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Stochastic Mechanics
Random Media
Signal Processing
and Image Synthesis
Mathematical Economics and Finance
Stochastic Optimization
Stochastic Control
Applications of Mathematics Stochastic Modelling and Applied Probability
6 Edited by I. Karatzas
M. Yor
Advisory Board P. Bremaud E. Carlen W. Fleming D. Geman G. Grimmett G. Papanicolaou J. Scheinkman
Springer-Verlag Berlin Heidelberg GmbH
Applications of Mathematics
Fleming/Rishel, Deterministic and Stochastic Optimal Control ( 197 5) 2 Marchuk, Methods of Numerical Mathematics (1975, 2nd. ed. 1982) 3 Balakrishnan, Applied Functional Analysis (1976, 2nd. ed. 1981) 4 Borovkov, Stochastic Processes in Queueing Theory (1976) 5 Liptser/Shiryaev, Statistics of Random Processes I: General Theory
(1977, 2nd. ed. 2001) 6 Liptser/Shiryaev, Statistics of Random Processes II: Applications
(1978,2nd.ed.2001) 7 V orob' ev, Game Theory: Lectures for Economists and Systems Scientists ( 1977) 8 Shiryaev, Optimal Stopping Rules ( 1978) 9 Ibragimov/Rozanov, Gaussian Random Processes (1978)
10 Wonham, Linear Multivariable Control: A Geometric Approach (1974,3rd.ed. 1985)
II Hida, Brownian Motion (1980) 12 Hestenes, Conjugate Direction Methods in Optimization (1980) 13 Kallianpur, Stochastic Filtering Theory (1980) 14 Krylov, Controlled Diffusion Processes (1980) 15 Prabhu, Stochastic Storage Processes: Queues, Insurance Risk, and Dams ( 1980) 16 Ibragimov/Has'minskii, Statistical Estimation: Asymptotic Theory (1981) 17 Cesari, Optimization: Theory and Applications (1982) 18 Elliott, Stochastic Calculus and Applications (1982) 19 Marchuk/Shaidourov, Difference Methods and Their Extrapolations (1983) 20 Hi jab, Stabilization of Control Systems ( 1986) 21 Protter, Stochastic Integration and Differential Equations (1990) 22 Benveniste/Metivier/Priouret, Adaptive Algorithms and Stochastic
Approximations (1990) 23 Kloeden/Platen, Numerical Solution of Stochastic Differential Equations
(1992, corr. 3rd. printing 1999) 24 Kushner/Dupuis, Numerical Methods for Stochastic Control Problems
in Continuous Time (1992) 25 Fleming/Soner, Controlled Markov Processes and Viscosity Solutions ( 1993) 26 Baccelli/Bremaud, Elements of Queueing Theory ( 1994) 27 Winkler, Image Analysis, Random Fields and Dynamic Monte Carlo Methods
(1995) 28 Kalpazidou, Cycle Representations of Markov Processes (1995) 29 Elliott/ Aggoun/Moore, Hidden Markov Models: Estimation and Control {1995) 30 Hernandez-Lerma/Lasserre, Discrete-Time Markov Control Processes ( 1995) 31 Devroye/Gyorfi/Lugosi, A Probabilistic Theory of Pattern Recognition {1996) 32 Maitra/Sudderth, Discrete Gambling and Stochastic Games (1996) 33 Embrechts/Kliippelberg/Mikosch, Modelling Extremal Events for Insurance
and Finance (1997, corr. 2nd printing 1999) 34 Duflo, Random Iterative Models ( 1997) 35 Kushner/Yin, Stochastic Approximation Algorithms and Applications (1997) 36 M usiela/Rutkowski, Martingale Methods in Financial Modelling ( 1997) 37 Yin, Continuous-Time Markov Chains and Applications (1998) 38 Dembo/Zeitouni, Large Deviations Techniques and Applications (1998) 39 Karatzas, Methods of Mathematical Finance (1998) 40 Fayolle/Iasnogorodski/Malyshev, Random Walks in the Quarter-Plane ( 1999) 41 Aves/Jensen, Stochastic Models in Reliability (1999) 42 Hernandez-Lerma/Lasserre, Further Topics on Discrete-Time Markov Control
Processes ( 1999) 43 Yong/Zhon, Stochastic Controls. Hamiltonian Systems and HJB Equations ( 1999) 44 Serfozo, Introduction to Stochastic Networks (1999) 45 Steele, Invitation to Stochastic Calculus (2000)
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: [email protected]
Translation Editor
Stephen S. Wilson
31 Harp Hill
Albert N. Shiryaev
Steklov Mathematical Institute, Russian Academy of Sciences Gubkina 8, 117966 Moscow, Russia e-mail: shiryaev@ genesis.mi.ras.ru
Cheltenham, Gloucestershire, GL52 6PY, United Kingdom e-mail: [email protected]
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
Preface to the Second Edition
At the end of 1960s and the beginning of 1970s, when the Russian version of this book was written, the 'general theory of random processes' did not operate widely with such notions as semimartingale, stochastic integral with respect to semimartingale, the Ito formula for semimartingales, etc. At that time in stochastic calculus (theory of martingales), the main object was the square integrable martingale. In a short time, this theory was applied to such areas as nonlinear filtering, optimal stochastic control, statistics for diffusiontype processes.
In the first edition of these volumes, the stochastic calculus, based on square integrable martingale theory, was presented in detail with the proof of the Doob-Meyer decomposition for submartingales and the description of a structure for stochastic integrals. In the first volume ('General Theory') these results were used for a presentation of further important facts such as the Girsanov theorem and its generalizations, theorems on the innovation processes, structure of the densities (Radon-Nikodym derivatives) for absolutely continuous measures being distributions of diffusion and ItO-type processes, and existence theorems for weak and strong solutions of stochastic differential equations.
All the results and facts mentioned above have played a key role in the derivation of 'general equations' for nonlinear filtering, prediction, and smoothing of random processes.
The second volume ('Applications') begins with the consideration of the so-called conditionally Gaussian model which is a natural 'nonlinear' extension of the Kalman-Bucy scheme. The conditionally Gaussian distribution of an unobservable signal, given observation, has permitted nonlinear filtering equations to be obtained, similar to the linear ones defined by the KalmanBucy filter. Parallel to the explicit filtering implementation this result has being applied in many cases: to establish the 'separation principle' in the LQG (linear model, quadratic cost functional, Gaussian noise) stochastic control problem, in some coding problems, and to estimate unknown parameters of random processes.
The square integrable martingales, involved in the above-mentioned models, were assumed to be continuous. The first English edition contained two additional chapters (18 and 19) dealing with point (counting) processes which
VI Preface to the Second Edition
are the simplest discontinuous ones. The martingale techniques, based on the Doob-Meyer decomposition, permitted, in this case as well, the investigation of the str~cture of discontinuous local martingales, to find the corresponding version of Girsanov's theorem, and to derive nonlinear stochastic filtering equations for discontinuous observations.
Over the long period of time since the publication of the Russian (1974) and English (1977, 1978) versions, the monograph 'Statistics of Random Processes' has remained a frequently cited text in the connection with the stochastic calculus for square integrable martingales and point processes, nonlinear filtering, and statistics of random processes. For this reason, the authors decided not to change the main material of the first volume. In the second volume ('Applications'), two subsections 14.6 and 16.5 and a new Chapter 20 have being added. In Subsections 14.6 and 16.5, we analyze the Kalman-Bucy filter under wrong initial conditions for cases of discrete and continuous time, respectively. In Chapter 20, we study an asymptotic optimality for linear and nonlinear filters, corresponding to filtering models presented in Chapters 8-11, when in reality filtering schemes are different from the above-mentioned but can be approximated by them in some sense.
Below we give a list of books, published after the first English edition and related to its content:
- Anulova, A., Veretennikov, A., Krylov, N., Liptser, R. and Shiryaev, A. (1998) Stochastic Calculus [4]
- Elliott, R. (1982) Stochastic Calculus and Applications [59] - Elliott, R.J., Aggoun, L. and Moore, J.B. (1995) Hidden Markov Models [60] - Dellacherie, C. and Meyer, P.A. (1980) Probabilites et Potentiel. Theorie des
Martingales [51] - Jacod, J. (1979) Calcul Stochastique et Problemes des Martingales [104] - Jacod, J. and Shiryaev, A.N. (1987) Limit Theorems for Stochastic Processes
[106] - Kallianpur, G. (1980) Stochastic Filtering Theory [135] - Karatzas, I. and Shreve, S.E. (1991) Brownian Motion and Stochastic Calculus
[142] - Krylov, N.V. (1980) Controlled Diffusion Processes [164] - Liptser, R.S. and Shiryaev, A.N. (1986, 1989) Theory of Martingales [214] - Meyer, P.A. (1989) A short presentation of stochastic calculus [230] - Metivier, M. and Pellaumail, J. (1980) Stochastic Integration [228] - 0ksendal, B. (1985, 1998) Stochastic Differential Equations [250] - Protter, P. (1990) Stochastic Integration and Differential Equations. A New Ap-
proach [257] - Revuz, D. and Yor, M. (1994) Continuous Martingales and Brownian Motion
[261] - Rogers, C. and Williams, D. (1987) Diffusions, Markov Processes and Martin-
gales: Ito Calculus [262] - Shiryaev, A.N. (1978) Optimal Stopping Rules [286] - Williams, D. (ed) (1981) Proc. Durham Symposium on Stochastic Integrals [308] - Shiryaev, A.N. (1999) Essentials of Stochastic Finance [288].
The topics gathered in these books are named 'general theory of random processes', 'theory of martingales', 'stochastic calculus', applications of the
Preface to the Second Edition VII
stochastic calculus, etc. It is important to emphasize that substantial progress in developing this theory was implied by the understanding of the fact that it is necessary to add to the Kolmogorov probability space ( !1, :F, P) the increasing family (filtration) of a-algebras (:Ft)t>o, where Ft can be interpreted as the set of events observed up to timet~ A new filtered probability space (!1, :F, (:Ft)t~o, P) is named the stochastic basis. The introduction of the stochastic basis has provided such notions as: 'to be adapted (optional, predictable) to filtration', semimartingale, and others. It is very natural that the old terminology also has changed for many cases. For example, the notion of the natuml process, introduced by P.A. Meyer for the description of the Doob-Meyer decomposition, was changed to predictable process. The importance of the notion of 'local martingale', introduced by K. Ito and S. Watanabe, was also realized.
In this publication, we have modernized the terminology as much as possible. The corresponding comments and indications of useful references and known results are given at the end of every chapter headed by 'Notes and References. 2'.
The authors are grateful to Dr. Stephen Wilson for the preparation of the Second Edition for publication. Our thanks are due to the member of the staff of the Mathematics Editorial of Springer-Verlag for their help during the preparation of this edition.
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 Pro-
cesses 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 Conti-
nuity 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
Table of Contents of Volume 5
Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V
Introduction.................................................. 1
1. Essentials of Probability Theory and Mathematical Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1 Main Concepts of Probability Theory . . . . . . . . . . . . . . . . . . . . . 11 1.2 Random Processes: Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3 Markov Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.4 Brownian Motion Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.5 Some Notions from Mathematical Statistics................ 34
2. Martingales and Related Processes: Discrete Time. . . . . . . . 39 2.1 Supermartingales and Submartingales on a Finite Time Interval 39 2.2 Submartingales on an Infinite Time Interval,
and the Theorem of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3 Regular Martingales: Levy's Theorem . . . . . . . . . . . . . . . . . . . . . 47 2.4 Invariance of the Supermartingale Property for Markov Times:
Riesz and Doob Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3. Martingales and Related Processes: Continuous Time .... 57 3.1 Right Continuous Supermartingales....................... 57 3.2 Basic Inequalities, the Theorem of Convergence, and Invari-
ance of the Supermartingale Property for Markov Times. . . . . 60 3.3 Doob-Meyer Decomposition for Supermartingales . . . . . . . . . . 64 3.4 Some Properties of Predictable Increasing Processes . . . . . . . . 74
4. The Wiener Process, the Stochastic Integral over the Wiener Process, and Stochastic Differential Equations . . . . . . . . . . . . 85 4.1 The Wiener Process as a Square Integrable Martingale . . . . . . 85 4.2 Stochastic Integrals: Ito Processes . . . . . . . . . . . . . . . . . . . . . . . . 92 4.3 Ito's Formula .......................................... 123 4.4 Strong and Weak Solutions of Stochastic Differential Equations132
XIV Table of Contents of Volume 5
5. Square Integrable Martingales and Structure of the Functionals on a Wiener Process ................... 161 5.1 Doob-Meyer Decomposition for Square Integrable Martingales 161 5.2 Representation of Square Integrable Martingales ............ 170 5.3 The Structure of Functionals of a Wiener Process ........... 174 5.4 Stochastic Integrals over Square Integrable Martingales ...... 182 5.5 Integral Representations of the Martingales which are Con-
ditional Expectations and the Fubini Theorem for Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.6 The Structure of Functionals of Processes of the Diffusion Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
6. Nonnegative Supermartingales and Martingales, and the Girsanov Theorem ............................... 219 6.1 Nonnegative Supermartingales ........................... 219 6.2 Nonnegative Martingales ................................ 228 6.3 The Girshanov Theorem and its Generalization ............ 238
7. Absolute Continuity of Measures corresponding to the Ito Processes and Processes of the Diffusion Type ............ 251 7.1 The Ito Processes, and the Absolute Continuity
of their Measures with respect to Wiener Measure .......... 251 7.2 Processes of the Diffusion Type: the Absolute Continuity
of their Measures with respect to Wiener Measure .......... 257 7.3 The Structure of Processes whose Measure is Absolutely
Continuous with Respect to Wiener Measure . . . . . . . . . . . . . . . 271 7.4 Representation of the Ito Processes as Processes
of the Diffusion Type, Innovation Processes, and the Struc-ture of Functionals on the Ito Process ..................... 273
7.5 The Case of Gaussian Processes .......................... 279 7.6 The Absolute Continuity of Measures of the Ito Processes
with respect to Measures Corresponding to Processes of the Diffusion Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
7.7 The Cameron-Martin Formula ........................... 297 7.8 The Cramer-Wolfowitz Inequality ........................ 299 7.9 An Abstract Version of the Bayes Formula ................. 303
8. General Equations of Optimal Nonlinear Filtering, Interpolation and Extrapolation of Partially Observable Random Processes ........................................ 317 8.1 Filtering: the Main Theorem ............................. 317 8.2 Filtering: Proof of the Main Theorem ..................... 319 8.3 Filtering of Diffusion Markov Processes .................... 326 8.4 Equations of Optimal Nonlinear Interpolation .............. 329 8.5 Equations of Optimal Nonlinear Extrapolation ............. 331
Table of Contents of Volume 5 XV
8.6 Stochastic Differential Equations with Partial Derivatives for the Conditional Density (the Case of Diffusion Markov Pro-cesses) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
9. Optimal Filtering, Interpolation and Extrapolation of Markov Processes with a Countable Number of States. 351 9.1 Equations of Optimal Nonlinear Filtering .................. 351 9.2 Forward and Backward Equations of Optimal Nonlinear
Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 9.3 Equations of Optimal Nonlinear Extrapolation ............. 368 9.4 Examples .............................................. 371
10. Optimal Linear Nonstationary Filtering .................. 375 10.1 The Kalman-Bucy Method .............................. 375 10.2 Martingale Proof of the Equations of Linear Nonstationary
Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 10.3 Equations of Linear Nonstationary Filtering:
the Multidimensional Case ............................... 392 10.4 Equations for an Almost Linear Filter for Singular BoB .... 400
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
Index ......................................................... 425
