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.
Reviews / Votes
From the reviews:
JOURNAL OF THE AMERICAN STOCHASTIC ASSOCIATION
"The material is accessible to researchers and advanced graduate students. These two classic volumes are very important resources for both probabilists and statisticians."
SIAM REVIEW
"Written by two renowned experts in the field, the books under review contain a thorough and insightful treatment of the fundamental underpinnings of various aspects of stochastic processes as well as a wide range of applications. Providing clear exposition, deep mathematical results, and superb technical representation, they are masterpieces of the subject of stochastic analysis and nonlinear filtering.What is special about these books is their broad coverage and in-depth study of optimal filtering problems.The books can be used by researchers in different areas who need to use stochastic calculus and who treat state estimation, detection, and stochastic control problems under incomplete information and partial observations.These two books are a comprehensive treatise on stochastic calculus, random processes, and filtering theory, and provide an excellent and illuminating introduction to these fields with a wide range of theoretical and practical issues. With the new additions and modifications of the first edition, they are to be welcomed and benefit not only the systems theory and control community but also mathematicians working on stochastic processes; engineers in control, communication, and signal processing; researchers in financial engineering; and scientists in many other related fields. It is conceivable that these books will have a significant impact on the aforementioned fields and will become classics."
SIAM REVIEW
".and the second volume can be used as a text for a special topics course in filtering."
Series
Edition
Language
Place of publication
Publishing group
Target group
Professional and scholarly
Research
Edition type
Illustrations
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 23 mm
Weight
ISBN-13
978-3-642-08365-5 (9783642083655)
DOI
10.1007/978-3-662-10028-8
Schweitzer Classification
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 Stochastic Integrals.- 19. The Structure of Local Martingales, Absolute Continuity of Measures for Point Processes, and Filtering.- 20. Asymptotically Optimal Filtering.
