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The Geometry of Filtering

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

Part of the Frontiers in Mathematics book series (FM)

Table of contents

  1. Front Matter
    Pages i-xi
  2. K. David Elworthy, Yves Le Jan, Xue-Mei Li
    Pages 1-10
  3. K. David Elworthy, Yves Le Jan, Xue-Mei Li
    Pages 11-32
  4. K. David Elworthy, Yves Le Jan, Xue-Mei Li
    Pages 33-59
  5. K. David Elworthy, Yves Le Jan, Xue-Mei Li
    Pages 61-86
  6. K. David Elworthy, Yves Le Jan, Xue-Mei Li
    Pages 87-99
  7. K. David Elworthy, Yves Le Jan, Xue-Mei Li
    Pages 101-114
  8. K. David Elworthy, Yves Le Jan, Xue-Mei Li
    Pages 115-120
  9. K. David Elworthy, Yves Le Jan, Xue-Mei Li
    Pages 121-133
  10. K. David Elworthy, Yves Le Jan, Xue-Mei Li
    Pages 135-158
  11. Back Matter
    Pages 159-169

About this book

Introduction

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.

Keywords

diffeomorphism diffusion operator diffusion process filtering manifold semigroup stochastic flow

Authors and affiliations

  • K. David Elworthy
    • 1
  • Yves Le Jan
    • 2
  • Xue-Mei Li
    • 3
  1. 1.Inst. MathematicsUniversity of WarwickCoventryUnited Kingdom
  2. 2.CNRS, Laboratoire de MathématiquesUniversité Paris-Sud XIOrsay CedexFrance
  3. 3.Inst. MathematicsUniversity of WarwickCoventryUnited Kingdom

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-0346-0176-4
  • Copyright Information Springer Basel AG 2010
  • Publisher Name Springer, Basel
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-0346-0175-7
  • Online ISBN 978-3-0346-0176-4
  • Series Print ISSN 1660-8046
  • Series Online ISSN 1660-8054
  • Buy this book on publisher's site