Stochastic Modelling and Applied Probability

By: Alan Bain, Dan Crisan

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Hardcover | 26 November 2008

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408 Pages

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24.13 x 15.88 x 2.54

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Many aspects of phenomena critical to our lives can not be measured directly. Fortunately models of these phenomena, together with more limited obs- vations frequently allow us to make reasonable inferences about the state of the systems that a?ect us. The process of using partial observations and a stochastic model to make inferences about an evolving system is known as stochastic ?ltering. The objective of this text is to assist anyone who would like to become familiar with the theory of stochastic ?ltering, whether graduate student or more experienced scientist. The majority of the fundamental results of the subject are presented using modern methods making them readily available for reference. The book may also be of interest to practitioners of stochastic ?ltering, who wish to gain a better understanding of the underlying theory. Stochastic ?ltering in continuous time relies heavily on measure theory, stochasticprocessesandstochasticcalculus.Whileknowledgeofbasicmeasure theory and probability is assumed, the text is largely self-contained in that the majority of the results needed are stated in two appendices. This should make it easy for the book to be used as a graduate teaching text. With this in mind, each chapter contains a number of exercises, with solutions detailed at the end of the chapter.

Industry Reviews

From the reviews: "This book provides a rigorous mathematical treatment of the nonlinear stochastic filtering problem with particular emphasis on numerical methods. ... The text is essentially self-contained ... . In an appendice the required results from measure theory and stochastic analysis are stated and proved. Intended readers are researchers and graduate students that have an interest in theoretical aspects of stochastic filtering. The text is supplemented with many exercises and detailed solutions. ... a standard reference for teaching and working in the field of stochastic filtering." (H. M. Mai, Zentralblatt MATH, Vol. 1176, 2010)

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You Can Find This Book In

Non-Fiction Mathematics Applied Mathematics Stochastics Calculus & Mathematical Analysis Functional Analysis & Transforms Probability & Statistics Economics, Finance, Business and Management Engineering & Technology Electronics & Communications Engineering Electronics Engineering Automatic Control Engineering

Numerical Analysis

Preface	p. v
Notation	p. xi
Introduction	p. 1
Foreword	p. 1
The Contents of the Book	p. 3
Historical Account	p. 5
Filtering Theory
The Stochastic Process [pi]	p. 13
The Observation [sigma]-algebra y[subscript t]	p. 16
The Optional Projection of a Measurable Process	p. 17
Probability Measures on Metric Spaces	p. 19
The Weak Topology on P(S)	p. 21
The Stochastic Process [pi]	p. 27
Regular Conditional Probabilities	p. 32
Right Continuity of Observation Filtration	p. 33
Solutions to Exercises	p. 41
Bibliographical Notes	p. 45
The Filtering Equations	p. 47
The Filtering Framework	p. 47
Two Particular Cases	p. 49
X a Diffusion Process	p. 49
X a Markov Process with a Finite Number of States	p. 51
The Change of Probability Measure Method	p. 52
Unnormalised Conditional Distribution	p. 57
The Zakai Equation	p. 61
The Kushner-Stratonovich Equation	p. 67
The Innovation Process Approach	p. 70
The Correlated Noise Framework	p. 73
Solutions to Exercises	p. 75
Bibliographical Notes	p. 93
Uniqueness of the Solution to the Zakai and the Kushner-Stratonovich Equations	p. 95
The PDE Approach to Uniqueness	p. 96
The Functional Analytic Approach	p. 110
Solutions to Exercises	p. 116
Bibliographical Notes	p. 125
The Robust Representation Formula	p. 127
The Framework	p. 127
The Importance of a Robust Representation	p. 128
Preliminary Bounds	p. 129
Clark's Robustness Result	p. 133
Solutions to Exercises	p. 139
Bibliographic Note	p. 139
Finite-Dimensional Filters	p. 141
The Benes Filter	p. 141
Another Change of Probability Measure	p. 142
The Explicit Formula for the Benes Filter	p. 144
The Kalman-Bucy Filter	p. 148
The First and Second Moments of the Conditional Distribution of the Signal	p. 150
The Explicit Formula for the Kalman-Bucy Filter	p. 154
Solutions to Exercises	p. 155
The Density of the Conditional Distribution of the Signal	p. 165
An Embedding Theorem	p. 166
The Existence of the Density of [rho subscript t]	p. 168
The Smoothness of the Density of [rho subscript t]	p. 174
The Dual of [rho subscript t]	p. 180
Solutions to Exercises	p. 182
Numerical Algorithms
Numerical Methods for Solving the Filtering Problem	p. 191
The Extended Kalman Filter	p. 191
Finite-Dimensional Non-linear Filters	p. 196
The Projection Filter and Moments Methods	p. 199
The Spectral Approach	p. 202
Partial Differential Equations Methods	p. 206
Particle Methods	p. 209
Solutions to Exercises	p. 217
A Continuous Time Particle Filter	p. 221
Introduction	p. 221
The Approximating Particle System	p. 223
The Branching Algorithm	p. 225
Preliminary Results	p. 230
The Convergence Results	p. 241
Other Results	p. 249
The Implementation of the Particle Approximation for [pi subscript t]	p. 250
Solutions to Exercises	p. 252
Particle Filters in Discrete Time	p. 257
The Framework	p. 257
The Recurrence Formula for [pi subscript t]	p. 259
Convergence of Approximations to [pi subscript t]	p. 264
The Fixed Observation Case	p. 264
The Random Observation Case	p. 269
Particle Filters in Discrete Time	p. 272
Offspring Distributions	p. 275
Convergence of the Algorithm	p. 281
Final Discussion	p. 285
Solutions to Exercises	p. 286
Appendices
Measure Theory	p. 293
Monotone Class Theorem	p. 293
Conditional Expectation	p. 293
Topological Results	p. 296
Tulcea's Theorem	p. 298
The Daniell-Kolmogorov-Tulcea Theorem	p. 301
Cadlag Paths	p. 303
Discontinuities of Cadlag Paths	p. 303
Skorohod Topology	p. 304
Stopping Times	p. 306
The Optional Projection	p. 311
Path Regularity	p. 312
The Previsible Projection	p. 317
The Optional Projection Without the Usual Conditions	p. 319
Convergence of Measure-valued Random Variables	p. 322
Gronwall's Lemma	p. 325
Explicit Construction of the Underlying Sample Space for the Stochastic Filtering Problem	p. 326
Stochastic Analysis	p. 329
Martingale Theory in Continuous Time	p. 329
Ito Integral	p. 330
Quadratic Variation	p. 332
Continuous Integrator	p. 338
Integration by Parts Formula	p. 341
Ito's Formula	p. 343
Localization	p. 343
Stochastic Calculus	p. 344
Girsanov's Theorem	p. 345
Martingale Representation Theorem	p. 348
Novikov's Condition	p. 350
Stochastic Fubini Theorem	p. 351
Burkholder-Davis-Gundy Inequalities	p. 353
Stochastic Differential Equations	p. 355
Total Sets in L[superscript 1]	p. 355
Limits of Stochastic Integrals	p. 358
An Exponential Functional of Brownian motion	p. 360
References	p. 367
Author Name Index	p. 383
Subject Index	p. 387
Table of Contents provided by Ingram. All Rights Reserved.

Stochastic Modelling and Applied Probability

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Hardcover

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