Reddy, Anugu Sumith; Budhiraja, Amarjit; Apte, Amit Some large deviation asymptotics in small noise filtering problems. (English) Zbl 1483.60046 SIAM J. Control Optim. 60, No. 1, 385-409 (2022). Summary: We consider nonlinear filters for diffusion processes when the observation and signal noises are small and of the same order. As the noise intensities approach zero, the nonlinear filter can be approximated by a certain variational problem that is closely related to R. E. Mortensen’s optimization problem [J. Optim. Theory Appl. 2, 386–394 (1968; Zbl 0177.36004)]. This approximation result can be made precise through a certain Laplace asymptotic formula. In this work we study probabilities of deviations of true filtering estimates from that obtained by solving the variational problem. Our main result gives a large deviation principle for Laplace functionals whose typical asymptotic behavior is described by Mortensen-type variational problems. Proofs rely on stochastic control representations for positive functionals of Brownian motions and Laplace asymptotics of the Kallianpur-Striebel formula.© 2022, Society for Industrial and Applied Mathematics Cited in 1 ReviewCited in 2 Documents MSC: 60F10 Large deviations 60G35 Signal detection and filtering (aspects of stochastic processes) 93E11 Filtering in stochastic control theory Keywords:Laplace asymptotics; large deviation principle; nonlinear filtering; small observation and signal noise; minimum energy estimate; 4DVAR Citations:Zbl 0177.36004 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] G. B. Arous and F. Castell, Flow decomposition and large deviations, J. Funct. Anal., 140 (1996), pp. 23-67. · Zbl 0861.60035 [2] J. S. Baras, A. Bensoussan, and M. R. James, Dynamic observers as asymptotic limits of recursive filters: Special cases, SIAM J. Appl. Math., 48 (1988), pp. 1147-1158, https://doi.org/10.1137/0148068. · Zbl 0658.93017 [3] A. 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