Numerical Integration of Stochastic Differential Equations

Transcription

1 Numerical Integration of Stochastic Differential Equations

2 Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 313

3 Numerical Integration of Stochastic Differential Equations by G.N. Milstein Department of Mathematics, Ural State University, Ekatarinburg, Russia Springer Science+Business Media, B.Y.

4 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 OA M dc ISBN DOI / ISBN (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.

5 Contents Introduction 1 Chapter 1. Mean-square approximation of solutions of systems of stochastic differential equations Theorem on the order of convergence (theorem on the reiat ion between approximation on a finite interval and one-step approximation) Statement of the theorem Lemmas Proof of Theorem Discussion Equations in the sense of Stratonovich Euler's method Examples Methods based on an analog of Taylor expansion of the solution Taylor expansion of the solution for systems of ordinary differential equations Expansion of the solution of a system of stochastic differential equations (Wagner-Platen expansion) Construction of implicit methods Explicit and implicit methods of order 3/2 for systems with additive noises Explicit methods based on Taylor-type expansion Implicit methods based on Taylor-type expansion Stiff systems of stochastic differential equations with additive noises. A-stability Runge-Kutta type methods (implicit and explicit) Two-step difference methods Optimal integrat ion methods for linear systems with additive noises Statement of the problem on numerical modeling of the Kalman-Bucy filter and on the optimal filter with discrete arrival of information 57

6 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 Construction of an auxiliary submartingale 5.3. The strenghtened convergence theorem Chapter 2. Modeling of Ita integrals Modeling Ita integrals depending on a single noise Auxiliary formulas for single Itâ integrals Reduction of repeated Itâ integrals to single Itâ integrals Exact modeling ofthe random variables w(h), fo h w(o) do, and f~ w 2 (O) do Approximate modeling of the random variables w(h), f~w(o) do, and ~~~~ M 7. Modeling Ita integrals depending on several noises Exact methods for modeling the random variables in a method of order 1 in the case of two noises Use of the numerical integration of special linear stochastic systems for modeling Itâ integrals 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 One-step approximation Initial assumptions and notations. Lemmas on properties ofremainders and Itâ integrals lo Forming one-step approximations of third order of accuracy Theorem on a method with one-step approximation of third order of accuracy Modeling of random variables and constructive formation of a one-step approximation of third order of accuracy The main theorem on convergence of weak approximations and methods of order of accuracy two A theorem on the relation between one-step approximation and approximation on a finite interval Theorem on a method of order of accuracy two Runge-Kutta type methods 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

7 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 Methods of order of accuracy two for computing Wiener integrals of functionals of integral type Statement of the problem Taylor expansions of mathematical expectations The trapezium method The rectangle method and other methods Generalisation of the trapezium formula to Wiener integrals of functionals of general form Methods of order of accuracy four for computing Wiener integrals of functionals of exponential type Introduction A fourth-order Runge-Kutta method for integrating the system (14.2) Reducing variances Examples of numerical experiments 156 Bibliography 165 Index