## White Noise Theory of Prediction, Filtering and SmoothingBased 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 |

589 | |

595 | |

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### Common terms and phrases

a-field absolutely continuous abstract Wiener space additive probability measure assume Bayes formula Borel measurable canonical Gauss measure Cauchy problem Chapter characteristic functional coefficients completes the proof conditional distribution conditional expectation conditions of Theorem continuous function corresponding countably additive probability cylinder probability cylindrical mapping defined definition denote easy exists filtering problem finite dimensional finitely additive fixed following result Fubini's theorem func functions f given Q hence Hilbert space Holder continuous holds implies integral Lemma Let f Let g Let H linear Markov process martingale measurable function notation Note observations obtain orthogonal projection Polish space probability space proof of Theorem properties prove Pt(y random variables relation Remark representation space respect satisfying 1.8 Section semigroup sequence signal process stochastic Suppose Theorem 3.1 Theorem VII theory tion topology unique classical solution unique solution unnormalized conditional density weak convergence white noise Wiener process