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AuthorLiptser, R. S. author
TitleStatistics of Random Processes I [electronic resource] : General Theory / by R. S. Liptser, A. N. Shiryayev
ImprintNew York, NY : Springer New York : Imprint: Springer, 1977
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Descript X, 395 p. online resource


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., ยฐ 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 estiยญ mate of lJ/ from ̃h is the a posteriori mean m/ = M(lJ/1 ff̃), where ff̃ = 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


of volume I -- 1 Essentials of probability theory and mathematical statistics -- 2 Martingales and semimartingales: discrete time -- 3 Martingales and semimartingales: continuous time -- 4 The Wiener process, the stochastic integral over the Wiener process, and stochastic 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 Ito 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

Mathematics Probabilities Statistics Mathematics Probability Theory and Stochastic Processes Statistics general


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