Statistics of Random Processes II: Applications
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, shastic integral with respect to semimartingale, the Ito formula for semimartingales, etc. At that time in shastic 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 shastic control, statistics for diffusion­ type processes. In the first edition of these volumes, the shastic 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 shastic 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 processes, 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 shastic 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.
1101634123
Statistics of Random Processes II: Applications
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, shastic integral with respect to semimartingale, the Ito formula for semimartingales, etc. At that time in shastic 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 shastic control, statistics for diffusion­ type processes. In the first edition of these volumes, the shastic 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 shastic 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 processes, 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 shastic 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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Statistics of Random Processes II: Applications

Statistics of Random Processes II: Applications

Statistics of Random Processes II: Applications

Statistics of Random Processes II: Applications

Overview

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, shastic integral with respect to semimartingale, the Ito formula for semimartingales, etc. At that time in shastic 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 shastic control, statistics for diffusion­ type processes. In the first edition of these volumes, the shastic 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 shastic 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 processes, 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 shastic 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.

Product Details

ISBN-13: 9783642083655
Publisher: Springer Berlin Heidelberg
Publication date: 12/01/2010
Series: Stochastic Modelling and Applied Probability , #6
Edition description: Second Edition 2001
Pages: 402
Product dimensions: 6.10(w) x 9.25(h) x 0.24(d)

Table of Contents

11. Conditionally Gaussian Processes.- 12. Optimal Nonlinear Filtering: Interpolation and Extrapolation of Components of Conditionally Gaussian Processes.- 13. Conditionally Gaussian Sequences: Filtering and Related Problems.- 14. Application of Filtering Equations to Problems of Statistics of Random Sequences.- 15. Linear Estimation of Random Processes.- 16. Application of Optimal Nonlinear Filtering Equations to some Problems in Control Theory and Estimation Theory.- 17. Parameter Estimation and Testing of Statistical Hypotheses for Diffusion-Type Processes.- 18. Random Point Processes: Stieltjes Shastic Integrals.- 19. The Structure of Local Martingales, Absolute Continuity of Measures for Point Processes, and Filtering.- 20. Asymptotically Optimal Filtering.
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