Numerical Integration of Stochastic Differential Equations
Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 313
Numerical Integration of Stochastic Differential Equations by G.N. Milstein Department of Mathematics, Ural State University, Ekatarinburg, Russia Springer Science+Business Media, B.Y.
Library of Congress Cataloging-in-Publication Data Mil 'shteln, G. N. <Grigoril Nolkhovich) [Ch i 5 1 ennoe i ntegr i rovan i e stokhast i chesk i kh d i fferenfs i a 1 'nykh uravnenil. Englishl Numerical integrat ion of stochastic differential equations / by G.N. Mi lstein. p. cm. -- (Mathematics and its applications ; v. 313) Includes bibliographical references and index. 1. Stochastic differential equations--numerical solutions. 2. Wiener integrals. 1. Title. II. Ser ies: Mathematics and its applications (Kluwer Academic Publishers) ; v. 313. OA274.23.M5513 1995 519.2--dc20 94-37674 ISBN 978-90-481-4487-7 DOI 10.1007/978-94-015-8455-5 ISBN 978-94-015-8455-5 (ebook) This is a revised and updated translation of the original Russian work Numericallntegration of Stochastic DijJerential Equations, Ural State University Press, Sverdlovsk 1988 Printed on acid-free paper AH Rights Reserved 1995 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1995 Softcover reprint ofthe hardcover 18t edition 1995 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
Contents Introduction 1 Chapter 1. Mean-square approximation of solutions of systems of stochastic differential equations 11 1. Theorem on the order of convergence (theorem on the reiat ion between approximation on a finite interval and one-step approximation) 11 1.1. Statement of the theorem 11 1.2. Lemmas 12 1.3. Proof of Theorem 1.1 15 1.4. Discussion 16 1.5. Equations in the sense of Stratonovich 17 1.6. Euler's method 18 1. 7. Examples 21 2. Methods based on an analog of Taylor expansion of the solution 23 2.1. Taylor expansion of the solution for systems of ordinary differential equations 23 2.2. Expansion of the solution of a system of stochastic differential equations (Wagner-Platen expansion) 25 2.3. Construction of implicit methods 35 3. Explicit and implicit methods of order 3/2 for systems with additive noises 37 3.1. Explicit methods based on Taylor-type expansion 37 3.2. Implicit methods based on Taylor-type expansion 41 3.3. Stiff systems of stochastic differential equations with additive noises. A-stability 45 3.4. Runge-Kutta type methods (implicit and explicit) 49 3.5. Two-step difference methods 52 4. Optimal integrat ion methods for linear systems with additive noises 56 4.1. Statement of the problem on numerical modeling of the Kalman-Bucy filter and on the optimal filter with discrete arrival of information 57
vi CONTENTS 4.2. Discretisation of the system (4.1), (4.2) 4.3. An optimal filter with discrete arrival of information 4.4. An optimal integration method of the first order of accuracy 5. A strengthening of the main convergence theorem 5.1. The theorem on convergence in the mean of order 4 5.2. Construction of an auxiliary submartingale 5.3. The strenghtened convergence theorem Chapter 2. Modeling of Ita integrals 75 6. Modeling Ita integrals depending on a single noise 75 6.1. Auxiliary formulas for single Itâ integrals 76 6.2. Reduction of repeated Itâ integrals to single Itâ integrals 79 6.3. Exact modeling ofthe random variables w(h), fo h w(o) do, and f~ w 2 (O) do 82 6.4. Approximate modeling of the random variables w(h), f~w(o) do, and ~~~~ M 7. Modeling Ita integrals depending on several noises 90 7.1. Exact methods for modeling the random variables in a method of order 1 in the case of two noises 90 7.2. Use of the numerical integration of special linear stochastic systems for modeling Itâ integrals 91 7.3. Modeling the!tâ integrals f~wi(s) dwj(s), i,j = 1,..., q 92 Chapter 3. Weak approximation of solutions of systems of stochastic differential equations 101 8. One-step approximation 101 8.1. Initial assumptions and notations. Lemmas on properties ofremainders and Itâ integrals lo 1 8.2. Forming one-step approximations of third order of accuracy 106 8.3. Theorem on a method with one-step approximation of third order of accuracy 109 8.4. Modeling of random variables and constructive formation of a one-step approximation of third order of accuracy 110 9. The main theorem on convergence of weak approximations and methods of order of accuracy two 112 9.1. A theorem on the relation between one-step approximation and approximation on a finite interval 112 9.2. Theorem on a method of order of accuracy two 115 9.3. Runge-Kutta type methods 116 10. A method of order of accuracy three for systems with additive noises 118 lo.1. Main lemmas 119 lo.2. Construction of a one-step approximation of order of accuracy four, and of a method of order three 122 59 60 61 63 63 70 72
CONTENTS vii 11. An implicit method 12. Reducing the error of the Monte-Carlo method Chapter 4. Application of the numerical integration of stochastic equations for the Monte-Carlo computation of Wiener integrals 135 13. Methods of order of accuracy two for computing Wiener integrals of functionals of integral type 135 13.1. Statement of the problem 135 13.2. Taylor expansions of mathematical expectations 136 13.3. The trapezium method 138 13.4. The rectangle method and other methods 140 13.5. Generalisation of the trapezium formula to Wiener integrals of functionals of general form 141 14. Methods of order of accuracy four for computing Wiener integrals of functionals of exponential type 145 14.1. Introduction 145 14.2. A fourth-order Runge-Kutta method for integrating the system (14.2) 147 14.3. Reducing variances 152 14.4. Examples of numerical experiments 156 Bibliography 165 Index 169 127 130