t 1 + + A(t−s) (n) +e (t)(t) Lp (;H ) ≤ p B (Xt0 + rXs ) + 0
t0
+ + × ( (t1 )(s), . . . , (tn )(s)) + +
2 dr
1 2
ds
Lp (;H S(U ,H ))
and hence (t)(t) Lp (;H ) ≤ Ln p
t
(t − s)2θ−1
1 + + + + + 1 + +Xt + rXs + 0 + H 0
t0
+ + × (t1 )(s) H . . . (tn )(s) H + +
2 dr
1 2
ds
Lp (;R)
for every t ∈ [t0 , T ]. Since 1 + + + + + 1 + +Xt + rXs + (t1 )(s) . . . (tn )(s) + p H H L (;R) dr 0 H 0
≤
1 0
+ + 1 + +Xt0 + rXs +Lp(n+1) (;H ) (t1 )(s) Lp(n+1) (;H )
· · · (tn )(s) Lp(n+1) (;H ) dr ≤ C(s − t0 )
ordt(t1 )+···+ordt(tn )
1 0
+ + 1 + +rXs + (1 − r)Xt0 +Lp(n+1) (;H ) dr
≤ C 1 + Rp(n+1) (s − t0 )ordt(t1 )+···+ordt(tn ) ≤ C (t)ordt(t1 )+···+ordt(tn )
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8.7. Proofs
171
for every s ∈ [t0 , t] by the induction hypothesis, it follows that (t)(t) Lp (;H ) ≤ C
t
(t − s)(2θ−1) ds
12
(t)ordt(t1 )+···+ordt(tn )
t0
≤ C (t)θ+ordt(t1 )+···+ordt(tn ) = C (t)ordt(t) for every t ∈ [t0 , T ]. A similar calculation shows the result when t (1) = 2.
8.7.4
Proof of Theorem 8.4
The equality
P Xt = Xt0 + (w)(t) = 1
in Theorem 8.4 can also be written as (w) = X
(8.62)
in the space . If w = w0 , then (8.62) follows from (8.30). Therefore, assume that w = w0 . Then by the definition of the set SW’, there are natural numbers n ∈ N, i1 , . . . , in ∈ N, and j1 , . . . , jn ∈ N with (ik , jk ) ∈ acn E(ik−1 ,jk−1 ) . . . E(i1 ,j1 ) w0 for every k ∈ {1, . . . , n} such that w = E(in ,jn ) . . . E(i1 ,j1 ) w0 holds. Hence, (w) = (E(in ,jn ) . . . E(i1 ,j1 ) w0 ) = (E(in−1 ,jn−1 ) . . . E(i1 ,j1 ) w0 ) = · · · = (w0 ) by Lemma 8.6. Finally, (w) = (w0 ) = X by (8.30). This shows (8.62). Now let p ∈ [1, ∞) be arbitrary. It remains to show the approximation estimate + + +Xt − Xt − (w)(t)+ p ≤ C (t)ord(w) 0 L (;H ) for every t ∈ [t0 , T ], which can also be written as (w)(t) − (w)(t) Lp (;H ) ≤ C (t)ord(w) by (8.62). Here and below C ∈ [0, ∞) is a constant which changes from line to line but depends on w, on p ≥ 1, on R given in (8.33), and on the sequence (Rn )n∈N given in (8.34) only. Let t1 , . . . , tn ∈ ST’ be such that w = (t1 , . . . , tn ). Then Lemma 8.7 give φ(ti )(t) Lp (;H ) ≤ C (t)ordt(ti )
(8.63)
for every t ∈ [t0 , T ] and i ∈ {1, 2, . . . , n}.
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172
Chapter 8. Taylor Approximations for SPDEs with Multiplicative Noise In the next step, the definitions of and give (w) = φ(t1 ) + · · · + φ(tn ),
(w) = ψ(t1 ) + · · · + ψ(tn ).
Hence, (w) − (w) = (φ(t1 ) − ψ(t1 )) + · · · + (φ(tn ) − ψ(tn )) =
φ(ti )
i=1,...,n ti is active
by the definition of ψ. Finally, by (8.63), it follows that (w)(t) − (w)(t) Lp (;H ) ≤ φ(ti )(t) Lp (;H ) i=1,...,n ti is active
≤
C (t)ordt(ti )
i=1,...,n ti is active
=
C (t)ordt(ti )−ord(w) (t)ord(w)
i=1,...,n ti is active
≤
C (T + 1)ordt(ti )−ord(w) (t)ord(w)
i=1,...,n ti is active
≤ C (t)ord(w) for every t ∈ [t0 , T ]. This completes the proof of Theorem 8.4.
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Appendix A
Regularity Estimates for SPDEs
The proof of Theorem 5.1 is presented here. It makes frequent use of some important inequalities, which will first be considered. Then the semigroup, drift, and diffusion terms of the SPDE (5.20) will be systematically estimated for use in the proof. The estimates here are very similar to those in the literature (see particularly Da Prato & Zabzcyk [24], Kruse & Larsson [87], and van Neerven, Veraar & Weis [121, 122]). But for completeness they are proved in detail below. The following notation is used in sections A.3–A.5. For two normed real vector spaces V1 , · V1 , V2 , · V2 and a mapping f : V1 → V2 from V1 to V2 , define f Lip(V1 ,V2 ) := f (0) V2 + sup
f (v) − f (w) V2
v,w∈V1
v − w V1
∈ [0, ∞].
(A.1)
In addition, for real numbers T ∈ (0, ∞), r ∈ (0, 1), a probability space (, F , P), a separable real Hilbert space (V , · V , ·, ·V ), a random variable Z : → V , and a stochastic process Y : [0, T ] × → V the notations # p $ 1 Z Lp (;V ) := E Z V p ∈ [0, ∞]
(A.2)
and Y C r ([0,T ],Lp (;V )) # p $ 1 := sup E Yt V p + t∈[0,T ]
sup
t1 ,t2 ∈[0,T ] t1 =t2
+p $ 1 #+ E +Yt2 − Yt1 +V p |t2 − t1 |r
∈ [0, ∞]
(A.3)
are used in what follows. 173
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174
Appendix A. Regularity Estimates for SPDEs
A.1
Some useful inequalities
A.1.1
Minkowski’s integral inequality
The following inequality is known as Minkowski’s integral inequality (see, for instance, Appendix A.1 in Stein [115] and Theorem 202 in Hardy, Littlewood & Pólya [49]; see also Lemma 18 in [62] for the next proof). Proposition 8. Let (1 , G1 , µ1 ) and (2 , G2 , µ2 ) be two finite measure spaces and let f : 1 × 2 → [0, ∞] be G1 ⊗ G2 /B([0, ∞])-measurable. Then
p f (x, y) µ2 (dy)
p1
1
p
|f (x, y)|p µ1 (dx)
≤
µ1 (dx)
2
1
µ2 (dy)
1
2
(A.4)
for all p ∈ [1, ∞).
Proof. In the case p = 1 equality holds and (A.4) follows by Fubini’s theorem. Therefore, let p > 1 and let q ∈ (1, ∞) such that p1 + q1 = 1. First, consider the case |f (x, y)| ≤ C < ∞
for all x ∈ 1 , y ∈ 2 ,
where f is bounded by a constant C > 0. Then p f (x, y) µ2 (dy) µ1 (dx) 1
2
(p−1)
=
f (x, y) µ2 (dy)
1
f (x, u) µ2 (du) µ1 (dx)
2
2 (p−1)
=
f (x, y) µ2 (dy) 2 1
)
f (x, u) µ1 (dx) µ2 (du)
2
(p−1)q
≤
f (x, y) µ2 (dy) 2
1
2
=
p f (x, y) µ2 (dy)
1
1 f (x, u)p µ1 (dx)
p
µ2 (du)
1
* 1 q µ1 (dx) ·
(1− p1 )
1 f (x, u)p µ1 (dx)
µ1 (dx)
2
2
p
µ2 (du)
1
due to Fubini’s theorem and Hölder’s inequality. Since
p f (x, y) µ2 (dy)
1
(1− p1 ) µ1 (dx)
<∞
2
is finite due to the boundedness of f , the assertion follows in the case of a bounded f . In the general case, one can approximate f : 1 × 2 → [0, ∞] by functions
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A.1. Some useful inequalities
175
fN : 1 × 2 → [0, ∞] given by x ∈ 1 , y ∈ 2 ,
fN (x, y) := min (f (x, y) , N ) , ?
for every N ∈ N. Of course, the fN are G1 ⊗G2 B([0, ∞])-measurable and bounded. Therefore, the first case implies
p fN (x, y) µ2 (dy)
p1 µ1 (dx)
|fN (x, y)| µ1 (dx)
≤
2
1
1
p
2
p
µ2 (dy)
1
for every N ∈ N. By letting N → ∞, the assertion in the general case follows from the theorem of monotone convergence. In the next step a special case of Minkowski’s integral inequality (A.4) is presented. CorollaryA.1. Let T ∈ (0, ∞), let (, F , P) be a probability space, and let Y : [0, T ]× → [0, ∞] be a product measurable stochastic process. Then ) E
T 0
p * Ys ds
1 p
≤
T
# $ p1 ds E |Ys |p
(A.5)
0
for all p ∈ [1, ∞). Corollary A.1 immediately follows from Proposition 8.
A.1.2
Burkholder–Davis–Gundy-type inequalities
Proposition 9. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × # T → H S(U0 , H ) be a predictable stochastic process with P 0 Ys 2H S(U0 ,H ) ds < $ ∞ = 1. Then +p p1 + t 1 t 2 12 + + p (p − 1) 2 p p + + E + Ys dWs + E Ys H S(U0 ,H ) ≤ ds (A.6) 2 0 0 H and ) +p p1 + t p2 * t + + 2 + Ys H S(U0 ,H ) ds E + ≤ kp E + Ys dWs + 0
H
1 p
(A.7)
0
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176
Appendix A. Regularity Estimates for SPDEs
for all t ∈ [0, T ] and all p ∈ [2, ∞), where kp ∈ [1, ∞), p ∈ [2, ∞), is a family of real numbers defined by kp :=
p (p − 1) 2
1 2
( p −1) 2 p (p − 1)
(A.8)
for all p ∈ [2, ∞). The proof of inequalities (A.6) and (A.7) can be found in Lemmas 7.7 and 7.2 in Da Prato and Zabczyk [24]. Corollary A.2. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × # T → H S(U0 , H ) be a predictable stochastic process with P 0 Ys 2H S(U0 ,H ) ds < $ ∞ = 1. Then ) E
+ t +p * + + + sup + Ys dWs + +
t∈[0,T ]
0
1 p
H
≤
p3 2 (p − 1)
12
T
0
2 12 p p ds E Ys H S(U0 ,H )
(A.9)
and ) E
+ t +p * + + + sup + + Ys dWs +
t∈[0,T ]
≤
0
1 p
H
p (p − 1) 2
1 2
p (p − 1)
p 2
E
0
T
Ys 2H S(U0 ,H ) ds
p2
1
p
(A.10)
for all p ∈ [2, ∞). Proof. Doob’s maximal inequality (see, e.g., Theorem 3.8 in Da Prato and Zabczyk [24]) implies ) E
+ t +p * + + + sup + Ys dWs + +
t∈[0,T ]
0
H
1 p
) + +p * + T + p + E + ≤ Ys dWs + + (p − 1) 0 H
1 p
(A.11)
for all p ∈ [2, ∞). Combining (A.11) and Proposition 9 then shows (A.9) and (A.10).
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A.2. Semigroup term
A.2
177
Semigroup term
Lemma A.3. Assume that the setting in Section 5.3 is fulfilled. Then + + + + ≤1 +(t (η − A))r e(A−η)t +
(A.12)
for all t ∈ (0, ∞) and all r ∈ [0, 2], and + + + + +(t (η − A))−r e(A−η)t − I +
(A.13)
L(H )
≤1
L(H )
for all t ∈ (0, ∞) and all r ∈ [0, 1]. Proof. The estimate x 2 ≤ ex for all x ∈ [0, ∞) shows + + + + = sup (t (η + λi ))r e−(η+λi )t ≤ sup x r e−x ≤ 1 +(t (η − A))r e(A−η)t + L(H )
i∈I
x∈(0,∞)
for all t ∈ (0, ∞) and all r ∈ [0, 2]. Moreover, the inequality 1 + x ≤ ex for all x ∈ R implies + + + + = sup (t (η + λi ))−r 1 − e−(η+λi )t +(t (η − A))−r e(A−η)t − I + L(H ) i∈I 1 − e−x ≤1 ≤ sup xr x∈(0,∞) for all t ∈ (0, ∞) and all r ∈ [0, 1]. Lemma A.4. Assume that the setting in Section 5.3 is fulfilled. Then + +r + + + + + + ≤ eηt ηr +(η − A)−1 + +1 +(t (η − A))−r eAt − I + L(H )
L(H )
for all t ∈ (0, ∞) and all r ∈ [0, 1]. Proof. Lemma A.3 implies + + + + +(t (η − A))−r eAt − I + L(H ) + + + (A−η)t + −r At e −e ≤ +(t (η − A)) + L(H ) + + + + (A−η)t −r e + +(t (η − A)) −I + L(H ) + + + + + + (A−η)t −r At ≤ +(t (η − A)) +L(H ) +e − e +
L(H )
+1
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178
Appendix A. Regularity Estimates for SPDEs
for all t ∈ (0, ∞) and all r ∈ [0, 1], and therefore the estimate 1 − e−x ≤ x r for all x ∈ [0, ∞) and all r ∈ [0, 1] gives + + + + + + ≤ +(t (η − A))−r +L(H ) eηt − 1 + 1 +(t (η − A))−r eAt − I + L(H ) + + + + −r + ηt + e 1 − e−ηt + 1 ≤ +(η − A)−r +L(H ) eηt ηr + 1 = (t (η − A)) L(H ) + + +r +r r+ −1 + −1 + ηt ηt r + η +(η − A) + +1 +1 ≤ e = e η +(η − A) + L(H )
L(H )
for all t ∈ (0, ∞) and all r ∈ [0, 1]. Lemma A.5. Assume that the setting in Section 5.3 is fulfilled. Then L(H , H1 ) and + + t + + As + +(η − A) e ds ≤ eηt + + 0
t 0
eAs ds ∈ (A.14)
L(H )
for all t ∈ [0, ∞) and + t + + + As + lim (η − A) e ds v + + =0 t!0 + 0
(A.15)
H
for all v ∈ H . Proof. Note that
(η + λi )2
i∈I
≤e
(η + λi )
2
2 |ei , vH |2
t
e
−(λi +η)s
0
i∈I
= e2ηt
e−λi s ds
0
2ηt
t
1 − e−(λi +η)t
2
2 ds
|ei , vH |2
(A.16)
|ei , vH |2 ≤ e2ηt v 2H < ∞
i∈I
for all t ∈ [0, ∞) and all v ∈ H . Inequality (A.16) implies (A.14) and (A.15). Lemma A.6. Assume that#the setting in Section 5.3 is fulfilled and let Y : → H be p$ a random variable with E Y H < ∞. Then + t +p + + As + lim E +(η − A) = 0. e ds Y + +
t!0
0
(A.17)
H
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A.2. Semigroup term
179
Proof. Lemma A.5 implies + + t + + As + e ds Y (ω)+ lim +(η − A) + =0
(A.18)
for all ω ∈ . Additionally, Lemma A.5 gives + + t + + As η +(η − A) e ds Y (ω)+ + + ≤ e Y (ω) H
(A.19)
t!0
0
H
0
H
# p$ for all t ∈ [0, 1] and all ω ∈ . Combining (A.18), (A.19), the assumption E Y H < ∞, and Lebesgue’s theorem of dominated convergence then shows (A.17). This completes the proof of Lemma A.6. Lemma A.7. Assume that the setting in Section 5.3 is fulfilled. Then + + t1 + + A(t2 −s) A(t1 −s) + +(η − A)(1−r) e v ds − e + + 0 H + + + −r A(t2 −t1 ) ηt1 + − I v+ e ≤ e +(η − A) H + +r + + + 1 v H (t2 − t1 )r ≤ eηt2 ηr +(η − A)−1 +
(A.20)
L(H )
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all v ∈ H , and all r ∈ [0, 1]. Proof. Lemma A.5 implies + t1 + + + A(t2 −s) A(t1 −s) + +(η − A)(1−r) − e e v ds + + 0 H + t1 + + + (1−r) =+ eAs ds eA(t2 −t1 ) − I v + +(η − A) + 0 H + t1 + + + + + + + As −r A(t2 −t1 ) + (η (η e − A) ≤+ − A) e ds − I v+ + + + 0
L(H )
+ + + + ≤ eηt1 +(η − A)−r eA(t2 −t1 ) − I v +
(A.21)
H
H
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all v ∈ H , and all r ∈ [0, 1]. Combining Lemma A.4 and (A.21) then shows (A.20). Lemma A.8. Assume that the setting in Section 5.3 is fulfilled. Then t + +2 12 1 eηt + + As +(η − A) 2 e v + ds ≤ √ v H H 2 0
(A.22)
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180
Appendix A. Regularity Estimates for SPDEs
for all t ∈ [0, ∞) and all v ∈ H and t + +2 1 + + lim +(η − A) 2 eAs v + ds = 0 t!0
(A.23)
H
0
for all v ∈ H . Proof. The assertion follows from t+ +2 1 + + +(η − A) 2 eAs v + ds H 0 / 2 t . 1 = ei , (η − A) 2 eAs v ds H
i∈I 0
=
|ei , vH | (η + λi )
i∈I
≤e
2
2ηt
t
e
−2λi s
|ei , vH |2 (η + λi )
=
2
ds
0 t
e−2(η+λi )s ds
0
i∈I
e2ηt
|ei , vH |2 1 − e−2(η+λi )t
i∈I
≤
e2ηt v 2H 2
for all t ∈ [0, ∞) and all v ∈ H . Lemma A.9. Assume that the setting in Section 5.3 is fulfilled. Then 12
t + +2 1 + + +(η − A) 2 eAs S +
H S(U ,H )
0
ds
for all t ∈ [0, ∞) and all S ∈ H S(U , H ) and t + +2 1 + + lim +(η − A) 2 eAs S + t!0
eηt ≤ √ S H S(U ,H ) 2
H S(U ,H )
0
ds = 0
(A.24)
(A.25)
for all S ∈ H S(U , H ). Proof. Let J be a set and let gj j ∈J ⊂ U be an orthonormal basis of U . Lemma A.8 then implies t+ +2 +2 t + 1 1 + + + + ds = +(η − A) 2 eAs S + +(η − A) 2 eAs Sgj + ds 0
H S(U ,H )
j ∈J 0
H
2ηt + e2ηt + +Sgj +2 = e S 2 ≤ H S(U ,H ) H 2 2
(A.26)
j ∈J
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A.2. Semigroup term
181
for all t ∈ [0, ∞) and all S ∈ H S(U , H ), and this shows (A.24). Combining (A.26) and Lemma A.8 then shows (A.25). This completes the proof of Lemma A.9. Lemma A.10. Assume that the setting in Section 5.3 is fulfilled. Then
+2 1 + + +(η − A)( 2 −r) eA(t2 −s) − eA(t1 −s) S +
12
t1 + 0
H S(U ,H )
eηt1 ≤ √ 2 ηt e 2 ≤ √ 2
ds
+ + + + +(η − A)−r eA(t2 −t1 ) − I S + H S(U ,H ) + +r + + + 1 S H S(U ,H ) (t2 − t1 )r ηr +(η − A)−1 +
(A.27)
L(H )
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all S ∈ H S(U , H ), and all r ∈ [0, 1]. Proof. Lemma A.9 implies t1 + +2 1 + + ds +(η − A)( 2 −r) eA(t2 −s) − eA(t1 −s) S + H S(U ,H ) 0 t1 + +2 1 + + = +(η − A) 2 eAs (η − A)−r eA(t2 −t1 ) − I S + ≤
0 e2ηt1
2
H S(U ,H )
+ +2 + + +(η − A)−r eA(t2 −t1 ) − I S +
ds
(A.28)
H S(U ,H )
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all S ∈ H S(U , H ), and all r ∈ [0, 12 ]. Combining Lemma A.4 and (A.28) then shows (A.27). Lemma A.11. Assume that the setting in Section 5.3 is fulfilled and let Y : → p H S(U , H ) be a random variable with E Y H S(U ,H ) < ∞. Then ) p * +2 t+ 2 1 + + As lim E +(η − A) 2 e Y + = 0. ds H S(U ,H ) t!0
(A.29)
0
Proof. Lemma A.9 implies t + +2 1 + + lim +(η − A) 2 eAs Y (ω)+ t!0
0
H S(U ,H )
ds = 0
(A.30)
for all ω ∈ . Additionally, Lemma A.9 gives p t + +2 2 + 1 eηp + p As +(η − A) 2 e Y (ω)+ ds ≤ p Y (ω) H S(U ,H ) H S(U ,H ) 2 0 2
(A.31)
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182
Appendix A. Regularity Estimates for SPDEs
for all t ∈ [0, 1] and all ω ∈ . Combining (A.30), (A.31), the assumption p E[ Y H S(U ,H ) ] < ∞, and Lebesgue’s theorem of dominated convergence then shows (A.29).
A.3
Drift term
Hölder continuity for the drift term Lemma A.12. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × → Hα be a product measurable stochastic process. Then t t + + p1 1 p + A(t−s) +p p (α−r) ηt (t − s) E +e E Ys Hα Ys + ds ≤ e ds H r 0 0 1 eηt t (1+α−r) p p ≤ sup E Ys Hα (1 + α − r) s∈[0,T ] (A.32) for all t ∈ [0, T ] and all r ∈ [α, α + 1). Proof. Lemma A.3 implies t+ + + A(t−s) + Ys + p ds +e L (;Hr ) 0 t+ + + + Ys Lp (;Hα ) ds ≤ +(η − A)(r−α) eA(t−s) + L(H ) 0 t (α−r) ηt Ys Lp (;Hα ) ds (t − s) ≤e 0 eηt t (1+α−r) ≤ sup Ys Lp (;Hα ) (1 + α − r) s∈[0,T ] for all t ∈ [0, T ] and all r ∈ [α, α + 1). Corollary A.13. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × → Hα be a product measurable stochastic process. Then 2 + p1 + t + A(t−s) +p E +e Ys + ds Hr
0
≤
e2ηt t (1+α−r) (1 + α − r)
t
(t − s)
0
(α−r)
2 p p E Ys Hα ds
(A.33)
for all t ∈ [0, T ] and all r ∈ [α, α + 1).
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A.3. Drift term
183
Proof. Inequality (A.33) immediately follows from Lemma A.12 and Hölder’s inequality. Lemma A.14. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × → Hα be a product measurable stochastic process. Then
t1
+ +p p1 + + A(t2 −s) A(t1 −s) E + e −e ds Ys + Hr 0 (1+α−r−u) + +u eηt2 t1 + + ≤ +1 ηu +(η − A)−1 + L(H ) (1 + α − r − u) 1 p p (t2 − t1 )u · sup E Yt Hα
(A.34)
t∈[0,T ]
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all u ∈ [0, 1 + α − r), and all r ∈ [α, α + 1). Proof. Lemmas A.3 and A.4 give
t1 +
+ + A(t2 −s) + − eA(t1 −s) Ys + p ds + e L (;Hr ) 0 t1 + + + + Ys Lp (;Hα ) ds ≤ +(η − A)(r−α) eA(t2 −s) − eA(t1 −s) + L(H ) 0 t1 + + + + ≤ eηt1 s (α−r−u) ds +(η − A)−u eA(t2 −t1 ) − I + L(H ) 0 sup Yt Lp (;Hα )
·
t∈[0,T ]
(1+α−r−u)
eηt2 t1 (1 + α − r − u)
≤ ·
(A.35)
+ +u + + ηu +(η − A)−1 +
L(H )
+1
sup Yt Lp (;Hα ) (t2 − t1 )u
t∈[0,T ]
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all u ∈ [0, 1 + α − r), and all r ∈ [α, α + 1). Corollary A.15. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × p → Hα be a product measurable stochastic process with supt∈[0,T ] E[ Yt Hα ] < ∞. Then
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184
Appendix A. Regularity Estimates for SPDEs )+ + E + + ≤
t2
e
A(t2 −s)
t1
Ys ds −
e
0
A(t1 −s)
0
+p * + Ys ds + +
1 p
Hr
+ p1 t2 + + A(t2 −s) +p E +e Ys + ds Hr
t1
t1 + +p p1 + + A(t2 −s) A(t1 −s) Ys + E + e −e ds + Hr 0 + +u eηT T (1+α−r−u) u+ −1 + ≤ +2 η +(η − A) + L(H ) (1 + α − r − u) 1 p p (t2 − t1 )u < ∞ · sup E Yt Hα
(A.36)
t∈[0,T ]
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all u ∈ [0, 1 + α − r), and all r ∈ [α, α + 1). Proof. Inequality (A.36) follows immediately from Corollary A.1, Lemma A.12, and Lemma A.14. Corollary A.16. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × p → Hδ be a product measurable stochastic process with supt∈[0,T ] E[ Yt Hδ ] < ∞. Then )+ + E + + ≤
≤
eA(t2 −s) F (Ys ) ds −
0
0
t1
+p * + eA(t1 −s) F (Ys ) ds + +
1 p
Hr
+p p1 t2 + + + E +eA(t2 −s) F (Ys )+ ds
t1
+
t2
Hr
+p p1 t1 + + A(t2 −s) + A(t1 −s) E + e F (Ys )+ −e ds Hr
0 eηT
(A.37)
+ +u max(T , 1) F Lip(Hδ ,Hα ) + + +2 ηu +(η − A)−1 + L(H ) (1 + α − r − u) 1 p p · max sup E Yt Hδ , 1 (t2 − t1 )u < ∞ t∈[0,T ]
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all u ∈ [0, 1 + α − r), and all r ∈ [α, α + 1). Proof. Inequality (A.37) follows immediately from Corollary A.15.
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A.3. Drift term
185
Corollary A.17. Assume that the setting in Section 5.3 is fulfilled, assume α ≤ δ, and let Y , Z : [0, T ] × → Hδ be two product measurable stochastic processes. Then + +p p1 t + A(t−s) + (F (Ys ) − F (Zs ))+ E +e ds Hδ
0
≤ ≤
2
e2ηT
F 2Lip(Hδ ,Hα ) T (1+α−δ) (1 + α − δ)
e2ηT
t
0
2 p p (t − s)(α−δ) E Ys − Zs Hδ ds
F 2Lip(Hδ ,Hα ) max(T 2 , 1) (1 + α − δ)
t
· 0
2 p p (t − s)min(α−δ,2β−2δ) E Ys − Zs Hδ ds (A.38)
for all t ∈ [0, T ]. Proof. Inequality (A.38) follows immediately from Corollary A.13. More Hölder continuity for the drift term Lemma A.18. Assume that the setting in Section 5.3 is fulfilled, let Y : [0, T ] × → H be a stochastic process, and let f : H → U be a mapping. Then f (Y ) C r ([0,T ],Lp (;U )) sup f (Yt ) Lp (;U )
=
t∈[0,T ]
+ + +f (Yt ) − f (Yt )+ p 2 1 L (;U ) + sup r |t2 − t1 | t1 ,t2 ∈[0,T ]
t1 =t2
≤ f Lip(H ,U ) max Y C r ([0,T ],Lp (;H )) , 1 for all r ∈ (0, 1). Proof. Note that f (Y ) C r ([0,T ],Lp (;U )) =
+ + +f (Yt ) − f (Yt )+ p 2 1 L (;U ) + sup r |t2 − t1 | t1 ,t2 ∈[0,T ]
sup f (Yt ) Lp (;U )
t∈[0,T ]
t1 =t2
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186
Appendix A. Regularity Estimates for SPDEs
f (v) − f (w) U ≤ f (0) U + sup v − w H v,w∈H
v =w
sup Yt Lp (;H )
t∈[0,T ]
+ + + + Yt2 − Yt1 Lp (;H ) f (v) − f (w) U + sup sup v − w H |t2 − t1 |r v,w∈H t1 ,t2 ∈[0,T ] v =w
t1 =t2
f (v) − f (w) U = f (0) U + sup Y C r ([0,T ],Lp (;H )) v − w H v,w∈H v =w
and hence f (Y ) C r ([0,T ],Lp (;U ))
f (v) − f (w) U ≤ f (0) U + sup max Y C r ([0,T ],Lp (;H )) , 1 v − w H v,w∈H v =w
= f Lip(H ,U ) max Y C r ([0,T ],Lp (;H )) , 1 for all r ∈ (0, 1). Lemma A.19. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × → Hα be a product measurable stochastic process. Then t1
0
+ 1 p + + A(t2 −s) +p A(t1 −s) E + e −e ds Yt1 − Ys + Hr
(1+u−v) + +(v+α−r) eηt2 t1 (v+α−r) + −1 + ≤ η +1 +(η − A) + L(H ) (1 + u − v) + +p p1 E +Ys2 − Ys1 +H α (t2 − t1 )(v+α−r) · sup u |s2 − s1 | s1 ,s2 ∈[0,T ] s1 =s2
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all r ∈ [α + v − 1, α + v], all u ∈ (v − 1, ∞), and all v ∈ [1, 2].
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A.3. Drift term
187
Proof. Lemma A.3 implies
+ + + A(t2 −s) − eA(t1 −s) Yt1 − Ys + + e
t1 + 0
t1 +
≤ 0
+ + A(t2 −s) + − eA(t1 −s) + +e
+ + + + ≤ eηt1 +eA(t2 −t1 ) − I +
Lp (;Hr )
ds
+ + +Yt − Ys + p ds 1 L (;Hα ) L(H ,H(r−α) ) + + t1 +Yt − Ys + p 1 L (;H )
α ds (t1 − s)v + + t1 +Ys − Ys + p + + 2 1 L (;Hα ) + + (u−v) ≤ eηt1 +eA(t2 −t1 ) − I + s ds sup L(H ,H(r−α−v) ) s1 ,s2 ∈[0,T ] |s2 − s1 |u 0
L(H ,H(r−α−v) )
0
s1 =s2
and Lemma A.4 therefore shows t1 + + + A(t2 −s) + − eA(t1 −s) Yt1 − Ys + p ds + e L (;Hr ) 0 (1+u−v) + +(v+α−r) eηt2 t1 −1 + (v+α−r) + ≤ +1 η +(η − A) + L(H ) (1 + u − v) + + +Ys − Ys + p 2 1 L (;Hα ) (v+α−r) · sup (t2 − t1 ) |s2 − s1 |u s1 ,s2 ∈[0,T ] s1 =s2
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all r ∈ [α + v − 1, α + v], all u ∈ (v − 1, ∞), and all v ∈ [1, 2]. Lemma A.20. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × p → Hα be a product measurable stochastic process with supt∈[0,T ] E[ Yt Hα ] < ∞. Then )+ +p * p1 t1 + + t2 A(t −s) e 2 Ys ds − eA(t1 −s) Ys ds + E + + + 0 0 Hr + +(1+α−r) eηT max(T , 1) (t2 − t1 )(1+α−r) + + η(1+α−r) +(η − A)−1 + ≤ +2 L(H ) min(1 + α − r, u) · Y C u ([0,T ],Lp (;Hα )) for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all u ∈ (0, 1), and all r ∈ [α, α + 1).
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188
Appendix A. Regularity Estimates for SPDEs
Proof. Corollary A.1 implies + + + +
t2 0
eA(t2 −s) Ys ds − t2 +
≤
t1
0
+ + A(t2 −s) + Ys + +e
+ + ++ +
0
t1
+ + eA(t1 −s) Ys ds + + ds +
Lp (;Hr ) t1 + + A(t2 −s)
+ e Lp (;Hr ) 0 + t1 + eA(t2 −s) − eA(t1 −s) Yt1 ds + + p
− eA(t1 −s)
+ + Yt1 − Ys +
Lp (;Hr )
ds
,
L (;Hr )
and Lemmas A.12, A.19, and A.7 therefore give + + + +
t2
e
A(t2 −s)
Ys ds −
0
≤
e 0
eηT
(t2 − t1 ) (1 + α − r)
t1
(1+α−r)
A(t1 −s)
+ + Ys ds + +
Lp (;Hr )
+(1+α−r) + −1 + (1+α−r) + +2 sup Yt Lp (;Hα ) η +(η − A) + L(H )
t∈[0,T ]
+(1+α−r) + eηT T u (t2 − t1 )(1+α−r) −1 + (1+α−r) + + +1 η +(η − A) + L(H ) u + + +Ys − Ys + p 2 1 L (;Hα ) · sup u |s − 2 s1 | s1 ,s2 ∈[0,T ]
s1 =s2
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all u ∈ (0, ∞), and all r ∈ [α, α + 1). This shows + + + + ≤
t2 0
eA(t2 −s) Ys ds −
0
t1
+ + eA(t1 −s) Ys ds + +
Lp (;Hr )
eηT max(T , 1) (t2 − t1 )(1+α−r) min(1 + α − r, u) + +(1+α−r) + + · η(1+α−r) +(η − A)−1 + + 2 Y C u ([0,T ],Lp (;Hα )) L(H )
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all u ∈ (0, 1), and all r ∈ [α, α + 1). Corollary A.21. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × p → Hδ be a product measurable stochastic process with supt∈[0,T ] E[ Yt Hδ ] < ∞. Then
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A.3. Drift term
189
)+ +p * p1 t1 + t2 A(t −s) + E + e 2 F (Ys ) ds − eA(t1 −s) F (Ys ) ds + + + 0 0 Hr (A.39) + +(1+α−r) (1+α−r) eηT max(T , 1) (t2 − t1 ) + + +2 η(1+α−r) +(η − A)−1 + ≤ L(H ) min(1 + α − r, u) · F Lip(Hδ ,Hα ) max Y C u ([0,T ],Lp (;Hδ )) , 1 for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all u ∈ (0, 1), and all r ∈ [α, α + 1). Proof. Inequality (A.39) immediately follows from Lemmas A.18 and A.20. Boundedness in Lp (; H(α+1) ) for the drift term Lemma A.22. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × → Hα be a product measurable stochastic process. Then t + +p p1 + + E +eA(t−s) (Yt − Ys )+ ds Hr 0 + + p1 +Ys − Ys +p ηt (1+α+u−r) E 2 1 e t Hα sup ≤ |s2 − s1 |u (1 + α + u − r) s1 ,s2 ∈[0,T ]
(A.40)
s1 =s2
for all t ∈ [0, T ], all r ∈ [α, 1 + α + u), and all u ∈ (0, 1]. Proof. Lemma A.3 implies t+ + + A(t−s) + (Yt − Ys )+ p +e 0
L (;Hr )
ds ≤ e
ηt
t
(t − s)
(α−r)
0
Yt − Ys Lp (;Hα ) ds
and therefore t+ + + A(t−s) + (Yt − Ys )+ p ds +e L (;Hr ) 0 + + t +Ys − Ys + p 2 1 L (;Hα ) ηt (u+α−r) ≤ e sup s ds |s2 − s1 |u 0 s1 ,s2 ∈[0,T ] s1 =s2
+ + +Ys − Ys + p 2 1 L (;Hα ) = sup |s2 − s1 |u (1 + α + u − r) s1 ,s2 ∈[0,T ]
eηt t (1+α+u−r)
s1 =s2
for all t ∈ [0, T ], all r ∈ [α, 1 + α + u), and all u ∈ (0, 1].
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190
Appendix A. Regularity Estimates for SPDEs
Lemma A.23. Assume that the setting in Section 5.3 is fulfilled, let r ∈ (0, 1) be a real number, and let Y : [0, T ] × → Hα be a product measurable stochastic process with Y C r ([0,T ],Lp (;Hα )) < ∞. Then *
)+ +p + + t A(t−s) + Ys ds + E + e + 0
H(α+1)
)+ +p + + t A(t−s) + Yt ds + ≤ E + e + 0
1 p
*
H(α+1)
1 p
t + +p + + E +eA(t−s) (Yt − Ys )+ +
1
p
ds
H(α+1)
0
≤ eηt max(t r /r, 1) Y C r ([0,T ],Lp (;Hα )) < ∞ (A.41) for all t ∈ [0, T ]. Proof. Corollary A.1 and Lemma A.22 imply + t + + + + eA(t−s) Ys ds + + + Lp (;H(α+1) )
0
+ t + + + A(t−s) + ds + + e Yt ds + + p p (;H L ) (α+1) 0 0 L (;H(α+1) ) + + ηt r +Yt − Yt + p e t 2 1 L (;Hα ) ≤ sup r |t r − 2 t1 | t1 ,t2 ∈[0,T ] t+ + + A(t−s) + (Yt − Ys )+ ≤ +e
t1 =t2
+ + t + + As α + (η (η e ds − A) Y ++ − A) t+ +
,
Lp (;H )
0
and Lemma A.5 hence shows t+ + + A(t−s) + (Yt − Ys )+ +e
+ t + + + A(t−s) + ds + + e Yt ds + + Lp (;H(α+1) )
0
0
Lp (;H(α+1) )
≤ e max(t /r, 1) Y C r ([0,T ],Lp (;Hα )) ηt
r
for all t ∈ [0, T ]. Continuity in Lp (; H(α+1) ) for the drift term Lemma A.24. Assume that the setting in Section 5.3 is fulfilled, let r ∈ (0, 1) be a real number, and let Y : [0, T ] × → Hα be a product measurable stochastic process
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A.3. Drift term
191
with Y C r ([0,T ],Lp (;Hα )) < ∞. Then )+ + lim E + +
t1 "t t2 !t
t2
t1
e
A(t2 −s)
+p + Ys ds + +
* =0
(A.42)
H(α+1)
for all t ∈ [0, T ]. Proof. The triangle inequality and Corollary A.1 imply + + t t2 + + + 2 A(t −s) + + + A(t2 −s) 2 + + Y ds − Y ds Y e ≤ + p +e s t s 2 + p + L (;H(α+1) ) t1 t 1 L (;H(α+1) ) + + + + (t2 −t1 ) + + eAs ds (η − A)α Yt2 − Yt + + +(η − A) + p + 0 L (;H ) + + + + (t2 −t1 ) + + eAs ds (η − A)α Yt + , + +(η − A) + p + 0 L (;H )
and Lemmas A.5 and A.22 hence show + + t + + 2 A(t −s) eηT 2 + + Y C r ([0,T ],Lp (;Hα )) (t2 − t1 )r e Y ds ≤ s + p + r t1 L (;H(α+1) ) + + + eηT +(η − A)α Yt2 − Yt +Lp (;H ) + + + + (t2 −t1 ) + As α + + +(η − A) e ds (η − A) Yt + + p + 0 L (;H )
for all t, t1 , t2 ∈ [0, T ] with t1 ≤ t ≤ t2 . Therefore + + t + + 2 A(t −s) 2eηT 2 + + Y C r ([0,T ],Lp (;Hα )) (t2 − t1 )r e Y ds ≤ s + p + r t1 L (;H(α+1) ) + + + + (t2 −t1 ) + As α + e ds (η − A) Yt + + +(η − A) + + 0
Lp (;H )
(A.43) for all t, t1 , t2 ∈ [0, T ] with t1 ≤ t ≤ t2 . Combining inequality (A.43) and Lemma A.6 then shows (A.42). Lemma A.25. Assume that the setting in Section 5.3 is fulfilled, let r ∈ (0, 1) be a real number, and let Y : [0, T ] × → Hα be a product measurable stochastic process
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192
Appendix A. Regularity Estimates for SPDEs
with Y C r ([0,T ],Lp (;Hα )) < ∞. Then )+ +p + t1 A(t −s) + A(t1 −s) 2 + lim E + Y − e ds e s + + t1 "t t2 !t
0
* =0
H(α+1)
for all t ∈ [0, T ]. Proof. Lemma A.7 implies + t + + 1 A(t −s) + A(t1 −s) 2 + + Y − e ds e t1 + + p 0 L (;H(α+1) ) + + + + + + + ηt1 + A(t2 −t1 ) ≤e + e − I Yt1 − Yt + p + eηt1 + eA(t2 −t1 ) − I Yt + p L (;Hα ) L (;Hα ) + + + + + + ηT + ηT A(t −t ) α 2 1 − I (η − A) Yt + p Yt − Yt1 +Lp (;H ) + e + e ≤e α
L (;H )
for all t, t1 , t2 ∈ [0, T ] with t1 ≤ t ≤ t2 . This shows + t + + 1 A(t −s) + A(t1 −s) 2 + lim + e Y − e ds t1 + + t1 "t t2 !t
=0
(A.44)
Lp (;H(α+1) )
0
for all t ∈ [0, T ]. Combining Corollary A.1, Lemma A.19, and (A.44) then shows + + t + + 1 A(t −s) A(t1 −s) 2 + lim + Y − e ds e s + p + t1 "t 0
t2 !t
≤ lim
t1 "t 0 t2 !t
L (;H(α+1) )
t1 + + + A(t2 −s) + − eA(t1 −s) Yt1 − Ys + + e
Lp (;H(α+1) )
+ + + lim + + t1 "t t2 !t
0
ds
+ + eA(t2 −s) − eA(t1 −s) Yt1 ds + +
t1
Lp (;H(α+1) )
=0 for all t ∈ [0, T ]. Corollary A.26. Assume that the setting in Section 5.3 is fulfilled, let r ∈ (0, 1) be a real number, and let Y : [0, T ] × → Hα be a product measurable stochastic process with Y C r ([0,T ],Lp (;Hα )) < ∞. Then )+ * +p t1 + t2 A(t −s) + A(t1 −s) 2 + + lim E + =0 (A.45) e Ys ds − e Ys ds + t1 "t t2 !t
0
0
H(α+1)
for all t ∈ [0, T ].
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A.4. Diffusion term
193
Proof. Equation (A.45) follows immediately from Lemmas A.24 and A.25. Corollary A.27. Assume that the setting in Section 5.3 is fulfilled, let r ∈ (0, 1) be a real number, and let Y : [0, T ] × → Hδ be a product measurable stochastic process with Y C r ([0,T ],Lp (;Hδ )) < ∞. Then )+ + lim E + +
t1 "t t2 !t
t2
eA(t2 −s) F (Ys ) ds −
0
t1
0
+p + A(t1 −s) e F (Ys ) ds + +
* =0
(A.46)
H(α+1)
for all t ∈ [0, T ]. Proof. Equation (A.46) follows immediately from Corollary A.26.
A.4
Diffusion term
Hölder continuity for the diffusion term Lemma A.28. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × → H S(U0 , Hβ ) be a product measurable stochastic process. Then +p + t + + E +eA(t−s) Ys + ηt
≤
t
(2β−2r)
(t − s)
0
1
(1 + 2β − 2r) 2
ds
2 12 p p ds E Ys H S(U0 ,Hβ )
1
eηt t ( 2 +β−r)
1 2
p
H S(U0 ,Hr )
0
≤e
2
sup s∈[0,T ]
(A.47)
1 p p E Ys H S(U0 ,Hβ )
for all t ∈ [0, T ] and all r ∈ [β, β + 12 ). Proof. Lemma A.3 implies t+ + + A(t−s) +2 Ys + p ds +e L (;H S(U0 ,Hr )) 0 t+ +2 + + Ys 2Lp (;H S(U0 ,Hβ )) ds ≤ +(η − A)(r−β) eA(t−s) + 0
L(H )
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194
Appendix A. Regularity Estimates for SPDEs t (t − s)(2β−2r) Ys 2Lp (;H S(U0 ,Hβ )) ds ≤ e2ηt 0 (1+2β−2r) 2ηt e t ≤ sup Ys 2Lp (;H S(U0 ,Hβ )) (1 + 2β − 2r) s∈[0,T ]
for all t ∈ [0, T ] and all r ∈ [β, β + 12 ). Lemma A.29. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × → H S(U0 , Hβ ) be a product measurable stochastic process. Then
t1 0
+ +p + + E + eA(t2 −s) − eA(t1 −s) Ys +
≤
( 1 +β−r−u)
eηt2 t1 2
(t2 − t1 )u 1
·
(1 + 2β − 2r − 2u) 2 sup
t∈[0,T ]
2
1 2
p
ds
H S(U0 ,Hr )
+ +u + ηu + +(η − A)−1 +
L(H )
+1
(A.48)
1 p p E Yt H S(U0 ,Hβ )
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all r ∈ [β − u, β + 12 − u), and all u ∈ [0, 1]. Proof. Lemmas A.3 and A.4 show
t1 +
+2 + A(t2 −s) + − eA(t1 −s) Ys + p ds + e L (;H S(U0 ,Hr )) 0 t1 + +2 + + Ys 2Lp (;H S(U0 ,Hβ )) ds ≤ +(η − A)(r−β) eA(t2 −s) − eA(t1 −s) + L(H ) 0 t1 + +2 + + 2ηt1 ≤e s (2β−2r−2u) ds +(η − A)−u eA(t2 −t1 ) − I + L(H ) 0 ·
sup Yt 2Lp (;H S(U0 ,Hβ ))
t∈[0,T ]
(1+2β−2r−2u)
e2ηt2 t1 ≤ (1 + 2β − 2r − 2u) ·
(A.49)
+ +u + η +(η − A)−1 + u+
L(H )
2 +1
sup Yt 2Lp (;H S(U0 ,Hβ )) (t2 − t1 )2u
t∈[0,T ]
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all r ∈ [β − u, β + 12 − u), and all u ∈ [0, 1].
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A.4. Diffusion term
195
Corollary A.30. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × → H S(U0 , Hβ ) be a product measurable stochastic process. Then 2 + +p t2 p + + E +eA(t2 −s) Ys + ds H S(U0 ,Hr )
t1
t1
+ ≤
+ +p + + E + eA(t2 −s) − eA(t1 −s) Ys +
2
1 2
p
ds
H S(U0 ,Hr )
0
1
eηT T ( 2 +β−r−u) (t2 − t1 )u 1
(1/2 + β − r − u) 2 1 p p · sup E Yt H S(U0 ,Hβ )
(A.50)
+ +u + + ηu +(η − A)−1 +
L(H )
+1
t∈[0,T ]
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all u ∈ [0, 12 + β − r), and all r ∈ [β, β + 12 ). Proof. Inequality (A.50) follows immediately from Lemmas A.28 and A.29. Corollary A.31. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × p → Hδ be a predictable stochastic process with supt∈[0,T ] E[ Yt Hδ ] < ∞. Then )+ + E + + ≤
t2
e 0
p (p − 1) 2
+ ≤
B(Ys ) dWs −
1 2
t1
e
A(t1 −s)
0
+p * + B(Ys ) dWs + +
t1
+ +p + + E +eA(t2 −s) B(Ys )+
2
p
· max
1 2
ds
H S(U0 ,Hr )
+ +p + + E + eA(t2 −s) − eA(t1 −s) B(Ys )+
p (p − 1) 2
2
1
1 2
p
H S(U0 ,Hr )
eηT max(T , 1) B Lip(Hδ ,Hβ )
1 p
Hr
t2
t1
0
A(t2 −s)
ds
+ +u + + ηu +(η − A)−1 +
(1/2 + β − r − u) 2 1 p p sup E Yt Hδ , 1 (t2 − t1 )u < ∞
L(H )
+1
t∈[0,T ]
(A.51) for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all u ∈ [0, 12 + β − r), and all r ∈ [β, β + 12 ). Proof. Inequality (A.51) follows immediately from Proposition 9 and Corollary A.30.
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196
Appendix A. Regularity Estimates for SPDEs
Corollary A.32. Assume that the setting in Section 5.3 is fulfilled, assume β ≤ δ, and let Y , Z : [0, T ] × → Hδ be two product measurable stochastic processes. Then 2 t + +p p + + A(t−s) (B(Ys ) − B(Zs ))+ E +e ds H S(U0 ,Hδ ) 0 t 2 p p (t − s)(2β−2δ) E Ys − Zs Hδ ds ≤ e2ηT B 2Lip(Hδ ,HS(U0 ,Hβ )) 0
≤e
2ηT
B 2Lip(Hδ ,HS(U0 ,Hβ )) max(T , 1)
t
(t − s)
min(α−δ,2β−2δ)
0
2 p p E Ys − Zs Hδ ds (A.52)
for all t ∈ [0, T ]. Proof. Inequality (A.52) follows immediately from Lemma A.28. More Hölder continuity for the diffusion term Lemma A.33. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × → H S(U0 , Hβ ) be a product measurable stochastic process. Then
t1 0
+ + + +p E + eA(t2 −s) − eA(t1 −s) Yt1 − Ys +
≤
( 1 +u−v)
eηt2 t1 2
(t2 − t1 )(v+β−r) 1
·
sup
2
ds
H S(U0 ,Hr )
1 2
p
+ +(v+β−r) + η(v+β−r) + +1 +(η − A)−1 +
(1 + 2u − 2v) 2 + +p E +Ys2 − Ys1 +H S(U
L(H )
1 p 0 ,Hβ ) |s2 − s1 |u
s1 ,s2 ∈[0,T ] s1 =s2
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all r ∈ [β + v − 1, β + v], all u ∈ (v − 12 , ∞), and all v ∈ [ 12 , 2]. Proof. Lemma A.3 implies
12
t1 + 0
+ + A(t2 −s) +2 − eA(t1 −s) Yt1 − Ys + p + e
+ + + + ≤ eηt1 +eA(t2 −t1 ) − I +
L(H ,H(r−β−v) )
L (;H S(U0 ,Hr ))
t1
0
ds
+ + +Yt − Ys +2 p 1 L (;H S(U
0 ,Hβ ))
(t1 − s)
2v
12 ds
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A.4. Diffusion term ≤
197
( 1 +u−v)
eηt1 t1 2
1 2
+ + + + +eA(t2 −t1 ) − I +
L(H ,H(r−β−v) ) (1 + 2u − 2v) + + +Ys − Ys + p 2 1 L (;H S(U0 ,Hβ )) · sup , |s − s1 |u 2 s1 ,s2 ∈[0,T ] s1 =s2
and Lemma A.4 therefore shows
12 + + A(t2 −s) +2 A(t1 −s) −e ds Yt1 − Ys + p + e L (;H S(U0 ,Hr )) 0 ( 1 +u−v) + +(v+β−r) (t2 − t1 )(v+β−r) eηt2 t1 2 (v+β−r) + −1 + η ≤ +1 +(η − A) + 1 L(H ) (1 + 2u − 2v) 2 + + +Ys − Ys + p 2 1 L (;H S(U0 ,Hβ )) · sup |s − s1 |u 2 s1 ,s2 ∈[0,T ] t1 +
s1 =s2
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all r ∈ [β + v − 1, β + v], all u ∈ (v − 12 , ∞), and all v ∈ [ 12 , 2]. Lemma A.34. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × → H S(U0 , Hβ ) be a product measurable stochastic process. Then ) E
p * 2 ds H S(U0 ,Hr )
t2 +
t1
+ + A(t2 −s) +2 Ys + +e
) + E ≤
0
t1 +
+ A(t2 −s) − eA(t1 −s) + e 1
eηT max(T , 1) (t2 − t1 )( 2 +β−r) min(1 + 2β − 2r, 2u)
1 p
p * 2 ds H S(U0 ,Hr )
+2 + Ys +
1 p
(A.53)
+ +( 1 +β−r) ( 12 +β−r) + −1 + 2 η +2 +(η − A) + L(H )
· Y C u ([0,T ],Lp (;H S(U0 ,Hβ ))) for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all r ∈ [β, β + 12 ), and all u ∈ (0, 1).
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198
Appendix A. Regularity Estimates for SPDEs
Proof. Corollary A.1 and the triangle inequality imply + + + +
+1 +2 ds + + p H S(U0 ,Hr )
t2 + t1
+ + A(t2 −s) +2 Ys + +e
+ + ++ +
+1 +2 ds + + p H S(U0 ,Hr )
t1 +
+2 + A(t2 −s) + − eA(t1 −s) Ys + + e
0
L 2 (;R)
+ + A(t2 −s) +2 Ys + p +e
t2 +
≤
L (;H S(U0 ,Hr ))
t1
0
0
ds 12
+ +2 + A(t2 −s) − eA(t1 −s) Yt1 − Ys + p + e
t1 +
+ + + ++ +
L 2 (;R)
12
L (;H S(U0 ,Hr ))
+1 +2 ds + + p H S(U0 ,Hr )
t1 +
+2 + A(t2 −s) + − eA(t1 −s) Yt1 + + e
ds
,
L 2 (;R)
and Lemmas A.28, A.33, and A.10 therefore show + + + +
+1 +2 ds + + p H S(U0 ,Hr )
+ + A(t2 −s) +2 Ys + +e
t2 + t1
L 2 (;R)
+1 + t + +2 +2 + 1 + A(t −s) + A(t1 −s) 2 + e − e ds Y ++ + + s + p + H S(U0 ,Hr ) 0 L 2 (;R) + + ( 12 +β−r) ηT ( 1 +β−r) e (t2 − t1 ) ( 12 +β−r) + −1 + 2 η ≤ +2 +(η − A) + 1 L(H ) (1 + 2β − 2r) 2 ·
sup Yt Lp (;H S(U0 ,Hβ ))
(A.54)
t∈[0,T ]
1 + +( 1 +β−r) eηT T u (t2 − t1 )( 2 +β−r) ( 12 +β−r) + −1 + 2 + η +1 √ +(η − A) + L(H ) 2u + + +Ys − Ys + p 2 1 L (;H S(U0 ,Hβ )) · sup |s2 − s1 |u s1 ,s2 ∈[0,T ]
s1 =s2
for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all r ∈ [β, β + 12 ), and all u ∈ (0, 1). Inequality (A.54) then shows (A.53). Corollary A.35. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × p → H S(U0 , Hδ ) be a predictable stochastic process with supt∈[0,T ] E[ Yt Hδ ] < ∞. Then
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A.4. Diffusion term )+ + E + +
t2
e
A(t2 −s)
199 B(Ys ) dWs −
t1
e
0
A(t1 −s)
0
) ≤ kp E
+p * + B(Ys ) dWs + +
1 p
Hr
p * +2 t2 + 2 + A(t2 −s) + B(Ys )+ ds +e H S(U0 ,Hr )
1 p
t1
) p * p1 +2 t1 + 2 + + A(t −s) A(t −s) ds B(Ys )+ + kp E + e 2 −e 1 H S(U0 ,Hr ) 0 1 +( 1 +β−r) + 1 eηT max(T , 1) kp (t2 − t1 )( 2 +β−r) +2 + η( 2 +β−r) +(η − A)−1 + +2 ≤ L(H ) min(1 + 2β − 2r, 2u) · B Lip(Hδ ,H S(U0 ,Hβ )) max Y C u ([0,T ],Lp (;Hδ )) , 1 (A.55) for all t1 , t2 ∈ [0, T ] with t1 ≤ t2 , all r ∈ [β, β + 12 ), and all u ∈ (0, 1). Proof. Inequality (A.55) follows immediately from Proposition 9 and Lemmas A.18 and A.34. Boundedness in Lp (; H (β+ 1 ) ) for the diffusion term 2
Lemma A.36. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × → H S(U0 , Hβ ) be a product measurable stochastic process. Then p p1 t + +2 2 + E + ds +eA(t−s) Ys + 0 H S(U0 ,H(β+ 1 ) ) 2
ηt − 12
≤e 2
√ max(t r / r, 1) Y C r ([0,T ],Lp (;H S(U0 ,Hβ )))
(A.56)
for all t ∈ [0, T ] and all r ∈ (0, 1). Proof. The triangle inequality shows + + +2 + t+ + β+ 12 + + A(t−s) + e Ys + + +(η − A) + 0
+1 +2 + ds + + p H S(U0 ,H )
L 2 (;R)
+ + +1 +2 + t+ +2 + β+ 12 + + + ≤ + +(η − A) eA(t−s) Yt + ds + + + 0 + p H S(U0 ,H )
L 2 (;R)
+ + +1 +2 + t+ +2 + β+ 12 + + + + + +(η − A) eA(t−s) (Yt − Ys )+ ds + + + 0 + p H S(U0 ,H )
L 2 (;R)
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200
Appendix A. Regularity Estimates for SPDEs
and therefore + + +1 +2 + t+ +2 + β+ 12 + + + A(t−s) + e Ys + ds + + +(η − A) + 0 + p H S(U0 ,H )
L 2 (;R)
+ t + +1 +2 + + +2 1 As β + + 2 ≤ + +(η − A) e (η − A) Yt + ds + + p H S(U0 ,H )
L 2 (;R)
0
+1 + t + +2 +2 + + 1 + A(t−s) 2 Yt − Ys H S(U0 ,Hβ ) ds + ++ + + p + +(η − A) 2 e L(H )
,
L 2 (;R)
0
and Lemmas A.9 and A.3 hence give + + +2 + t+ + β+ 12 + + A(t−s) + e Ys + + +(η − A) +
+1 +2 + ds + + p H S(U0 ,H )
0
L 2 (;R)
eηt ≤ √ Yt Lp (;H S(U0 ,Hβ )) 2 + + t 12 +Ys − Ys + p 1 L (;H S(U0 ,Hβ )) 2 ηt (2r−1) s ds + e sup |s2 − s1 |r 0 s1 ,s2 ∈[0,T ] s1 =s2
√ ≤ √ max(t r / r, 1) Y C r ([0,T ],Lp (;H S(U0 ,Hβ ))) 2 eηt
for all t ∈ [0, T ] and all r ∈ (0, 1). Corollary A.37. Assume that the setting in Section 5.3 is fulfilled, let r ∈ (0, 1) be a real number, and let Y : [0, T ] × → H S(U0 , Hβ ) be a predictable stochastic process with Y C r ([0,T ],Lp (;H S(U0 ,Hβ ))) < ∞. Then 1
+p + t + + E+ eA(t−s) Ys dWs + + + 0
p
ds
H(β+ 1 ) 2
p p1 +2 t + 2 + + ≤ kp E +eA(t−s) Ys + ds 0 H S(U0 ,H(β+ 1 ) ) ηt − 12
≤ kp e 2
√
(A.57)
2
max(t / r, 1) Y C r ([0,T ],Lp (;H S(U0 ,Hβ ))) < ∞ r
for all t ∈ [0, T ]. Proof. Inequality (A.57) follows immediately from Proposition 9 and Lemma A.36.
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A.4. Diffusion term
201
Continuity in Lp (; H (β+ 1 ) ) for the diffusion term 2
Lemma A.38. Assume that the setting in Section 5.3 is fulfilled and let Y : [0, T ] × → H S(U0 , Hβ ) be a product measurable stochastic process. Then +p + t + + E +eA(t−s) (Yt − Ys )+
2
ds
H S(U0 ,Hr )
0
≤
1 2
p
eηt t
( 12 +β+u−r)
(1 + 2β + 2u − 2r)
1 2
sup
1 p 0 ,Hβ ) u |s2 − s1 |
+ +p E +Ys2 − Ys1 +H S(U
s1 ,s2 ∈[0,T ] s1 =s2
for all t ∈ [0, T ], all r ∈ [β, 12 + β + u) with r ≤ β + 2, and all u ∈ (0, ∞). Proof. Lemma A.3 implies 12
t + +2 + A(t−s) + (Yt − Ys )+ p +e 0
L (;H S(U0 ,Hr ))
ds
12 (t − s)(2β−2r) Yt − Ys 2Lp (;H S(U ,Hβ )) ds 0 + 1 +p p t 12 + + E Ys2 − Ys1 H S(U ,H ) 0 β ηt (2β+2u−2r) ≤ e sup s ds |s2 − s1 |u 0 s1 ,s2 ∈[0,T ]
≤ eηt
t
s1 =s2
=
eηt t
( 12 +β+u−r)
(1 + 2β + 2u − 2r)
1 2
sup
1 p 0 ,Hβ ) u |s2 − s1 |
+ +p E +Ys2 − Ys1 +H S(U
s1 ,s2 ∈[0,T ] s1 =s2
for all t ∈ [0, T ], all r ∈ [β, 12 + β + u) with r ≤ β + 2, and all u ∈ (0, ∞). Lemma A.39. Assume that the setting in Section 5.3 is fulfilled, let r ∈ (0, 1) be a real number, and let Y : [0, T ] × → H S(U0 , Hβ ) be a predictable stochastic process with Y C r ([0,T ],Lp (;H S(U0 ,Hβ ))) < ∞. Then
+ + lim E + + t1 "t t2 !t
t2
t1
+p + A(t2 −s) e Ys dWs + +
= 0
(A.58)
H(β+ 1 ) 2
for all t ∈ [0, T ].
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202
Appendix A. Regularity Estimates for SPDEs
Proof. Proposition 9 and Corollary A.1 imply + + + +
t2 t1
+ + eA(t2 −s) Ys dWs + +
Lp (;H(β+ 1 ) ) 2
+ +1 + + t2 + +2 + A(t2 −s) +2 + + ≤ kp + Ys + ds + +e + t1 + p H S(U0 ,H(β+ 1 ) ) ≤ kp
2
L 2 (;R) 1 2
+ +2 + A(t2 −s) Yt2 − Ys + p +e
t2 +
L (;H S(U0 ,H(β+ 1 ) ))
t1
ds
2
+1 + +2 + (t2 −t1 ) + 2 + 1 + + + + As β ds + + kp + +(η − A) 2 e (η − A) Yt2 − Yt + + p + 0 H S(U0 ,H ) + +1 + (t2 −t1 ) + +2 +2 1 + + + + As β ds + + kp + +(η − A) 2 e (η − A) Yt + + 0 + p H S(U0 ,H )
L 2 (;R)
,
L 2 (;R)
and Lemmas A.38 and A.9 therefore show + + t + + 2 A(t −s) 2 + + e Y dW s s+ + p t1
≤
L (;H(β+ 1 ) )
kp
(t2 − t1 ) √ 2r
eηT
r
2
sup
s1 ,s2 ∈[0,T ] s1 =s2
+ + +Ys − Ys + 2 1 H S(U
0 ,Hβ )
|s2 − s1 |r
+ kp eηT + + √ +Yt2 − Yt +Lp (;H S(U ,H )) 0 β 2 + +1 + (t2 −t1 ) + +2 +2 1 + + + + As β + kp + ds + +(η − A) 2 e (η − A) Yt + + 0 + p H S(U0 ,H ) L 2 (;R) r ηT 2 kp e (t2 − t1 ) Y C r ([0,T ],Lp (;H S(U0 ,Hβ ))) ≤ r + +1 + (t2 −t1 ) + +2 +2 1 + + + + + kp + ds + +(η − A) 2 eAs (η − A)β Yt + + 0 + p H S(U0 ,H )
(A.59)
L 2 (;R)
for all t, t1 , t2 ∈ [0, T ] with t1 ≤ t ≤ t2 . Combining (A.59) and Lemma A.11 then shows (A.58).
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A.4. Diffusion term
203
Lemma A.40. Assume that the setting in Section 5.3 is fulfilled, let r ∈ (0, 1) be a real number, and let Y : [0, T ] × → H S(U0 , Hβ ) be a predictable stochastic process with Y C r ([0,T ],Lp (;H S(U0 ,Hβ ))) < ∞. Then +p + t + + 1 A(t −s) = 0 e 2 − eA(t1 −s) Ys dWs + (A.60) lim E+ + + t1 "t 0
t2 !t
H(β+ 1 ) 2
for all t ∈ [0, T ]. Proof. Proposition 9 and Lemma A.10 imply + + t + + 1 A(t −s) A(t1 −s) 2 + e Yt1 dWs + −e + p + 0
L (;H(β+ 1 ) ) 2
+1 + +2 + t1 + +2 + + A(t2 −s) + + A(t1 −s) Yt1 + −e ds + ≤ kp + + e + p + 0 H S(U0 ,H(β+ 1 ) ) 2 L 2 (;R) + + + ηt1 + A(t2 −t1 ) − I Yt1 + p ≤ kp e + e L (;H S(U0 ,Hβ )) + + + + + + ≤ kp eηT +Yt1 − Yt +Lp (;H S(U ,H )) + + eA(t2 −t1 ) − I Yt + p 0
β
L (;H S(U0 ,Hβ ))
for all t, t1 , t2 ∈ [0, T ] with t1 ≤ t ≤ t2 . This shows + + t + + 1 A(t −s) A(t1 −s) 2 + −e e Yt1 dWs + lim + +
=0
(A.61)
for all t ∈ [0, T ]. Moreover, Proposition 9 and Lemma A.33 yield + t +1 + 1 A(t −s) +2 A(t1 −s) 2 + e Yt1 − Ys dWs + −e =0 lim + +
(A.62)
t1 "t t2 !t
t1 "t t2 !t
Lp (;H(β+ 1 ) )
0
2
Lp (;H(β+ 1 ) )
0
2
for all t ∈ [0, T ]. Combining (A.61) and (A.62) then shows (A.60). Corollary A.41. Assume that the setting in Section 5.3 is fulfilled, let r ∈ (0, 1) be a real number, and let Y : [0, T ] × → H S(U0 , Hβ ) be a predictable stochastic process with Y C r ([0,T ],Lp (;H S(U0 ,Hβ ))) < ∞. Then + t +p t1 + 2 A(t −s) + = 0 (A.63) e 2 Ys dWs − eA(t1 −s) Ys dWs + lim E+ + + t1 "t t2 !t
0
0
H(β+ 1 ) 2
for all t ∈ [0, T ].
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204
Appendix A. Regularity Estimates for SPDEs
Proof. Equation (A.63) immediately follows from Lemmas A.39 and A.40. Corollary A.42. Assume that the setting in Section 5.3 is fulfilled, let r ∈ (0, 1) be a real number, and let Y : [0, T ] × → Hδ be a predictable stochastic process with Y C r ([0,T ],Lp (;Hδ )) < ∞. Then
+ + lim E+ + t1 "t t2 !t
t2
eA(t2 −s) B(Ys ) dWs −
0
0
t1
+p + eA(t1 −s) B(Ys ) dWs + +
= 0
H(β+ 1 ) 2
(A.64)
for all t ∈ [0, T ]. Proof. Equation (A.64) immediately follows from Corollary A.41.
A.5
Existence and uniqueness
Proposition 10. Assume that the setting in Section 5.3 is fulfilled. Then there exists an up-to-modifications unique predictable stochastic process X : [0, T ] × → Hδ p which satisfies supt∈[0,T ] E[ Xt Hδ ] < ∞ and (5.19). Proof. First, assume α ≤ δ and β ≤ δ without loss of generality. Moreover, throughout this proof let H be the real vector space of equivalence classes of Hδ -valued predictable stochastic processes Y : [0, T ] × → Hδ that satisfy p sup E Yt Hδ < ∞, (A.65) t∈[0,T ]
where two stochastic processes lie in one equivalence class if and only if they are modifications of each other. As usual, it is not distinguished between a predictable stochastic process Y : [0, T ] × → Hδ satisfying (A.65) and its equivalence class in H . In the next step the vector space H is equipped with the norms Y H,r := sup ert Yt Lp (;Hδ ) t∈[0,T ]
for all Y ∈ H and all r ∈ R. Note that the pairs H , · H,r for r ∈ R are real Banach spaces. In the next step consider the mapping : H → H defined by t t At A(t−s) ( Y )t := e ξ + e F (Ys ) ds + eA(t−s) B(Ys ) dWs (A.66) 0
0
P-a.s. for all t ∈ [0, T ] and all Y ∈ H . The mapping : H → H is indeed welldefined due to Proposition 3.6 in [24] and Corollaries A.16 and A.31. Now it
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A.5. Existence and uniqueness
205
is proved for each r ∈ R that : H → H is globally Lipschitz continuous from (H , · H,r ) to (H , · H,r ). More formally, Corollary A.1 and Proposition 9 imply + + +( Y )t − ( Z)t + p L (;Hδ ) t+ + + A(t−s) + (F (Ys ) − F (Zs ))+ p ≤ ds +e L (;Hδ )
0
12 1 t + +2 p (p − 1) 2 + A(t−s) + (B(Ys ) − B(Zs ))+ p ds + +e L (;H S(U0 ,Hδ )) 2 0
for all t ∈ [0, T ] and all Y , Z ∈ H . Lemma A.12 and Corollary A.32 therefore show + + +( Y )t − ( Z)t + p L (;Hδ ) t (t − s)(α−δ) Ys − Zs Lp (;Hδ ) ds ≤ eηT F Lip(Hδ ,Hα ) 0
+pe
ηT
B Lip(Hδ ,H S(U0 ,Hβ ))
t
(t − s)
(2β−2δ)
0
Ys − Zs 2Lp (;Hδ ) ds
12
for all t ∈ [0, T ] and all Y , Z ∈ H . Therefore, (Y ) − (Z) H,r ≤ eηT F Lip(Hδ ,Hα )
0
t
(t − s)(α−δ) er(t−s) ds Y − Z H,r
+ p eηT B Lip(Hδ ,H S(U0 ,Hβ ))
t
(t − s)(2β−2δ) e2r(t−s) ds
12
0
Y − Z H,r
and hence (Y ) − (Z) H,r ≤ p eηT max F Lip(Hδ ,Hα ) , B Lip(Hδ ,H S(U0 ,Hβ )) Y − Z H,r 1 t
·
t
s (α−δ) ers ds +
0
s (2β−2δ) e2rs ds
2
0
for all Y , Z ∈ H and all r ∈ R. Moreover, note that T 12 T s (α−δ) ers ds + s (2β−2δ) e2rs ds = 0. lim r→−∞
(A.67)
0
(A.68)
0
Combining then shows that : H → H is a contraction from (A.67) and (A.68) H , · H,r to H , · H,r for a sufficiently small r ∈ (−∞, 0). Hence there exists a unique X : [0, T ] × → Hδ ∈ H which fulfills (X) = X, i.e., which fulfills (5.19).
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206
Appendix A. Regularity Estimates for SPDEs
A.6
Regularity
Now Theorem 5.1 is proved by using the results in Sections A.2–A.5. Proof. Proposition 10 shows the existence of an up-to-modifications unique prep dictable stochastic process Y : [0, T ]× → Hδ which satisfies supt∈[0,T ] E[ Yt Hδ ] < ∞ and (5.19). In the next step Lemma A.4 implies (A.69) eAt ξ , t ∈ [0, T ], ∈ ∩r∈(−∞,γ ] C min(γ −r,1) ([0, T ], Lp (; Hr )) . Moreover, Corollary A.16 shows
t
0
eA(t−s) F (Ys ) ds, t ∈ [0, T ], ∈ ∩r∈[α,α+1) ∩u∈[0,1+α−r) C u ([0, T ], Lp (; Hr )) = ∩r∈(−∞,α+1) ∩u∈[0,min(1+α−r,1)) C u ([0, T ], Lp (; Hr )) .
(A.70)
Additionally, Proposition 9 and Lemmas A.28 and A.29 yield
t 0
eA(t−s) B(Ys ) dWs , t ∈ [0, T ], ∈ ∩r∈(−∞,β+ 1 ) ∩u∈[0, 1 ]∩[0, 1 +β−r) C u ([0, T ], Lp (; Hr )) . 2
2
2
(A.71)
Combining (A.69)–(A.71) then shows
eAt ξ +
t
eA(t−s) F (Ys ) ds +
0
t
eA(t−s) B(Ys ) dWs
0
t∈[0,T ]
∈ ∩r∈(−∞,γ ]∩(−∞,min(α+1,β+ 1 )) ∩u∈[0,min(γ −r, 1 )]∩[0,min(α+1,β+ 1 )−r) 2 2 2 p u C ([0, T ], L (; Hr )) . (A.72) Now a distinction between the two cases γ < min(α + 1, β + 12 ) and γ = min(α + 1, β + 12 ) is made. First, consider the case γ < min(α + 1, β + 12 ). Relation (A.72) then reduces to
eAt ξ + 0
t
eA(t−s) F (Ys ) ds +
t
eA(t−s) B(Ys ) dWs
0
∈ ∩r∈(−∞,γ ] C
t∈[0,T ]
min(γ −r, 12 )
([0, T ], Lp (; Hr )) .
(A.73)
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A.6. Regularity
207
In the next step, (A.73) and Proposition 3.6 in Da Prato and Zabczyk [24] imply the existence of a predictable modification X : [0, T ] × → Hγ of Y : [0, T ] × → Hδ and, in particular, 1 (A.74) X ∈ ∩r∈(−∞,γ ] C min(γ −r, 2 ) ([0, T ], Lp (; Hr )) . This proves Theorem 5.1 in the case γ < min(α + 1, β + 12 ). Second, consider the case γ = min(α + 1, β + 12 ). Note that (A.72) then shows Y ∈ ∩u∈[0,min(γ −δ, 1 )) C u ([0, T ], Lp (; Hδ )) . 2
(A.75)
Corollary A.21, Lemma A.23, and Corollary A.27 therefore imply
t 0
eA(t−s) F (Ys ) ds, t ∈ [0, T ], ∈ ∩r∈(−∞,α+1] C min(1+α−r,1) ([0, T ], Lp (; Hr )) .
(A.76)
Moreover, (A.75) and Corollaries A.35, A.37, and A.42 give
t 0
eA(t−s) B(Ys ) dWs , t ∈ [0, T ], 1 1 ∈ ∩r∈(−∞,β+ 1 ] C min( 2 +β−r, 2 ) ([0, T ], Lp (; Hr )) . 2
(A.77)
Combining (A.69), (A.76), and (A.77) then yields
eAt ξ + 0
t
eA(t−s) F (Ys ) ds +
t
eA(t−s) B(Ys ) dWs
0
∈ ∩r∈(−∞,γ ] C
t∈[0,T ]
min(γ −r, 12 )
([0, T ], Lp (; Hr )) .
(A.78)
Proposition 3.6 in Da Prato and Zabczyk [24] therefore implies the existence of a predictable modification X : [0, T ] × → Hγ of Y : [0, T ] × → Hδ and, in particular, 1 X ∈ ∩r∈(−∞,γ ] C min(γ −r, 2 ) ([0, T ], Lp (; Hr )) . (A.79) This shows Theorem 5.1 in the case γ = min(α +1, β + 12 ). The proof of Theorem 5.1 is thus complete.
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Index Burkholder–Davis–Gundy-type inequalities, 175 Cahn–Hilliard–Cook stochastic PDE, 67 convergence global, 157 local, 157 pathwise, 35 strong order, 27 weak, 47 weak order, 27 Cox–Ingersoll–Ross stochastic ODE, 37, 46 cylindrical Q-Wiener process, 57
Hilbert space random variable, 56 stochastic process, 56 Hilbert–Schmidt operator, 55 integral operator, 98, 136 Laplacian semigroup, 91 linear growth condition, 26 linear-implicit Crank–Nicolson approximation, 118 linear-implicit Crank–Nicolson scheme, 157 linear-implicit Euler approximation, 118 linear-implicit Euler scheme, 157 Milstein scheme, 156 Minkowski’s integral inequality, 174 multi-index, 14
discretization error global, 3 local, 3 of RODE–Taylor schemes, 17
Nemytskii operator, 92 Euler scheme, 3 averaged, 12 Euler–Maruyama scheme, 27 exponential Euler scheme, 123, 155
ODE integral representation, 4 Taylor expansion, 1 Taylor scheme, 3
Feller diffusion with logistic growth stochastic ODE, 37 Fisher–Wright stochastic ODE, 37, 47
random ODE (RODE), 9 Taylor scheme, 15 Taylor-like expansion, 13 with Brownian motion, 22 with fractional Brownian motion, 24 random PDE, 53
Ginzburg–Landau stochastic ODE, 37 Hasminskii condition, 27 Heun scheme, 4 219
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220 stochastic convolution, 93 stochastic heat equation, 133, 151 stochastic ODE (SODE) existence and uniqueness, 26 finite dimensional, 60, 152 general stochastic Taylor expansions, 31 Itô stochastic differential equation, 25 Itô–Taylor expansion, 29 Itô–Taylor numerical scheme, 32 modified Itô–Taylor scheme, 45 on restricted regions, 44 without uniformly bounded coefficients, 43 stochastic PDE (SPDE), 53 autonomization, 88 convergence, 74 existence and uniqueness, 59 exponential Euler scheme, 73 fourth order, 65 higher order Taylor expansion, 139 mild form, 55 mild solution, 54 numerical method, 69 numerical scheme, 154 numerical simulation, 158 regularity, 59 Runge–Kutta scheme, 124 second order, 60 space–time white noise, 61 Taylor approximation nonlinear equation, 116 smoother noise, 121 space–time white noise, 114 Taylor expansion, 96, 135 abstract examples, 104, 149 trace–class noise, 63 with additive noise, 79 with multiplicative noise, 127 with time-dependent coefficients, 67
Index stochastic trees and woods, 140 construction, 142 order, 146 subtrees, 145 Taylor expansion, 147 Volterra–Lotka stochastic ODE, 46 Wiener process, 56
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