Statistics of Random Processes I

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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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.

• 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

• Institute for Problems of Control Theory, Moscow, USSR

  R. S. Liptser

• Institute of Control Sciences, Moscow, USSR

  A. N. Shiryayev

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