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Monte Carlo algorithms and asymptotic problems in nonlinear filtering. (English) Zbl 1129.93535

Hida, Takeyuki (ed.) et al., Stochastics in finite and infinite dimensions. Conference in honor of Gopinath Kallianpur, Calcutta, India, December 18–23, 2000. Boston: Birkhäuser (ISBN 0-8176-4137-8). 59-87 (2000).
Summary: We are concerned with numerically feasible approximations to nonlinear filtering problems which are of interest over a very long time interval. The cost of concern is the pathwise error per unit time. In [SIAM J. Control Optim. 37, No. 6, 1946–1979 (1999; Zbl 0934.93064)], the authors have shown, under reasonable conditions, that (as time, noise bandwidth, process and filter approximation, etc.) go to their limit in any way at all, the limit of the pathwise average costs per unit time is what one gets with the optimal filter. When good approximations cannot be constructed due to excessive computational requirements, approximations based on random sampling methods (or, perhaps, combinations of sampling and analytical methods) become attractive. Extensions of the previous work to a wide class of such algorithms is dealt with, with similar results. For brevity, we confine ourselves to discrete time, but the same results hold for the continuous time case [the authors, SIAM J. Control Optim. 38, No. 6, 1874–1908 (2000; Zbl 1031.93148)].
For the entire collection see [Zbl 0952.00068].

MSC:

93E11 Filtering in stochastic control theory
60G35 Signal detection and filtering (aspects of stochastic processes)
65C05 Monte Carlo methods
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