Synopses & Reviews
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.
Review
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)
Synopsis
Filtering is the science of nding the law of a process given a partial observation of it. The main objects we study here are di usion processes. These are naturally associated with second-order linear di erential operators which are semi-elliptic and so introduce a possibly degenerate Riemannian structure on the state space. In fact, much of what we discuss is simply about two such operators intertwined by a smooth map, the \projection from the state space to the observations space, and does not involve any stochastic analysis. From the point of view of stochastic processes, our purpose is to present and to study the underlying geometric structure which allows us to perform the ltering in a Markovian framework with the resulting conditional law being that of a Markov process which is time inhomogeneous in general. This geometry is determined by the symbol of the operator on the state space which projects to a symbol on the observation space. The projectible symbol induces a (possibly non-linear and partially de ned) connection which lifts the observation process to the state space and gives a decomposition of the operator on the state space and of the noise. As is standard we can recover the classical ltering theory in which the observations are not usually Markovian by application of the Girsanov- Maruyama-Cameron-Martin Theorem. This structure we have is examined in relation to a number of geometrical topics.
Synopsis
The geometry used in 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. That geometry is considered in situations of geometric, stochastic analytic or filtering interest.
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.