Nonlinear Filtering: Interacting Particle Solution

P. Del Moral

1996, v.2, №4, 555-580

ABSTRACT

This paper covers stochastic particle methods for the numerical solution of the nonlinear filtering equations based on the simulation of interacting particle systems. The main contribution of this paper is to prove convergence of such approximations to the optimal filter, thus yielding what seems to be the first convergence results for such approximations of the nonlinear filtering equations. This new treatment has been influenced primarily by the development of genetic algorithms [J.H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, 1975; R. Cerf, Une thГ©orie asymptotique des Algorithmes GГ©nГ©tiques, UniversitГ© Montpellier II, Sciences et techniques du Languedoc, 1994] and secondarily by the papers of Kunita and Stettner [H. Kunita, Asymptotic behavior of nonlinear filtering errors of Markov processes, J. Multivar. Anal., 1971, v.1, N4, 365-393; L. Stettner, On Invariant Measures of Filtering Processes, Stochastic Differential Systems, Lect. Notes in Control and Inform. Sci., 1989, 126, Springer Verlag]. Such interacting particle solutions encompass genetic algorithms. Incidentally, our models provide essential insight for the analysis of genetic algorithms with a non-homogeneous fitness function with respect to time.

Keywords: nonlinear filtering,Bayesian estimating,Monte Carlo sampling,interacting particle systems,sampling-resampling,evolutionary processes,genetic algorithms

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