© 2001

Statistics of Random Processes

II. Applications


  • In the second edition, two new subsections devoted to the Kalman filter under wrong initial conditions, and a new chapter on asymptotically optimal filtering under diffusion approximation have been added

  • Moreover in each chapter a comment is added about the progress of recent years


Part of the Stochastic Modelling and Applied Probability book series (SMAP, volume 6)

Table of contents

  1. Front Matter
    Pages I-XV
  2. Robert S. Liptser, Albert N. Shiryaev
    Pages 1-16
  3. Robert S. Liptser, Albert N. Shiryaev
    Pages 55-97
  4. Robert S. Liptser, Albert N. Shiryaev
    Pages 145-175
  5. Robert S. Liptser, Albert N. Shiryaev
    Pages 261-308
  6. Robert S. Liptser, Albert N. Shiryaev
    Pages 355-382
  7. Back Matter
    Pages 383-402

About this book


The subject of these two volumes is non-linear filtering (prediction and smoothing) theory and its application to the problem of optimal estimation, control with incomplete data, information theory, and sequential testing of hypothesis. The required mathematical background is presented in the first volume: the theory of martingales, stochastic differential equations, the absolute continuity of probability measures for diffusion and Ito processes, elements of stochastic calculus for counting processes. The book is not only addressed to mathematicians but should also serve the interests of other scientists who apply probabilistic and statistical methods in their work. The theory of martingales presented in the book has an independent interest in connection with problems from financial mathematics.
In the second edition, the authors have made numerous corrections, updating every chapter, adding two new subsections devoted to the Kalman filter under wrong initial conditions, as well as a new chapter devoted to asymptotically optimal filtering under diffusion approximation. Moreover, in each chapter a comment is added about the progress of recent years.


Filtering Incomplete Data Control Martingale Point Process STATISTICA conditionally Gaussian information theory

Authors and affiliations

  1. 1.Department of Electrical Engineering SystemsTel Aviv UniversityTel AvivIsrael
  2. 2.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

Bibliographic information

  • Book Title Statistics of Random Processes
  • Book Subtitle II. Applications
  • Authors Robert S. Liptser
    Albert N. Shiryaev
  • Series Title Stochastic Modelling and Applied Probability
  • DOI
  • Copyright Information Springer-Verlag Berlin Heidelberg 2001
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-3-540-63928-2
  • Softcover ISBN 978-3-642-08365-5
  • eBook ISBN 978-3-662-10028-8
  • Series ISSN 0172-4568
  • Edition Number 2
  • Number of Pages XV, 402
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Additional Information Original Russian edition published by Nauka, Moscow, 1974
  • Topics Probability Theory and Stochastic Processes
    Statistical Theory and Methods
  • Buy this book on publisher's site
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From the reviews:


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


"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 stochastic 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 stochastic 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 stochastic calculus and who treat state estimation, detection, and stochastic control problems under incomplete information and partial observations…These two books are a comprehensive treatise on stochastic 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 stochastic 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."


"…and the second volume can be used as a text for a special topics course in filtering."