Branching and interacting particle systems approximations of feynman-kac formulae with applications to non-linear filtering

  • P. Del Moral
  • L. Miclo
Cours Spécialisé
Part of the Lecture Notes in Mathematics book series (LNM, volume 1729)

Abstract

This paper focuses on interacting particle systems methods for solving numerically a class of Feynman-Kac formulae arising in the study of certain parabolic differential equations, physics, biology, evolutionary computing, nonlinear filtering and elsewhere. We have tried to give an “exposé” of the mathematical theory that is useful for analyzing the convergence of such genetic-type and particle approximating models including law of large numbers, large deviations principles, fluctuations and empirical process theory as well as semigroup techniques and limit theorems for processes.

In addition, we investigate the delicate and probably the most important problem of the long time behavior of such interacting measure valued processes.

We will show how to link this problem with the asymptotic stability of the corresponding limiting process in order to derive useful uniform convergence results with respect to the time parameter.

Several variations including branching particle models with random population size will also be presented. In the last part of this work we apply these results to continuous time and discrete time filtering problems.

Keywords

Interacting and branching particle systems genetic algorithms weighted sampling Moran processes measure valued dynamical systems defined by Feynman-Kac formulae asymptotic stability chaos weak propagation large deviations principles central limit theorems nonlinear filtering A.M.S. codes 60G35 60F10 60H10 60G57 60K35 60F05 62L20 92D25 92D15 93E10 93E11 

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Authors and Affiliations

  • P. Del Moral
    • 1
  • L. Miclo
    • 1
  1. 1.Laboratoire de Statistiques et Probabilités CNRS UMR C5583Université Paul SabatierToulouse cedexFrance

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