Frontiers in Mathematics

The Geometry of Filtering

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

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About this book

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.

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)

Table of contents (9 chapters)

  • Diffusion Operators

    Elworthy, K. David (et al.)

    Pages 1-10

  • Decomposition of Diffusion Operators

    Elworthy, K. David (et al.)

    Pages 11-32

  • Equivariant Diffusions on Principal Bundles

    Elworthy, K. David (et al.)

    Pages 33-59

  • Projectible Diffusion Processes and Markovian Filtering

    Elworthy, K. David (et al.)

    Pages 61-86

  • Filtering with non-Markovian Observations

    Elworthy, K. David (et al.)

    Pages 87-99

Buy this book

eBook $54.99
price for USA (gross)
  • ISBN 978-3-0346-0176-4
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $69.99
price for USA
  • ISBN 978-3-0346-0175-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
The Geometry of Filtering
Authors
Series Title
Frontiers in Mathematics
Copyright
2010
Publisher
Birkhäuser Basel
Copyright Holder
Springer Basel AG
eBook ISBN
978-3-0346-0176-4
DOI
10.1007/978-3-0346-0176-4
Softcover ISBN
978-3-0346-0175-7
Series ISSN
1660-8046
Edition Number
1
Number of Pages
XI, 169
Topics