The Geometry of Filtering by K. David ElworthyThe Geometry of Filtering by K. David Elworthy

The Geometry of Filtering

byK. David Elworthy, Yves Le Jan, Xue-Mei Li

Paperback | November 30, 2010

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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.
Title:The Geometry of FilteringFormat:PaperbackDimensions:169 pagesPublished:November 30, 2010Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3034601751

ISBN - 13:9783034601757


Table of Contents

1 Diffusion Operators.- Representations of Diffusion Operators .- The Associated First Order Operator.- Diffusion Operators Along a Distribution.- Lifts of Diffusion Operators .- Notes.- 2 Decomposition of Diffusion Operators.- The Horizontal Lift Map.- Example: The Horizontal Lift Map of SDEs .- Lifts of Cohesive Operators and The Decomposition Theorem.- Diffusion Operators with Projectible Symbols.- Horizontal lifts of paths and completeness of semi-connections.- Topological Implications.- 3 Equivariant Diffusions on Principal Bundles.- Invariant Semi-connections on Principal Bundles.- Decompositions of Equivariant Operators.- Derivative Flows and Adjoint Connections.- Vector Bundles and Generalised Weitzenböck Formulae.- 4 Projectible Diffusion Processes.- Integration of predictable processes.- Horizontality and filtrations.- The Filtering Equation.- A family of Markovian kernels.- The filtering equation.- Approximations.- Krylov-Veretennikov Expansion.- Conditional Laws.- Equivariant case: skew product decomposition.- Conditional expectations of induced processes on vector bundles.- 5 Filtering with non-Markovian Observations.- Signals with Projectible Symbol.- Innovations and innovations processes.- Classical Filtering.- Examples.- 6 The Commutation Property.- Commutativity of Diffusion Semigroups.- Consequences for the Horizontal Flow.- 7 Example: Riemannian Submersions and Symmetric Spaces.- Riemannian Submersions.- Riemannian Symmetric Spaces.- 8 Example: Stochastic Flows.- Semi-connections on the Bundle of Diffeomorphisms.- Semi-connections Induced by Stochastic Flows.- Semi-connections on Natural Bundles.- 9 Appendices.- Girsanov-Maruyama-Cameron-Martin Theorem.-Stochastic differential equations for degenerate diffusions.- Semi-martingales and G-martingales along a Sub-bundle.

Editorial Reviews

From the reviews:"The book provides a unified treatment of geometric structures related to filtering and extends in particular the earlier lecture notes of the authors . . The methods described are of essential interest for any researcher in the field of random dynamical systems and stochastic differential equations." (Anton Thalmaier, Mathematical Reviews, Issue 2012 e)