Statistics of Random Processes

Transcription

1 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

2 Authors Robert S. Liptser Tel Aviv University Department of Electrical Engineering Systems Ramat Aviv, P.O. Box Tel Aviv, Israel Albert N. Shiryaev Steklov Mathematical Institute, Russian Academy of Sciences Gubkina 8, Moscow, Russia genesis.mi.ras.ru Translation Editor Stephen S. Wilson 31 Harp Hill Cheltenham, Gloucestershire, GL52 6PY, United Kingdom 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 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 ; 5-6) Includes bibliographical references and indexes. Contents: 1. General theory- 2. Applications. ISBN ISBN (ebook) DOI / Stochastic processes. 2. Mathematical statistics. I. Shiriaev, Al'bert Nikolaevich. II. Title III. Series QA274.L5713 2ooo 519.2'3-dc21 ISSN ISBN 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: /3142CK o

3 Table of Contents Preface to the Second Edition V 11. Conditionally Gaussian Processes Assumptions and Formulation of the Theorem of Conditional Gaussian Behavior Auxiliary Lemmas Proof of the Theorem of Conditional Gaussian Behavior Optimal Nonlinear Filtering: Interpolation and Extrapolation of Components of Conditionally Gaussian Processes Optimal Filtering Equations Uniqueness of Solutions of Filtering Equations: Equivalence of a-algebras Ftt; and Ftt;o,W Optimal Filtering Equations in Several Dimensions Interpolation of Conditionally Gaussian Processes Optimal Extrapolation Equations Conditionally Gaussian Sequences: Filtering and Related Problems Theorem on Normal Correlation Recursive Filtering Equations for Conditionally Gaussian Sequences Forward and Backward Interpolation Equations Recursive Equations of Optimal Extrapolation Examples Application of Filtering Equations to Problems of Statistics of Random Sequences Optimal Linear Filtering of Stationary Sequences with Rational Spectra Maximum Likelihood Estimates for Coefficients of Linear Regression A Control Problem with Incomplete Data (Linear System with Quadratic Performance Index)

4 X Table of Contents 14.4 Asymptotic Properties of the Optimal Linear Filter Recursive Computation of the Best Approximate Solutions (Pseudo-solutions) of Linear Algebraic Systems Kalman Filter under Wrong Initial Conditions Linear Estimation of Random Processes Wide-Sense Wiener Processes Optimal Linear Filtering for some Classes of Nonstationary Processes Linear Estimation of Wide-Sense Stationary Random Processes with Rational Spectra Comparison of Optimal Linear and Nonlinear Estimates Application of Optimal Nonlinear Filtering Equations to some Problems in Control Theory and Estimation Theory An Optimal Control Problem Using Incomplete Data Asymptotic Properties of Kalman-Bucy Filters Computation of Mutual Information and Channel Capacity of a Gaussian Channel with Feedback Optimal Coding and Decoding for Transmission of a Gaussian Signal Through a Channel with Noiseless Feedback Asymptotic Properties of the Linear Filter under Wrong Initial Conditions Parameter Estimation and Testing of Statistical Hypotheses for Diffusion-Type Processes Maximum Likelihood Method for Coefficients of Linear Regression Parameter Estimation of the Drift Coefficient for Diffusion-Type Processes Parameter Estimation of the Drift Coefficient for a One-Dimensional Gaussian Process Two-Dimensional Gaussian Markov Processes: Parameter Estimation Sequential Maximum Likelihood Estimates Sequential Testing of Two Simple Hypotheses for Ito Processes Some Applications to Stochastic Approximation Random Point Processes: Stieltjes Stochastic Integrals Point Processes and their Compensators Minimal Representation of a Point Process: Processes of the Poisson Type Construction of Point Processes with Given Compensators: Theorems on Existence and Uniqueness

5 Table of Contents XI 18.4 Stieltjes Stochastic Integrals The Structure of Point Processes with Deterministic and Continuous Compensators The Structure of Local Martingales, Absolute Continuity of Measures for Point Processes, and Filtering The Structure of Local Martingales Nonnegative Supermartingale: Analog of Girsanov's Theorem Optimal Filtering from the Observations of Point Processes The Necessary and Sufficient Conditions for Absolute Continuity of the Measures Corresponding to Point Processes Calculation of the Mutual Information and the Cramer-Rao- Wolfowitz Inequality (the Point Observations) Asymptotically Optimal Filtering Total Variation Norm Convergence and Filtering Robust Diffusion Approximation for Filtering Bibliography Index