The Geometry of Filtering

1. Front Matter

   Pages i-xi

   PDF

2. Diffusion Operators

   • K. David Elworthy, Yves Le Jan, Xue-Mei Li

   Pages 1-10

3. Decomposition of Diffusion Operators

   • K. David Elworthy, Yves Le Jan, Xue-Mei Li

   Pages 11-32

4. Equivariant Diffusions on Principal Bundles

   • K. David Elworthy, Yves Le Jan, Xue-Mei Li

   Pages 33-59

5. Projectible Diffusion Processes and Markovian Filtering

   • K. David Elworthy, Yves Le Jan, Xue-Mei Li

   Pages 61-86

6. Filtering with non-Markovian Observations

   • K. David Elworthy, Yves Le Jan, Xue-Mei Li

   Pages 87-99

7. The Commutation Property

   • K. David Elworthy, Yves Le Jan, Xue-Mei Li

   Pages 101-114

8. Example: Riemannian Submersions and Symmetric Spaces

   • K. David Elworthy, Yves Le Jan, Xue-Mei Li

   Pages 115-120

9. Example: Stochastic Flows

   • K. David Elworthy, Yves Le Jan, Xue-Mei Li

   Pages 121-133

10. Appendices

    • K. David Elworthy, Yves Le Jan, Xue-Mei Li

    Pages 135-158

11. Back Matter

    Pages 159-169

    PDF

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.

• diffeomorphism
• diffusion operator
• diffusion process
• filtering
• manifold
• semigroup
• stochastic flow

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)

• Inst. Mathematics, University of Warwick, Coventry, United Kingdom

  K. David Elworthy, Xue-Mei Li

• CNRS, Laboratoire de Mathématiques, Université Paris-Sud XI, Orsay Cedex, France

  Yves Le Jan

Policies and ethics