Fundamentals of Stochastic Filtering

Transcription

1 Alan Bain Dan Crisan Fundamentals of Stochastic Filtering Sprin ger

2 Contents Preface Notation v xi 1 Introduction Foreword The Contents of the Book Historical Account 5 Part I Filtering Theory 2 The Stochastic Process 7Г The Observation ст-algebra y t The Optional Projection of a Measurable Process Probability Measures on Metric Spaces The Weak Topology on P(S) The Stochastic Process 7r Regular Conditional Probabilities Right Continuity of Observation Filtration Solutions to Exercises Bibliographical Notes 45 3 The Filtering Equations The Filtering Framework Two Particular Cases X a Diffusion Process X a Markov Process with a Finite Number of States The Change of Probability Measure Method Unnormalised Conditional Distribution The Zakai Equation 61

3 viii Contents 3.6 The Kushner-Stratonovich Equation The Innovation Process Approach The Correlated Noise Framework Solutions to Exercises Bibliographical Notes 93 4 Uniqueness of the Solution to the Zakai and the Kushner-Stratonovich Equations The PDE Approach to Uniqueness The Functional Analytic Approach Solutions to Exercises Bibliographical Notes The Robust Representation Formula The Framework The Importance of a Robust Representation Preliminary Bounds Clark's Robustness Result Solutions to Exercises Bibliographic Note Finite-Dimensional Filters The Benes Filter Another Change of Probability Measure The Explicit Formula for the Benes Filter The Kalman-Bucy Filter The First and Second Moments of the Conditional Distribution of the Signal The Explicit Formula for the Kalman-Bucy Filter Solutions to Exercises The Density of the Conditional Distribution of the Signal An Embedding Theorem The Existence of the Density of p t The Smoothness of the Density of pt The Dual of p t Solutions to Exercises 182 Part II Numerical Algorithms 8 Numerical Methods for Solving the Filtering Problem The Extended Kaiman Filter Finite-Dimensional Non-linear Filters The Projection Filter and Moments Methods The Spectral Approach 202

4 Contents ix 8.5 Partial Differential Equations Methods Particle Methods Solutions to Exercises A Continuous Time Particle Filter Introduction The Approximating Particle System The Branching Algorithm Preliminary Results The Convergence Results Other Results The Implementation of the Particle Approximation for it t Solutions to Exercises Particle Filters in Discrete Time The Framework The Recurrence Formula for 7r t Convergence of Approximations to щ The Fixed Observation Case The Random Observation Case Particle Filters in Discrete Time Offspring Distributions Convergence of the Algorithm Final Discussion Solutions to Exercises 286 Part III Appendices A Measure Theory 293 A.l Monotone Class Theorem 293 A.2 Conditional Expectation 293 A.3 Topological Results 296 A.4 Tulcea's Theorem 298 A.4.1 The Daniell-Kolmogorov-Tulcea Theorem 301 A.5 Cädläg Paths 303 A.5.1 Discontinuities of Cädläg Paths 303 A.5.2 Skorohod Topology 304 A.6 Stopping Times 306 A.7 The Optional Projection 311 A.7.1 Path Regularity '. 312 A.8 The Previsible Projection 317 A.9 The Optional Projection Without the Usual Conditions 319 A. 10 Convergence of Measure-valued Random Variables 322 A.11 GronwalPs Lemma 325

5 x Contents A. 12 Explicit Construction of the Underlying Sample Space for the Stochastic Filtering Problem 326 В Stochastic Analysis 329 B.l Martingale Theory in Continuous Time 329 B.2 Ito Integral 330 B.2.1 Quadratic Variation 332 B.2.2 Continuous Integrator 338 B.2.3 Integration by Parts Formula 341 B.2.4 Ito's Formula 343 B.2.5 Localization 343 B.3 Stochastic Calculus 344 B.3.1 Girsanov's Theorem 345 B.3.2 Martingale Representation Theorem 348 B.3.3 Novikov's Condition 350 B.3.4 Stochastic Fubini Theorem 351 B.3.5 Burkholder-Davis-Gundy Inequalities 353 B.4 Stochastic Differential Equations 355 B.5 Total Sets in L B.6 Limits of Stochastic Integrals 358 B.7 An Exponential Functional of Brownian motion 360 References 367 Author Name Index 383 Subject Index 387