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 J. H. Holland
, “Adaptation in natural and artificial systems” (1975; Zbl 0317.68006
) and R. Cerf
, “Une théorie asymptotique des algorithms génétiques” (Montpellier, 1994)] and secondarily by the papers of H. Kunita
[J. Multivariate Anal. 1, 365-393 (1971; Zbl 0245.93027
)] and 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.