Statistics of Random Processes: I. General Theory / Edition 2

by Robert S. Liptser, B. Aries, Albert N. Shiryaev

Overview

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, shastic integral with respect to semimartingale, the ItO formula for semimartingales, etc. At that time in shastic 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 shastic control, statistics for diffusion­ type processes. In the first edition of these volumes, the shastic 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 shastic 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 Itô-type processes, and existence theorems for weak and strong solutions of shastic 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.

Product Details

ISBN-13:	9783540639299
Publisher:	Springer Berlin Heidelberg
Publication date:	12/12/2000
Series:	Stochastic Modelling and Applied Probability , #5
Edition description:	2nd rev. and exp. ed. 2001
Pages:	427
Product dimensions:	6.14(w) x 9.21(h) x 0.04(d)

Table of Contents

1. Essentials of Probability Theory and Mathematical Statistics.- 2. Martingales and Related Processes: Discrete Time.- 3. Martingales and Related Processes: Continuous Time.- 4. The Wiener Process, the Shastic Integral over the Wiener Process, and Shastic Differential Equations.- 5. Square Integrable Martingales and Structure of the Functionals on a Wiener Process.- 6. Nonnegative Supermartingales and Martingales, and the Girsanov Theorem.- 7. Absolute Continuity of Measures corresponding to the Itô Processes and Processes of the Diffusion Type.- 8. General Equations of Optimal Nonlinear Filtering, Interpolation and Extrapolation of Partially Observable Random Processes.- 9. Optimal Filtering, Interpolation and Extrapolation of Markov Processes with a Countable Number of States.- 10. Optimal Linear Nonstationary Filtering.

Editorial Reviews

From the reviews:

JOURNAL OF THE AMERICAN STOCHASTIC ASSOCIATION

"The material is accessible to researchers and advanced graduate students. These two classic volumes are very important resources for both probabilists and statisticians."

SIAM REVIEW

"Written by two renowned experts in the field, the books under review contain a thorough and insightful treatment of the fundamental underpinnings of various aspects of shastic processes as well as a wide range of applications. Providing clear exposition, deep mathematical results, and superb technical representation, they are masterpieces of the subject of shastic analysis and nonlinear filtering…What is special about these books is their broad coverage and in-depth study of optimal filtering problems…The books can be used by researchers in different areas who need to use shastic calculus and who treat state estimation, detection, and shastic control problems under incomplete information and partial observations…These two books are a comprehensive treatise on shastic calculus, random processes, and filtering theory, and provide an excellent and illuminating introduction to these fields with a wide range of theoretical and practical issues. With the new additions and modifications of the first edition, they are to be welcomed and benefit not only the systems theory and control community but also mathematicians working on shastic processes; engineers in control, communication, and signal processing; researchers in financial engineering; and scientists in many other related fields. It is conceivable that these books will have a significant impact on the aforementioned fields and will become classics."

SIAM REVIEW

"The first volume of the books may also be used as an advanced graduate-level textbook for a course in shastic processes…"

From the reviews of the second edition:

"This is the revised and expanded second edition of the first version in Russian of (1974) and in English (1997, 1978). The ambitious program of the authors was to give for the first time a systematic account of the shastic calculus and the unifying power and efficiency of its methods for the study of statistics of random process. … The very detailed exposition of the text will in particular appeal to the mathematically interested reader and scientist." (Metrika, July, 2002)

"For several reasons shastic analysis has been among the scientific hits of the last few decades. … The reference list is also updated with essential recently published papers and books. Thus the comprehensive content and the masterly written text make the book attractive for researchers in shastic analysis and its applications as well as for graduate students in the area. … this book will continue to be among the most useful and popular books on the subject in the decades to come." (Jordan M. Stoyanov, Zentralblatt MATH, Issue 1008, 2003)

From the Publisher