Some Large Deviations Asymptotics in Small Noise Filtering Problems

TL;DR

This paper investigates the probabilities of deviations in small noise nonlinear filtering, deriving large deviation principles that connect the true filter behavior to variational approximations related to Mortensen's problem.

Contribution

It establishes a large deviation principle for nonlinear filters in small noise regimes, linking the asymptotics to Mortensen-type variational problems with rigorous proofs.

Findings

Large deviation principles for filtering estimates

Asymptotic behavior described by Mortensen's variational problem

Laplace asymptotics applied to Kallianpur-Striebel formula

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(1968). 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.