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Discrete filtering using branching and interacting particle systems. (English) Zbl 0967.93088

The aim of this paper is to provide a unified and simple approach for proving the validity of a series of numerical algorithms designed for solving discrete-time filtering problems.

It is known, that the stochastic filtering problem consists in effectively estimating the conditional distribution of a process (the signal) given the “noisy” information obtained from a related process (the observation). The basic problem can be identified in many applications: signal processing, radar control, satellite tracking, weather forecasting, speech recognition, etc. Several recently suggested approaches are based on the simulation of interacting and branching particle systems.

The authors introduce the general description of a branching and interacting particle system (BIPS) model and also present various types of branching mechanisms for these systems. The objective of the paper is achieved by proving the convergence of the empirical measures associated with the BIPS as the initial number of particles tends to $\infty ·$ The first result states that statistics of the BIPS converge in ${L}^{2}$ to the corresponding statistics of the solution to some measure valued dynamical system. The second result states that the convergence is pointwise. The problem of nonlinear filtering is introduced and one applies in this context the results stated above for abstract measure valued dynamical systems. In this case, the empirical measure associated to the BIPS converges to the posterior measure of the signal.

This publication contains also further remarks on multinomial BIPS and connections between particular classes of BIPS.

##### MSC:
 93E11 Filtering in stochastic control 60G57 Random measures 93C55 Discrete-time control systems 93C10 Nonlinear control systems 65C35 Stochastic particle methods (numerical analysis)