Format:
Online-Ressource
,
v.: digital
Edition:
Online-Ausg. Springer eBook Collection. Mathematics and Statistics Electronic reproduction; Available via World Wide Web
ISBN:
9783034601764
Series Statement:
Frontiers in Mathematics
Content:
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.
Note:
Includes bibliographical references and index
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The Geometry of Filtering; Contents; Introduction; Chapter 1 Diffusion Operators; Chapter 2 Decomposition of Diffusion Operators; Chapter 3 Equivariant Diffusions onPrincipal Bundles; Chapter 4 Projectible Diffusion Processes and Markovian Filtering; Chapter 5 Filtering with non-Markovian Observations; Chapter 6 The Commutation Property; Chapter 7 Example: Riemannian Submersions and Symmetric Spaces; Chapter 8 Example: Stochastic Flows; Chapter 9 Appendices; Bibliography; Index
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Cover13;The Geometry of Filtering -- Contents -- Introduction -- Chapter 1 Diffusion Operators -- 1.1 Representations of Diffusion Operators -- 1.2 The Associated First-Order Operator -- 1.3 Diffusion Operators Along a Distribution -- 1.4 Lifts of Diffusion Operators -- 1.5 Notes -- On symbols -- Chapter 2 Decomposition of Diffusion Operators -- 2.1 The Horizontal Lift Map -- 2.2 Lifts of Cohesive Operators and The Decomposition Theorem -- 2.3 The Lift Map for SDEs and Decomposition of Noise -- 2.3.1 Decomposition of Stratonovich SDE's -- 2.3.2 Decomposition of the noise and It244; SDE's -- 2.4 Diffusion Operators with Projectible Symbols -- 2.5 Horizontal lifts of paths and completeness of semi-connections -- 2.6 Topological Implications -- 2.7 Notes -- Intertwined stochastic differential equations -- Chapter 3 Equivariant Diffusions onPrincipal Bundles -- 3.1 Invariant Semi-connections on Principal Bundles -- 3.2 Decompositions of Equivariant Operators -- 3.3 Derivative Flows and Adjoint Connections -- 3.4 Associated Vector Bundles and Generalised Weitzenb246;ock Formulae -- 3.5 Notes -- G-invariant diffusion operators -- Canonical vertical diffusions on GLM -- Chapter 4 Projectible Diffusion Processes and Markovian Filtering -- 4.1 Integration of predictable processes -- 4.2 Horizontality and filtrations -- 4.3 Intertwined diffusion processes -- 4.4 A family of Markovian kernels -- 4.5 The filtering equation -- 4.6 Approximations -- 4.7 Krylov-Veretennikov Expansion -- 4.8 Conditional Laws -- 4.9 An SPDE example -- 4.10 Equivariant case: skew-product decomposition -- 4.11 Conditional expectations of induced processes on vector bundles -- 4.12 Notes -- Noise-free observations -- Krylov-Veretennikov formula -- Skew-product decompositions, regular conditional probabilities and SDE -- Operators on differential forms -- Chapter 5 Filtering with non-Markovian Observations -- 5.1 Signals with Projectible Symbol -- 5.2 Innovations and innovations processes -- 5.3 Classical Filtering -- 5.4 Example: Another SPDE -- 5.5 Notes -- Noise-free observations -- Chapter 6 The Commutation Property -- 6.1 Commutativity of Diffusion Semigroups -- 6.2 Consequences for the Horizontal Flow -- Chapter 7 Example: Riemannian Submersions and Symmetric Spaces -- 7.1 Riemannian Submersions -- 7.2 Riemannian Symmetric Spaces -- 7.3 Notes -- Intertwining of Laplacians etc. on functions and forms -- Chapter 8 Example: Stochastic Flows -- 8.1 Semi-connections on the Bundle of Diffeomorphisms -- 8.2 Semi-connections Induced by Stochastic Flows -- 8.3 Semi-connections on Natural Bundles -- Flows on Non-compact Manifolds -- Chapter 9 Appendices -- 9.1 Girsanov-Maruyama-Cameron-Martin Theorem -- 9.2 Stochastic differential equations for degenerate diffusions -- 9.2.1 Square roots of symbols of constant rank -- 9.2.2 A smooth diffusion operator with no smooth H246;rmander form -- 9.2.3 Non-existence of C178; flow-like couplings -- 9.2.4 Locally Lipschitz square roots and It244; equations -- 9.2.5 Miscellaneous results -- 9.3 Semi-martingales and 1043;-martingales along a Subbundle -- 9.4 Second fundamental forms and shape operators -- 9.5 Intertwined stochastic flows -- 9.5.1 Intertwined reproducing kernels and Gaussian spaces of vector 12;fields -- 9.5.2 Intertwined stochastic flows that induce Levi-Civita co.
,
Electronic reproduction; Available via World Wide Web
Additional Edition:
ISBN 9783034601757
Additional Edition:
Erscheint auch als Druck-Ausgabe Elworthy, Kenneth D. The Geometry of filtering Basel [u.a.] : Birkhäuser, 2010 ISBN 9783034601757
Language:
English
Subjects:
Mathematics
Keywords:
Differentialgeometrie
;
Filtertheorie
DOI:
10.1007/978-3-0346-0176-4
URL:
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