Nonlinear Filtering and Large Deviations: A PDE-Control Theoretic Approach.

Abstract

We consider the asymptotic nonlinear filtering problem dx = f(x)dt + SQRT EPSILONdw = h(x)dt + SQRT EPSILONdv, and obtain lim_EPSILON-->0 EPSILONlog q^EPSILON(x, t) = -W(x, t) for unnormalized conditional densities q^EPSILON(x, t) using PDE methods. Here, W(x, t) is the value function for a deterministic optimal control problem arising in Mortensen's deterministic estimation, and is the unique viscocity solution of a Hamiltonian-Jacobi-Bellman equation. Hijab has also studied this filtering problem, and we extend his large deviation result for certain unnormalized conditional measures. The resulting variational problem corresponds to the above control problem.