Peter E. Kloeden Eckhard Platen Numerical Solution of Stochastic Differential Equations
Peter E. Kloeden School of Computing and Mathematics, Deakin Universit y Geelong 3217, Victoria, Australia Eckhard Platen Managing Editors A 4 1A03 Institute of Advanced Studies, SMS, CFM, Australian National University Canberra, ACT 0200, Australia gttladt4clt-banherafer. ;R~~ff(s^~, Ott79-tieSi331-Mliitahv k 9sf oz I. Karatzas Departments of Mathematics and Statistics, Columbia University New York, NY 10027, US A M. Yo r Laboratoire de Probabilites, University Pierre et Marie Curie 4 Place Jussieu, Tour 56, F-75230 Paris Cedex, Franc e Second Corrected Printing 199 5 Mathematics Subject Classification (1991) : 60H10, 65C0 5 ISBN 3-540-54062-8 Springer-Verlag Berlin Heidelberg New Yor k ISBN 0-387-54062-8 Springer-Verlag New York Berlin Heidelber g Library of Congress Cataloging-in-Publication Data. Kloeden, Peter E. Numerical solution of stochastic differential equations/peter E. Kloeden, Eckhard Platen. p. cm. - (Applications of mathematics ; 23) "Second corrected printing" - T. p. verso. Includes bibliographical references (p. - ) and index. ISBN 0-387-54062 (acid-free). - ISBN 3-540-54062-8 (acid-free ) 1. Stochastic differential equations - Numerical solutions. I. Platen. Eckhard. H. Title. III. Series. QA274.23.K56 1992b 519.2.- dc20 95-463 CIP This work is subject to copyright. All rights are reserved, whether the whole or part of the materia l is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplicatio n of this publication or parts thereof is permitted only under the provisions of the German Copyrigh t Law of September 9, 1965, in its current version, and permission for use must always be obtaine d from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1992 Printed in the United States of America SPIN : 10502282 41/3143-5 4 3 2 1 0 - Printed on acid-free paper
Suggestions for the Reader xvii Basic Notation xxi Brief Survey of Stochastic Numerical Methods xxiii Part I. Preliminarie s Chapter 1. Probability and Statistics 1 1.1 Probabilities and Events 1 1.2 Random Variables and Distributions 5 1.3 Random Number Generators 1 1 1.4 Moments 1 4 1.5 Convergence of Random Sequences 2 2 1.6 Basic Ideas About Stochastic Processes 2 6 1.7 Diffusion Processes 34 1.8 Wiener Processes and White Noise 40 1.9 Statistical Tests and Estimation 44 Chapter 2. Probability and Stochastic Processes 5 1 2.1 Aspects of Measure and Probability Theory 5 1 2.2 Integration and Expectations 5 5 2.3 Stochastic Processes 6 3 2.4 Diffusion and Wiener Processes 6 8 Part II. Stochastic Differential Equations Chapter 3. Ito Stochastic Calculus 7 5 3.1 Introduction 7 5 3.2 The Ito Stochastic Integral 8 1 3.3 The Ito Formula 9 0 3.4 Vector Valued Ito Integrals 9 6 3.5 Other Stochastic Integrals 9 9 Chapter 4. Stochastic Differential Equations 10 3 4.1 Introduction 10 3 4.2 Linear Stochastic Differential Equations 110
4.3 Reducible Stochastic Differential Equations 11 3 4.4 Some Explicitly Solvable Equations 117 4.5 The Existence and Uniqueness of Strong Solutions 12 7 4.6 Strong Solutions as Diffusion Processes 14 1 4.7 Diffusion Processes as Weak Solutions 144 4.8 Vector Stochastic Differential Equations 14 8 4.9 Stratonovich Stochastic Differential Equations 154 Chapter 5. Stochastic Taylor Expansions 16 1 5.1 Introduction 16 1 5.2 Multiple Stochastic Integrals 16 7 5.3 Coefficient Functions 17 7 5.4 Hierarchical and Remainder Sets 18 0 5.5 Ito-Taylor Expansions 18 1 5.6 Stratonovich-Taylor Expansions 18 7 5.7 Moments of Multiple Ito Integrals 19 0 5.8 Strong Approximation of Multiple Stochastic Integrals 19 8 5.9 Strong Convergence of Truncated Ito-Taylor Expansions 20 6 5.10 Strong Convergenc e of Truncated Stratonovich-Taylor Expansions 21 0 5.11 Weak Convergence of Truncated Ito-Taylor Expansions 21 1 5.12 Weak Approximations of Multiple Ito Integrals 22 1 Part III. Applications of Stochastic Differential Equations Chapter 6. Modelling with Stochasti c Differential Equations 227 6.1 Ito Versus Stratonovich 22 7 6.2 Diffusion Limits of Markov Chains 22 9 6.3 Stochastic Stability 23 2 6.4 Parametric Estimation 24 1 6.5 Optimal Stochastic Control 24 4 6.6 Filtering 24 8 Chapter 7. Applications of Stochastic Differential Equations 25 3 7.1 Population Dynamics, Protein Kinetics and Genetics 25 3 7.2 Experimental Psychology and Neuronal Activity 25 6 7.3 Investment Finance and Option Pricing 25 7 7.4 Turbulent Diffusion and Radio-Astronomy 25 9 7.5 Helicopter Rotor and Satellite Orbit Stability 26 1 7.6 Biological Waste Treatment, Hydrology and Air Quality 26 3 7.7 Seismology and Structural Mechanics 26 6 7.8 Fatigue Cracking, Optical Bistability and Nemantic Liquid Crystals 26 9 7.9 Blood Clotting Dynamics and Cellular Energetics 271
7.10 Josephson Tunneling Junctions, Communications and Stochastic Annealing 273 Part IV. Time Discrete Approximation s Chapter 8. Time Discrete Approximatio n of Deterministic Differential Equations 27 7 8.1 Introduction 27 7 8.2 Taylor Approximations and Higher Order Methods 28 6 8.3 Consistency, Convergence and Stability 292 8.4 Roundoff Error 30 1 Chapter 9. Introduction to Stochastic Time Discrete Approximation 30 5 9.1 The Euler Approximation 30 5 9.2 Example of a Time Discrete Simulation 30 7 9.3 Pathwise Approximations 31 1 9.4 Approximation of Moments 31 6 9.5 General Time Discretizations and Approximations 32 1 9.6 Strong Convergence and Consistency 323 9.7 Weak Convergence and Consistency 32 6 9.8 Numerical Stability 33 1 Part V. Strong Approximation s Chapter 10. Strong Taylor Approximations 33 9 10.1 Introduction 33 9 10.2 The Euler Scheme 34 0 10.3 The Milstein Scheme 34 5 10.4 The Order 1.5 Strong Taylor Scheme 35 1 10.5 The Order 2.0 Strong Taylor Scheme 356 10.6 General Strong Ito-Taylor Approximations 360 10.7 General Strong Stratonovich-Taylor Approximations 36 5 10.8 A Lemma on Multiple Ito Integrals 36 9 Chapter 11. Explicit Strong Approximations 37 3 11.1 Explicit Order 1.0 Strong Schemes 37 3 11.2 Explicit Order 1.5 Strong Schemes 37 8 11.3 Explicit Order 2.0 Strong Schemes 38 3 11.4 Multistep Schemes 38 5 11.5 General Strong Schemes 390
Chapter 12. Implicit Strong Approximations 39 5 12.1 Introduction 39 5 12.2 Implicit Strong Taylor Approximations 39 6 12.3 Implicit Strong Runge-Kutta Approximations 40 6 12.4 Implicit Two-Step Strong Approximations 41 1 12.5 A-Stability of Strong One-Step Schemes 41 7 12.6 Convergence Proofs 420 Chapter 13. Selected Applications of Strong Approximations 42 7 13.1 Direct Simulation of Trajectories 42 7 13.2 Testing Parametric Estimators 43 5 13.3 Discrete Approximations for Markov Chain Filters 44 2 13.4 Asymptotically Efficient Schemes 45 3 Part VI. Weak Approximation s Chapter 14. Weak Taylor Approximations 457 14.1 The Euler Scheme 457 14.2 The Order 2.0 Weak Taylor Scheme 464 14.3 The Order 3.0 Weak Taylor Scheme 46 8 14.4 The Order 4.0 Weak Taylor Scheme 47 0 14.5 General Weak Taylor Approximations 47 2 14.6 Leading Error Coefficients 48 0 Chapter 15. Explicit and Implicit Weak Approximations 48 5 15.1 Explicit Order 2.0 Weak Schemes 48 5 15.2 Explicit Order 3.0 Weak Schemes 48 8 15.3 Extrapolation Methods 49 1 15.4 Implicit Weak Approximations 495 15.5 Predictor-Corrector Methods 50 1 15.6 Convergence of Weak Schemes 506 Chapter 16. Variance Reduction Methods 51 1 16.1 Introduction 51 1 16.2 The Measure Transformation Method 51 3 16.3 Variance Reduced Estimators 51 6 16.4 Unbiased Estimators 52 2 Chapter 17. Selected Applications of Weak Approximations 52 9 17.1 Evaluation of Functional Integrals 529 17.2 Approximation of Invariant Measures 540 17.3 Approximation of Lyapunov Exponents 545
Solutions of Exercises 549 Bibliographical Notes 58 7 References 59 7 Index 625