Nonlinear Filtering and Large Deviations

We consider the nonlinear filtering problem dx = f(x)dt + ( ∈ )dw, dy = h(x)dt + ( ∈ )^1/2 dv, and obtain lim  ∈ ->0 log q^t (x,t) = -W (x, t) for unnormalised conditional densities q'(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.