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Nonlinear filtering: Interacting particle solution. (English) Zbl 0879.60042
Summary: 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 [see {\it J. H. Holland}, “Adaptation in natural and artificial systems” (1975; Zbl 0317.68006) and {\it R. Cerf}, “Une théorie asymptotique des algorithms génétiques” (Montpellier, 1994)] and secondarily by the papers of {\it H. Kunita} [J. Multivariate Anal. 1, 365-393 (1971; Zbl 0245.93027)] and {\it Ł. Stettner} [in: Stochastic differential systems. Lect. Notes Control Inf. Sci. 126, 279-292 (1989; Zbl 0683.93082)]. 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.

60G35Signal detection and filtering (stochastic processes)
93E11Filtering in stochastic control
60F10Large deviations
60J10Markov chains (discrete-time Markov processes on discrete state spaces)
62D05Statistical sampling theory, sample surveys
65C05Monte Carlo methods
62F12Asymptotic properties of parametric estimators
62G05Nonparametric estimation
62L20Stochastic approximation
62M05Markov processes: estimation
92D15Problems related to evolution
93E10Estimation and detection in stochastic control