The Geometry of Filtering

by Elworthy, K. David; Le Jan, Yves; Li, Xue-mei

• ISBN13: 9783034601757
• ISBN10: 3034601751
• Format: Paperback
• Copyright: 2010-12-30
• Publisher: Birkhauser

Summary

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 stochastic 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, stochastic 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 stochastic flows, and generalised Weitzenbock formulae.

Table of Contents

Introduction	p. vii
Diffusion Operators	p. 1
Representations of Diffusion Operators	p. 1
The Associated First-Order Operator	p. 4
Diffusion Operators Along a Distribution	p. 5
Lifts of Diffusion Operators	p. 7
Notes	p. 10
Decomposition of Diffusion Operators	p. 11
The Horizontal Lift Map	p. 11
Lifts of Cohesive Operators and The Decomposition Theorem	p. 17
The Lift Map for SDEs and Decomposition of Noise	p. 23
Decomposition of Stratonovich SDE's	p. 24
Decomposition of the noise and Itô SDE's	p. 25
Diffusion Operators with Projectible Symbols	p. 26
Horizontal lifts of paths and completeness of semi-connections	p. 28
Topological Implications	p. 30
Notes	p. 31
Equivariant Diffusions on Principal Bundles	p. 33
Invariant Semi-connections on Principal Bundles	p. 34
Decompositions of Equivariant Operators	p. 36
Derivative Flows and Adjoint Connections	p. 41
Associated Vector Bundles and Generalised Weitzenböck Formulae	p. 46
Notes	p. 58
Projectible Diffusion Processes and Markovian Filtering	p. 61
Integration of predictable processes	p. 62
Horizontality and filtrations	p. 66
Intertwined diffusion processes	p. 66
A family of Markovian kernels	p. 70
The filtering equation	p. 71
Approximations	p. 73
Krylov-Veretennikov Expansion	p. 74
Conditional Laws	p. 75
An SPDE example	p. 79
Equivariant case: skew-product decomposition	p. 81
Conditional expectations of induced processes on vector bundles	p. 83
Notes	p. 85
Filtering with non-Markovian Observations	p. 87
Signals with Projectible Symbol	p. 88
Innovations and innovations processes	p. 91
Classical Filtering	p. 94
Example: Another SPDE	p. 95
Notes	p. 99
The Commutation Property	p. 101
Commutativity of Diffusion Semigroups	p. 103
Consequences for the Horizontal Flow	p. 105
Example: Riemannian Submersions and Symmetric Spaces	p. 115
Riemannian Submersions	p. 115
Riemannian Symmetric Spaces	p. 116
Notes	p. 119
Example: Stochastic Flows	p. 121
Semi-connections on the Bundle of Diffeomorphisms	p. 121
Semi-connections Induced by Stochastic Flows	p. 125
Semi-connections on Natural Bundles	p. 131
Appendices	p. 135
Girsanov-Maruyama-Cameron-Martin Theorem	p. 135
Stochastic differential equations for degenerate diffusions	p. 139
Semi-martingales and ¿ -martingales along a Subbundle	p. 145
Second fundamental forms and shape operators	p. 147
Intertwined stochastic flows	p. 148
Bibliography	p. 159
Index	p. 167
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