Statistics of Random Processes: I. General Theory
Robert S. Liptser, Robert Ševilevič Lipcer, Alʹbert Nikolaevič Šir‚ev, Alʹbert Nikolaevich Shiri︠a︡ev, Albert N. Shiryaev
Springer Science & Business Media, 2001 - Mathematics - 427 pages
At the end of 1960s and the beginning of 1970s, when the Russian version of this book was written, the 'general theory of random processes' did not operate widely with such notions as semimartingale, stochastic integral with respect to semimartingale, the ItO formula for semimartingales, etc. At that time in stochastic calculus (theory of martingales), the main object was the square integrable martingale. In a short time, this theory was applied to such areas as nonlinear filtering, optimal stochastic control, statistics for diffusion type processes. In the first edition of these volumes, the stochastic calculus, based on square integrable martingale theory, was presented in detail with the proof of the Doob-Meyer decomposition for submartingales and the description of a structure for stochastic integrals. In the first volume ('General Theory') these results were used for a presentation of further important facts such as the Girsanov theorem and its generalizations, theorems on the innovation pro cesses, structure of the densities (Radon-Nikodym derivatives) for absolutely continuous measures being distributions of diffusion and ItO-type processes, and existence theorems for weak and strong solutions of stochastic differential equations. All the results and facts mentioned above have played a key role in the derivation of 'general equations' for nonlinear filtering, prediction, and smoothing of random processes.
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absolute continuity according Applying assumed assumption bounded called Chapter condition Consequently consider construction Control convergence Corollary corresponding deduce defined definition Denote density diffusion type distribution equality equivalent establish estimate example exists expectation fact follows function Further Gaussian given Hence independent inequality introduce ItŰ formula Lemma limit linear Liptser Markov processes mathematical matrix mean measurable modification nonanticipative nonnegative Note o-algebras observable obtain optimal particular permits predictable probability space problems PROOF properties proving random process random variable representation respect result right continuous satisfied sequence Shiryaev shown simple functions solution square integrable martingale Statistics stochastic differential equations stochastic integrals strong solution sufficient supermartingale system of equations taking Theorem theory tion trajectories uniformly unique values vector Wiener process zero
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