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An Introduction to Stochastic Filtering Theory
Jie Xiong
Department of Mathematics
University of Tennessee
Knoxville, TN 37996-1300, USA
Contents
1 Introduction 1
1.1 Examples 1
1.2 Basic definitions and the filtering equation 6
1.3 An overview 8
2 Brownian motion and martingales 15
2.1 Martingales 15
2.2 Doob–Meyer decomposition 25
2.3 Meyer’s processes 32
2.4 Brownian motion 34
3 Stochastic integrals and Itô’s formula 36
3.1 Predictable processes 36
3.2 Stochastic integral 37
3.3 Itô’s formula 41
3.4 Martingale representation in terms of Brownian motion 46
3.5 Change of measures 52
3.6 Stratonovich integral 57
4 Stochastic differential equations 61
4.1 Basic definitions 62
4.2 Existence and uniqueness of a solution 67
4.3 Martingale problem 70
4.4 A stochastic flow 72
4.5 Markov property 79
5 Filtering model and Kallianpur–Striebel formula 82
5.1 The filtering model 82
5.2 The optimal filter 83
5.3 Filtering equation 86
5.4 Particle-system representation 93
5.5 Notes 95
6 Uniqueness of the solution for Zakai’s equation 96
6.1 Hilbert space 96
6.2 Transformation to a Hilbert space 98
6.3 Some useful inequalities 103
6.4 Uniqueness for Zakai’s equation 109
6.5 A duality representation 111
6.6 Notes 120
7 Uniqueness of the solution for the filtering equation 121
7.1 An interacting particle system 121
7.2 The uniqueness of the system 124
7.3 Uniqueness for the filtering equation 129
7.4 Notes 131
8 Numerical methods 132
8.1 Monte-Carlo method 132
8.2 A branching particle system 137
8.3 Convergence of Vnt 143
8.4 Convergence of Vn 149
8.5 Notes 155
9 Linear filtering 157
9.1 Gaussian system 157
9.2 Kalman–Bucy filtering 160
9.3 Discrete-time approximation of the Kalman–Bucy filtering 164
9.4 Some basic facts for a related deterministic control problem 165
9.5 Stability for Kalman–Bucy filtering 180
9.6 Notes 185
10 Stability of non-linear filtering 186
10.1 Markov property of the optimal filter 187
10.2 Ergodicity of the optimal filter 198
10.3 Finite memory property 205
10.4 Asymptotic stability for non-linear filtering with compact state space 211
10.5 Exchangeability of union intersection for σ-fields 223
10.6 Notes 230
11 Singular filtering 231
11.1 A special example 231
11.2 A general singular filtering model 236
11.3 Optimal filter with discrete support 240
11.4 Optimal filter supported on manifolds 245
11.5 Filtering model with Ornstein–Uhlenbeck noise 252
11.6 Notes 254
Bibliography 255
List of Notations 266
Index 269
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