Statistics of Random Processes: I. General TheorySpringer Science & Business Media, 17 avr. 2013 - 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. |
Table des matières
1 | |
6 | |
12 | |
Discrete Time | 34 |
Continuous Time | 57 |
The Wiener Process the Stochastic Integral over the Wiener | 85 |
Application of Filtering Equations to Problems of Statistics | 129 |
Linear Estimation of Random Processes | 145 |
Absolute Continuity of Measures corresponding to the Itô | 250 |
Stieltjes Stochastic Integrals | 261 |
The Structure of Local Martingales Absolute Continuity | 309 |
General Equations of Optimal Nonlinear Filtering | 317 |
Optimal Filtering Interpolation and Extrapolation | 351 |
Asymptotically Optimal Filtering | 355 |
Optimal Linear Nonstationary Filtering 375 | 374 |
Bibliography | 383 |
Square Integrable Martingales and Structure | 161 |
Application of Optimal Nonlinear Filtering Equations | 177 |
Signal Through a Channel with Noiseless Feedback | 195 |
Nonnegative Supermartingales and Martingales | 219 |
Index | 399 |
Bibliography | 409 |
424 | |
Autres éditions - Tout afficher
Statistics of Random Processes: I. General Theory Robert Liptser,Albert N. Shiryaev Aucun aperçu disponible - 2010 |
Expressions et termes fréquents
absolute continuity according to Theorem ao(t assumed assumption Brownian motion conditional mathematical expectation continuous modification continuous P-a.s continuous trajectories convergence Corollary deduce defined definition Denote diffusion type estimate exists F-measurable follows Fubini theorem function ƒ Gaussian given Hence inequality It(f Itô formula Itô process Lemma linear Liptser Markov processes matrix measurable function nonanticipative function nonlinear filtering nonnegative Note obtain predictable increasing process probability space proving random process random variable representation right continuous satisfied sequence Shiryaev simple functions solution of Equation square integrable martingale ẞt stochastic differential equations stochastic integrals strong solution sub-o-algebras submartingale supermartingale system of equations uniformly integrable unique vector Veroyatn w)ds w)dt w)dW W₁ Wiener process άμε