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Nonlinear filtering and measure-valued processes. (English) Zbl 0888.93056

The authors construct a sequence of branching particle systems with widely varying space and time distributional branching generating function. This sequence is used to prove the existence of a measure-valued branching process which is a weak solution to the “Zakai equation”, that is to say the evolution equation to the unnormalized filter of a standard nonlinear filtering problem (the signal process and the observation process are driven by a standard Brownian motion). Actually, the particle systems approximation can be used to solve numerically the filtering problem: the idea is to approach the Zakai equation by creating a sample from the posterior measure. The authors confess not a plain success but are able to produce arbitrarily good approximations. The hope is to improve the more often used method of the extended Kalman filter, despite the fact that the convergence could be quite slow.

MSC:

93E11 Filtering in stochastic control theory
60G57 Random measures
65C99 Probabilistic methods, stochastic differential equations
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