White Noise Theory of Prediction, Filtering and Smoothing

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CRC Press, Jan 1, 1988 - Mathematics - 614 pages
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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.

 

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Contents

Probabilistic preliminaries
6
Probability measures in function spaces
16
Gaussian white noise
23
CHAPTER n Markov Processes
31
Diffusion processes
43
The FeynmanKac formula
49
CHAPTER HI Cylinder Probabilities
57
Integration with respect to cylinder probabilities
68
Uniqueness of solution unbounded coefficients
301
Measure Valued Equations
327
Filtering when signal and noise arc infinite dimensional
363
Markov property of the optimal filter as a measure valued
369
A semigroup description of the white noise filtering theory
404
Prediction and Smoothing
411
The general case
424
Consistency of the unnormalized conditional densities
449

Representation and lifting maps
80
Examples of representations of the canonical Gauss measure
100
Relation to the DunfordSchwartz theory
125
Definition
145
Cylindrical mappings
151
Conditional expectation
166
QuasiCylinder Probabilities
179
Representation of a QCPand the lifting map
185
Polish space valued mappings on E e P
197
Absolute continuity for QCPs quasi cylindrical mappings
210
Independence
228
More on canonical Gauss measure
235
The abstract statistical model and the Bayes formula
247
Uniqueness
270
Consistency of the measure valued optimal filter
466
Robustness Palhwise and statistical
478
Smoothness properties of the conditional expectation
499
Statistical Applications
511
Likelihood ratios and signal detection
518
The filtering problem for countable state Markov processes
530
Filtering for infinite dimensional processes
540
Quasilinear filtering
554
General case
560
Appendix
571
Notes
583
References
589
Index
595
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