Robert S. Liptser Albert N. Shiryaev Statistics of Random Processes II. Applications Translated by A. B. Aries Translation Editor: Stephen S. Wilson Second, Revised and Expanded Edition Springer
Authors Robert S. Liptser Tel Aviv University Department of Electrical Engineering Systems Ramat Aviv, P.O. Box 39040 69978 Tel Aviv, Israel e-mail: liptser@eng.tau.ac.il Albert N. Shiryaev Steklov Mathematical Institute, Russian Academy of Sciences Gubkina 8, 117966 Moscow, Russia e-mail: shiryaev@ genesis.mi.ras.ru Translation Editor Stephen S. Wilson 31 Harp Hill Cheltenham, Gloucestershire, GL52 6PY, United Kingdom e-mail: techtrans@cwcom.net Managing Editors I. Karatzas Departments of Mathematics and Statistics Columbia University New York, NY 10027, USA M.Yor CNRS, Laboratoire de Probabilites Universite Pierre et Marie Curie 4 Place Jussieu, Tour 56 F-75230 Paris Cedex os, France Mathematics Subject Classification (2ooo): 6oGxx, 6oHxx, 6oJxx, 62Lxx, 62Mxx, 62Nxx, 93Exx, 94A05 Title of the Russian Original Edition: Statistika slucha!nykh protsessov. Nauka, Moscow, 1974 Cover pattern by courtesy of Rick Durrett (Cornell University, Ithaca) Library of Congress Cataloging-in-Publication Data Liptser, R. Sh. (Robert Shevilevich) [Statistika sluchainykh protsessov. English] Statistics of random processes I Robert Liptser, Albert N. Shiryaev; translated by A. B. Aries; translation editor, Stephen S. Wilson.- 2nd, rev. and expanded ed. p. em.- (Applications of mathematics, ISSN 0172-4568; 5-6) Includes bibliographical references and indexes. Contents: 1. General theory- 2. Applications. ISBN 978-3-642-08365-5 ISBN 978-3-662-10028-8 (ebook) DOI 10.1007/978-3-662-10028-8 1. Stochastic processes. 2. Mathematical statistics. I. Shiriaev, Al'bert Nikolaevich. II. Title III. Series QA274.L5713 2ooo 519.2'3-dc21 ISSN 0172-4568 ISBN 978-3-642-08365-5 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, re~itation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 2001 Originally published by Springer-Verlag Berlin Heidelberg New York in 2001 Softcover reprint of the hardcover 1st edition 2001 Printed on acid-free paper SPIN: 10640137 41/3142CK - 5 4 3 2 1 o
Table of Contents Preface to the Second Edition................................ V 11. Conditionally Gaussian Processes......................... 1 11.1 Assumptions and Formulation of the Theorem of Conditional Gaussian Behavior...................................... 1 11.2 Auxiliary Lemmas... 3 11.3 Proof of the Theorem of Conditional Gaussian Behavior... 9 12. Optimal Nonlinear Filtering: Interpolation and Extrapolation of Components of Conditionally Gaussian Processes. 17 12.1 Optimal Filtering Equations............................. 17 12.2 Uniqueness of Solutions of Filtering Equations: Equivalence of a-algebras Ftt; and Ftt;o,W............................. 25 12.3 Optimal Filtering Equations in Several Dimensions......... 32 12.4 Interpolation of Conditionally Gaussian Processes.......... 38 12.5 Optimal Extrapolation Equations......................... 49 13. Conditionally Gaussian Sequences: Filtering and Related Problems................................................. 55 13.1 Theorem on Normal Correlation... 55 13.2 Recursive Filtering Equations for Conditionally Gaussian Sequences..................................... 67 13.3 Forward and Backward Interpolation Equations............ 77 13.4 Recursive Equations of Optimal Extrapolation............. 88 13.5 Examples.............................................. 91 14. Application of Filtering Equations to Problems of Statistics of Random Sequences..................................... 99 14.1 Optimal Linear Filtering of Stationary Sequences with Rational Spectra................................... 99 14.2 Maximum Likelihood Estimates for Coefficients of Linear Regression... 107 14.3 A Control Problem with Incomplete Data (Linear System with Quadratic Performance Index)... 113
X Table of Contents 14.4 Asymptotic Properties of the Optimal Linear Filter... 121 14.5 Recursive Computation of the Best Approximate Solutions (Pseudo-solutions) of Linear Algebraic Systems... 132 14.6 Kalman Filter under Wrong Initial Conditions... 138 15. Linear Estimation of Random Processes... 145 15.1 Wide-Sense Wiener Processes... 145 15.2 Optimal Linear Filtering for some Classes of Nonstationary Processes... 157 15.3 Linear Estimation of Wide-Sense Stationary Random Processes with Rational Spectra... 161 15.4 Comparison of Optimal Linear and Nonlinear Estimates... 170 16. Application of Optimal Nonlinear Filtering Equations to some Problems in Control Theory and Estimation Theory 177 16.1 An Optimal Control Problem Using Incomplete Data... 177 16.2 Asymptotic Properties of Kalman-Bucy Filters... 184 16.3 Computation of Mutual Information and Channel Capacity of a Gaussian Channel with Feedback... 190 16.4 Optimal Coding and Decoding for Transmission of a Gaussian Signal Through a Channel with Noiseless Feedback... 195 16.5 Asymptotic Properties of the Linear Filter under Wrong Initial Conditions... 214 17. Parameter Estimation and Testing of Statistical Hypotheses for Diffusion-Type Processes... 219 17.1 Maximum Likelihood Method for Coefficients of Linear Regression... 219 17.2 Parameter Estimation of the Drift Coefficient for Diffusion-Type Processes............................. 225 17.3 Parameter Estimation of the Drift Coefficient for a One-Dimensional Gaussian Process... 230 17.4 Two-Dimensional Gaussian Markov Processes: Parameter Estimation................................... 236 17.5 Sequential Maximum Likelihood Estimates... 244 17.6 Sequential Testing of Two Simple Hypotheses for Ito Processes........................................ 248 17.7 Some Applications to Stochastic Approximation... 256 18. Random Point Processes: Stieltjes Stochastic Integrals... 261 18.1 Point Processes and their Compensators... 261 18.2 Minimal Representation of a Point Process: Processes of the Poisson Type.......................................... 269 18.3 Construction of Point Processes with Given Compensators: Theorems on Existence and Uniqueness... 277
Table of Contents XI 18.4 Stieltjes Stochastic Integrals... 286 18.5 The Structure of Point Processes with Deterministic and Continuous Compensators........................... 305 19. The Structure of Local Martingales, Absolute Continuity of Measures for Point Processes, and Filtering... 309 19.1 The Structure of Local Martingales... 309 19.2 Nonnegative Supermartingale: Analog of Girsanov's Theorem 315 19.3 Optimal Filtering from the Observations of Point Processes.. 325 19.4 The Necessary and Sufficient Conditions for Absolute Continuity of the Measures Corresponding to Point Processes..... 336 19.5 Calculation of the Mutual Information and the Cramer-Rao- Wolfowitz Inequality (the Point Observations)... 345 20. Asymptotically Optimal Filtering... 355 20.1 Total Variation Norm Convergence and Filtering... 355 20.2 Robust Diffusion Approximation for Filtering... 371 Bibliography.................................................. 383 Index... 399