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

We consider the nonlinear filtering problem dx = f(x)dt + √(є)dw, dy = h(x)dt + √(є)dv, and obtain lim[є→0] є log qє (x,t) = -W(x,t) for unnormalized conditional densities qє (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 viscosity solution of a Hamilton-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.