# Statistics of Random Processes I

## General Theory

• R. S. Liptser
• A. N. Shiryayev Book

Part of the Applications of Mathematics book series (SMAP, volume 5)

1. Front Matter
Pages i-x
2. R. S. Liptser, A. N. Shiryayev
Pages 1-10
3. R. S. Liptser, A. N. Shiryayev
Pages 11-36
4. R. S. Liptser, A. N. Shiryayev
Pages 37-54
5. R. S. Liptser, A. N. Shiryayev
Pages 55-81
6. R. S. Liptser, A. N. Shiryayev
Pages 82-151
7. R. S. Liptser, A. N. Shiryayev
Pages 152-206
8. R. S. Liptser, A. N. Shiryayev
Pages 207-235
9. R. S. Liptser, A. N. Shiryayev
Pages 236-296
10. R. S. Liptser, A. N. Shiryayev
Pages 297-328
11. R. S. Liptser, A. N. Shiryayev
Pages 329-350
12. R. S. Liptser, A. N. Shiryayev
Pages 351-380
13. Back Matter
Pages 381-395

### Introduction

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.

### Keywords

Markov process Martingal Semimartingal Semimartingale functional analysis mathematical statistics observable probability probability space probability theory statistics stochastic differential equation stochastic process stochastic processes stochastischer Prozess

#### Authors and affiliations

• R. S. Liptser
• 1
• A. N. Shiryayev
• 2
1. 1.Institute for Problems of Control TheoryMoscowUSSR
2. 2.Institute of Control SciencesMoscowUSSR

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4757-1665-8
• Copyright Information Springer-Verlag New York 1977
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4757-1667-2
• Online ISBN 978-1-4757-1665-8
• Series Print ISSN 0172-4568
• Buy this book on publisher's site
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