White Noise Theory of Prediction, Filtering and Smoothing

Based on the author's own research, this book rigorously and systematically develops the theory of Gaussian white noise measures on Hilbert spaces to provide a comprehensive account of nonlinear filtering theory. Covers Markov processes, cylinder and quasi-cylinder probabilities and conditional expectation as well as predictio0n and smoothing and the varied processes used in filtering. Especially useful for electronic engineers and mathematical statisticians for explaining the systematic use of finely additive white noise theory leading to a more simplified and direct presentation.

Table of Contents:
Part 1 Introduction: probabilistic preliminaries; probability measures in Polish spaces; probability measures in function spaces; Gaussian white noise. Part 2 Markov processes: definitions; the associated semigroup and its generator; diffusion processes; the Feynman-Kac formula. Part 3 Cylinder probabilities: cylinder probabilities; integration with respect to cylinder probabilities; representation and lifting maps; examples of representations of the canonical Gauss measure; relation to the Dunford- Schwartz theory. Part 4 Conditional expectation - definition and basic properties: absolute continuity of cylinder probabilities; cylindrical mappings; conditional expectation. Part 5 Quasi-cylinder probabilities: quasi-cylinder probabilities and integration; representation of a QCP and the lifting map; polish space valued mappings on (E); absolute continuity for QCP's, quasi-cylindrical mappings and conditional expectation; independence. Part 6 The abstract statistical model and Bayes' formula: measurable representation; more on canonical Gauss measure; the abstract statistical model and the Bayes formula. Part 7 Differential equations of filtering - finite dimensional noise and signal: nonlinear filtering model and the Bayes formula; white noise analogues of the Zakai and FKK equations; Zakai equation for the unnormalized density - uniqueness of solution (h bounded); uniqueness of solution (unbounded co-efficients). Part 8 Measure valued equations of filtering: multiplicative functionals of Markov processes and corresponding induced measures; filtering when signal and noise are infinite dimensional; Markov property of the optimal filter as a measure valued process; a semigroup description of the white noise filtering theory. Part 9 Prediction and smoothing: prediction and smoothing for the finite dimensional case; the general case. Part 10 Consistency and robustness of the white noise theory: general consistency results for filtering, prediction and smoothing; consistency of the unnormalized conditional densities for diffusion signal processes; consistency of the measure valued optimal filter; robustness - pathwise and statistical; smoothness properties of the conditional expectation (part contents). Part 11 Statistical applications. Part 12 Linear and quasilinear filtering theory.