Optimal Filtering, 1999
Volume I: Filtering of Stochastic Processes
Mathematics and Its Applications Series, Vol. 457

This book is devoted to an investigation of some important problems of mod­ ern filtering theory concerned with systems of 'any nature being able to per­ ceive, store and process an information and apply it for control and regulation'. (The above quotation is taken from the preface to [27]). Despite the fact that filtering theory is l'argely worked out (and its major issues such as the Wiener-Kolmogorov theory of optimal filtering of stationary processes and Kalman-Bucy recursive filtering theory have become classical) a development of the theory is far from complete. A great deal of recent activity in this area is observed, researchers are trying consistently to generalize famous results, extend them to more broad classes of processes, realize and justify more simple procedures for processing measurement data in order to obtain more efficient filtering algorithms. As to nonlinear filter­ ing, it remains much as fragmentary. Here much progress has been made by R. L. Stratonovich and his successors in the area of filtering of Markov processes. In this volume an effort is made to advance in certain of these issues. The monograph has evolved over many years, coming of age by stages. First it was an impressive job of gathering together the bulk of the impor­ tant contributions to estimation theory, an understanding and moderniza­ tion of some of its results and methods, with the intention of applying them to recursive filtering problems.

1 Introduction to estimation and filtering theory.- 1.1 Basic notions of probability theory.- 1.2 Introduction to estimation theory.- 1.3 Examples of estimation problems.- 1.4 Estimation and filtering: similarity and distinction.- 1.5 Basic notions of filtering theory.- 1.6 Appendix: Proofs of Lemmas and Theorems.- 2 Optimal filtering of stochastic processes in the context of the Wiener-Kolmogorov theory.- 2.1 Linear filtering of stochastic processes.- 2.2 Filtering of stationary processes.- 2.3 Recursive filtering.- 2.4 Linear filters maximizing a signal to noise ratio.- 2.5 Appendix: Proofs of Lemmas and Theorems.- 2.6 Bibliographical comments.- 3 Abstract optimal filtering theory.- 3.1 Random elements.- 3.2 Linear stable estimation.- 3.3 Resolution space and relative finitary transformations.- 3.4 Extended resolution space and linear transformations in it.- 3.5 Abstract version of the Wiener-Kolmogorov filtering theory.- 3.6 Optimal estimation in discrete resolution space.- 3.7 Spectral factorization.- 3.8 Optimal filter structure for discrete time case.- 3.9 Abstract Wiener problem.- 3.10 Appendix: Proofs of Lemmas and Theorems.- 3.11 Bibliographical comments.- 4 Nonlinear filtering of time series.- 4.1 Statement of nonlinear optimal filtering problem.- 4.2 Optimal filtering of conditionally Gaussian time series.- 4.3 Connection of linear and nonlinear filtering problems.- 4.4 Minimax filtering.- 4.5 Proofs of Lemmas and Theorems.- 4.6 Bibliographical comments.- References.- Notation.