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Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to nonlinear filtering. (English) Zbl 0963.60040

Azéma, J. (ed.) et al., Séminaire de Probabilités XXXIV. Berlin: Springer. Lect. Notes Math. 1729, 1-145 (2000).
This review paper gives a very complete approach of approximation results for a class of Feynman-Kac formulae by some branching and interacting particle systems, in cases of discrete time or continuous time. These problems are motivated by nonlinear filtering questions, but can also be applied in some physics, biology or economic modelling. The particle algorithms discussed in these notes belong to the class of Monte Carlo methods and many convergence results are available. In both situations of discrete time or continuous time, the authors investigate the asymptotic behaviour of the interacting particle systems they are interested in, when the number of particles growths. This asymptotic behaviour includes law of large numbers for the empirical measure-valued processes, large deviation principles, fluctuations, by use of different techniques, as semigroup techniques or limit theorems for processes.
The other point of interest concerns the long time behaviour of such interacting measure-valued processes. The authors link this problem with the asymptotic stability of the corresponding nonlinear limiting process in order to derive uniform convergence results with respect to the time parameter. Several variations and applications are then developed concerning principally continuous time or discrete time filtering problems.
For the entire collection see [Zbl 0940.00007].

MSC:

60G35 Signal detection and filtering (aspects of stochastic processes)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
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