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Some Large Deviation Asymptotics In Small Noise Filtering Problems
Journal
SIAM Journal on Control and Optimization
ISSN
03630129
Date Issued
2022-01-01
Author(s)
Reddy, Anugu Sumith
Budhiraja, Amarjit
Apte, Amit
Abstract
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 Mortensen's optimization problem [R. Mortensen, J. Optim. Theory Appl., 2 (1968), pp. 386-394]. 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.
Volume
60
Publication link
Subjects