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Statistics of Random Processes I

General Theory

  • Book
  • © 1977

Overview

Authors:
  1. R. S. Liptser

    1. Institute for Problems of Control Theory, Moscow, USSR

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    You can also search for this author in PubMed   Google Scholar

  2. A. N. Shiryayev

    1. Institute of Control Sciences, Moscow, USSR

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    You can also search for this author in PubMed   Google Scholar

Part of the book series: Stochastic Modelling and Applied Probability (SMAP, volume 5)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-x
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  2. Introduction

    • R. S. Liptser, A. N. Shiryayev
    Pages 1-10

  3. Essentials of probability theory and mathematical statistics

    • R. S. Liptser, A. N. Shiryayev
    Pages 11-36

  4. Martingales and semimartingales: discrete time

    • R. S. Liptser, A. N. Shiryayev
    Pages 37-54

  5. Martingales and semimartingales: continuous time

    • R. S. Liptser, A. N. Shiryayev
    Pages 55-81

  6. The Wiener process, the stochastic integral over the Wiener process, and stochastic differential equations

    • R. S. Liptser, A. N. Shiryayev
    Pages 82-151

  7. Square integrable martingales, and structure of the functionals on a Wiener process

    • R. S. Liptser, A. N. Shiryayev
    Pages 152-206

  8. Nonnegative supermartingales and martingales, and the Girsanov theorem

    • R. S. Liptser, A. N. Shiryayev
    Pages 207-235

  9. Absolute continuity of measures corresponding to the Ito processes and processes of the diffusion type

    • R. S. Liptser, A. N. Shiryayev
    Pages 236-296

  10. General equations of optimal nonlinear filtering, interpolation and extrapolation of partially observable random processes

    • R. S. Liptser, A. N. Shiryayev
    Pages 297-328

  11. Optimal filtering, interpolation and extrapolation of Markov processes with a countable number of states

    • R. S. Liptser, A. N. Shiryayev
    Pages 329-350

  12. Optimal linear nonstationary filtering

    • R. S. Liptser, A. N. Shiryayev
    Pages 351-380

  13. Back Matter

    Pages 381-395
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Keywords

About this book

A considerable number of problems in the statistics of random processes are formulated within the following scheme. On a certain probability space (Q, ff, P) a partially observable random process (lJ,~) = (lJ ~/), t :;::-: 0, is given with only the second component n ~ = (~/), t:;::-: 0, observed. At any time t it is required, based on ~h = g., ° s sst}, to estimate the unobservable state lJ/. This problem of estimating (in other words, the filtering problem) 0/ from ~h will be discussed in this book. It is well known that if M(lJ;) < 00, then the optimal mean square esti­ mate of lJ/ from ~h is the a posteriori mean m/ = M(lJ/1 ff~), where ff~ = CT{ w: ~., sst} is the CT-algebra generated by ~h. Therefore, the solution of the problem of optimal (in the mean square sense) filtering is reduced to finding the conditional (mathematical) expectation m/ = M(lJ/lffa. In principle, the conditional expectation M(lJ/lff;) can be computed by Bayes' formula. However, even in many rather simple cases, equations obtained by Bayes' formula are too cumbersome, and present difficulties in their practical application as well as in the investigation of the structure and properties of the solution.

Authors and Affiliations

  • Institute for Problems of Control Theory, Moscow, USSR

    R. S. Liptser

  • Institute of Control Sciences, Moscow, USSR

    A. N. Shiryayev

Bibliographic Information

  • Book Title: Statistics of Random Processes I

  • Book Subtitle: General Theory

  • Authors: R. S. Liptser, A. N. Shiryayev

  • Series Title: Stochastic Modelling and Applied Probability

  • DOI: https://doi.org/10.1007/978-1-4757-1665-8

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag New York 1977

  • eBook ISBN: 978-1-4757-1665-8Published: 11 November 2013

  • Series ISSN: 0172-4568

  • Series E-ISSN: 2197-439X

  • Edition Number: 1

  • Number of Pages: X, 395

  • Additional Information: Title of the original Russian edition: Statistika slucajnych processov

  • Topics: Probability Theory and Stochastic Processes, Statistics, general

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