Peter E. Kloeden Eckhard Platen. Numerical Solution of Stochastic Differential Equations

Transcription

1 Peter E. Kloeden Eckhard Platen Numerical Solution of Stochastic Differential Equations

2 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 Paris Cedex, Franc e Second Corrected Printing Mathematics Subject Classification (1991) : 60H10, 65C0 5 ISBN Springer-Verlag Berlin Heidelberg New Yor k ISBN 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 (acid-free). - ISBN (acid-free ) 1. Stochastic differential equations - Numerical solutions. I. Platen. Eckhard. H. Title. III. Series. QA K b dc 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 : / Printed on acid-free paper

3 Suggestions for the Reader xvii Basic Notation xxi Brief Survey of Stochastic Numerical Methods xxiii Part I. Preliminarie s Chapter 1. Probability and Statistics Probabilities and Events Random Variables and Distributions Random Number Generators Moments Convergence of Random Sequences Basic Ideas About Stochastic Processes Diffusion Processes Wiener Processes and White Noise Statistical Tests and Estimation 44 Chapter 2. Probability and Stochastic Processes Aspects of Measure and Probability Theory Integration and Expectations Stochastic Processes Diffusion and Wiener Processes 6 8 Part II. Stochastic Differential Equations Chapter 3. Ito Stochastic Calculus Introduction The Ito Stochastic Integral The Ito Formula Vector Valued Ito Integrals Other Stochastic Integrals 9 9 Chapter 4. Stochastic Differential Equations Introduction Linear Stochastic Differential Equations 110

4 4.3 Reducible Stochastic Differential Equations Some Explicitly Solvable Equations The Existence and Uniqueness of Strong Solutions Strong Solutions as Diffusion Processes Diffusion Processes as Weak Solutions Vector Stochastic Differential Equations Stratonovich Stochastic Differential Equations 154 Chapter 5. Stochastic Taylor Expansions Introduction Multiple Stochastic Integrals Coefficient Functions Hierarchical and Remainder Sets Ito-Taylor Expansions Stratonovich-Taylor Expansions Moments of Multiple Ito Integrals Strong Approximation of Multiple Stochastic Integrals Strong Convergence of Truncated Ito-Taylor Expansions Strong Convergenc e of Truncated Stratonovich-Taylor Expansions Weak Convergence of Truncated Ito-Taylor Expansions Weak Approximations of Multiple Ito Integrals 22 1 Part III. Applications of Stochastic Differential Equations Chapter 6. Modelling with Stochasti c Differential Equations Ito Versus Stratonovich Diffusion Limits of Markov Chains Stochastic Stability Parametric Estimation Optimal Stochastic Control Filtering 24 8 Chapter 7. Applications of Stochastic Differential Equations Population Dynamics, Protein Kinetics and Genetics Experimental Psychology and Neuronal Activity Investment Finance and Option Pricing Turbulent Diffusion and Radio-Astronomy Helicopter Rotor and Satellite Orbit Stability Biological Waste Treatment, Hydrology and Air Quality Seismology and Structural Mechanics Fatigue Cracking, Optical Bistability and Nemantic Liquid Crystals Blood Clotting Dynamics and Cellular Energetics 271

5 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 Introduction Taylor Approximations and Higher Order Methods Consistency, Convergence and Stability Roundoff Error 30 1 Chapter 9. Introduction to Stochastic Time Discrete Approximation The Euler Approximation Example of a Time Discrete Simulation Pathwise Approximations Approximation of Moments General Time Discretizations and Approximations Strong Convergence and Consistency Weak Convergence and Consistency Numerical Stability 33 1 Part V. Strong Approximation s Chapter 10. Strong Taylor Approximations Introduction The Euler Scheme The Milstein Scheme The Order 1.5 Strong Taylor Scheme The Order 2.0 Strong Taylor Scheme General Strong Ito-Taylor Approximations General Strong Stratonovich-Taylor Approximations A Lemma on Multiple Ito Integrals 36 9 Chapter 11. Explicit Strong Approximations Explicit Order 1.0 Strong Schemes Explicit Order 1.5 Strong Schemes Explicit Order 2.0 Strong Schemes Multistep Schemes General Strong Schemes 390

6 Chapter 12. Implicit Strong Approximations Introduction Implicit Strong Taylor Approximations Implicit Strong Runge-Kutta Approximations Implicit Two-Step Strong Approximations A-Stability of Strong One-Step Schemes Convergence Proofs 420 Chapter 13. Selected Applications of Strong Approximations Direct Simulation of Trajectories Testing Parametric Estimators Discrete Approximations for Markov Chain Filters Asymptotically Efficient Schemes 45 3 Part VI. Weak Approximation s Chapter 14. Weak Taylor Approximations The Euler Scheme The Order 2.0 Weak Taylor Scheme The Order 3.0 Weak Taylor Scheme The Order 4.0 Weak Taylor Scheme General Weak Taylor Approximations Leading Error Coefficients 48 0 Chapter 15. Explicit and Implicit Weak Approximations Explicit Order 2.0 Weak Schemes Explicit Order 3.0 Weak Schemes Extrapolation Methods Implicit Weak Approximations Predictor-Corrector Methods Convergence of Weak Schemes 506 Chapter 16. Variance Reduction Methods Introduction The Measure Transformation Method Variance Reduced Estimators Unbiased Estimators 52 2 Chapter 17. Selected Applications of Weak Approximations Evaluation of Functional Integrals Approximation of Invariant Measures Approximation of Lyapunov Exponents 545

7 Solutions of Exercises 549 Bibliographical Notes 58 7 References 59 7 Index 625