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The Geometry of Filtering

The geometry which is the topic of this book is that determined by a map of one space N onto another, M, mapping a diffusion process, or operator, on N to one on M.

Filtering theory is the science of obtaining or estimating information about a system from partial and possibly flawed observations of it. The system itself may be random, and the flaws in the observations can be caused by additional noise. In this volume the randomness and noises will be of Gaussian white noise type so that the system can be modelled by a diffusion process; that is it evolves continuously in time in a Markovian way, the future evolution depending only on the present situation.

This is the standard situation of systems governed by Ito type shastic differential equations. The state space will be the smooth manifold, N, possibly infinite dimensional, and the "observations" will be obtained by a smooth map onto another manifold, N, say. We emphasise that the geometry is important even when both manifolds are Euclidean spaces. This can also be viewed from a purely partial differential equations viewpoint as one smooth second order elliptic partial differential operator lying above another, both with no zero order term.

We consider the geometry of this situation with special emphasis on situations of geometric, shastic analytic, or filtering interest. The most well studied case is of one Brownian motion being mapped to another with a consequent skew product decomposition (or equivalently the case of Riemannian submersions). This sort of decomposition is generalised and a key to the rest of the book. It is used to study in particular, classical filtering, (semi-)connections determined by shastic flows, and generalised Weitzenbock formulae.

1018297354
The Geometry of Filtering

The geometry which is the topic of this book is that determined by a map of one space N onto another, M, mapping a diffusion process, or operator, on N to one on M.

Filtering theory is the science of obtaining or estimating information about a system from partial and possibly flawed observations of it. The system itself may be random, and the flaws in the observations can be caused by additional noise. In this volume the randomness and noises will be of Gaussian white noise type so that the system can be modelled by a diffusion process; that is it evolves continuously in time in a Markovian way, the future evolution depending only on the present situation.

This is the standard situation of systems governed by Ito type shastic differential equations. The state space will be the smooth manifold, N, possibly infinite dimensional, and the "observations" will be obtained by a smooth map onto another manifold, N, say. We emphasise that the geometry is important even when both manifolds are Euclidean spaces. This can also be viewed from a purely partial differential equations viewpoint as one smooth second order elliptic partial differential operator lying above another, both with no zero order term.

We consider the geometry of this situation with special emphasis on situations of geometric, shastic analytic, or filtering interest. The most well studied case is of one Brownian motion being mapped to another with a consequent skew product decomposition (or equivalently the case of Riemannian submersions). This sort of decomposition is generalised and a key to the rest of the book. It is used to study in particular, classical filtering, (semi-)connections determined by shastic flows, and generalised Weitzenbock formulae.

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The Geometry of Filtering

The Geometry of Filtering

The Geometry of Filtering

The Geometry of Filtering

Overview

The geometry which is the topic of this book is that determined by a map of one space N onto another, M, mapping a diffusion process, or operator, on N to one on M.

Filtering theory is the science of obtaining or estimating information about a system from partial and possibly flawed observations of it. The system itself may be random, and the flaws in the observations can be caused by additional noise. In this volume the randomness and noises will be of Gaussian white noise type so that the system can be modelled by a diffusion process; that is it evolves continuously in time in a Markovian way, the future evolution depending only on the present situation.

This is the standard situation of systems governed by Ito type shastic differential equations. The state space will be the smooth manifold, N, possibly infinite dimensional, and the "observations" will be obtained by a smooth map onto another manifold, N, say. We emphasise that the geometry is important even when both manifolds are Euclidean spaces. This can also be viewed from a purely partial differential equations viewpoint as one smooth second order elliptic partial differential operator lying above another, both with no zero order term.

We consider the geometry of this situation with special emphasis on situations of geometric, shastic analytic, or filtering interest. The most well studied case is of one Brownian motion being mapped to another with a consequent skew product decomposition (or equivalently the case of Riemannian submersions). This sort of decomposition is generalised and a key to the rest of the book. It is used to study in particular, classical filtering, (semi-)connections determined by shastic flows, and generalised Weitzenbock formulae.

Product Details

ISBN-13: 9783034601757
Publisher: Springer Basel
Publication date: 11/30/2010
Series: Frontiers in Mathematics Series
Edition description: 2010
Pages: 169
Product dimensions: 6.61(w) x 9.45(h) x 0.02(d)

Table of Contents

Introduction vii

1 Diffusion Operators 1

1.1 Representations of Diffusion Operators 1

1.2 The Associated First-Order Operator 4

1.3 Diffusion Operators Along a Distribution 5

1.4 Lifts of Diffusion Operators 7

1.5 Notes 10

2 Decomposition of Diffusion Operators 11

2.1 The Horizontal Lift Map 11

2.2 Lifts of Cohesive Operators and The Decomposition Theorem 17

2.3 The Lift Map for SDEs and Decomposition of Noise 23

2.3.1 Decomposition of Stratonovich SDE's 24

2.3.2 Decomposition of the noise and Itô SDE's 25

2.4 Diffusion Operators with Projectible Symbols 26

2.5 Horizontal lifts of paths and completeness of semi-connections 28

2.6 Topological Implications 30

2.7 Notes 31

3 Equivariant Diffusions on Principal Bundles 33

3.1 Invariant Semi-connections on Principal Bundles 34

3.2 Decompositions of Equivariant Operators 36

3.3 Derivative Flows and Adjoint Connections 41

3.4 Associated Vector Bundles and Generalised Weitzenböck Formulae 46

3.5 Notes 58

4 Projectible Diffusion Processes and Markovian Filtering 61

4.1 Integration of predictable processes 62

4.2 Horizontality and filtrations 66

4.3 Intertwined diffusion processes 66

4.4 A family of Markovian kernels 70

4.5 The filtering equation 71

4.6 Approximations 73

4.7 Krylov-Veretennikov Expansion 74

4.8 Conditional Laws 75

4.9 An SPDE example 79

4.10 Equivariant case: skew-product decomposition 81

4.11 Conditional expectations of induced processes on vector bundles 83

4.12 Notes 85

5 Filtering with non-Markovian Observations 87

5.1 Signals with Projectible Symbol 88

5.2 Innovations and innovations processes 91

5.3 Classical Filtering 94

5.4 Example: Another SPDE 95

5.5 Notes 99

6 The Commutation Property 101

6.1 Commutativity of Diffusion Semigroups 103

6.2 Consequences for the Horizontal Flow 105

7 Example: Riemannian Submersions and Symmetric Spaces 115

7.1 Riemannian Submersions 115

7.2 Riemannian Symmetric Spaces 116

7.3 Notes 119

8 Example: Stochastic Flows 121

8.1 Semi-connections on the Bundle of Diffeomorphisms 121

8.2 Semi-connections Induced by Stochastic Flows 125

8.3 Semi-connections on Natural Bundles 131

9 Appendices 135

9.1 Girsanov-Maruyama-Cameron-Martin Theorem 135

9.2 Stochastic differential equations for degenerate diffusions 139

9.3 Semi-martingales and Γ -martingales along a Subbundle 145

9.4 Second fundamental forms and shape operators 147

9.5 Intertwined stochastic flows 148

Bibliography 159

Index 167

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