Optimal control and pathwise nonlinear filtering of nondegenerate diffusions

A linear parabolic partial differential equation describing the pathwise filter for a nondegenerate diffusion is changed, by an exponential substitution, into the dynamic programming equation of an optimal stochastic control problem. This substitution is applied to obtain results about the rate of decay as |x| → ∞ of solutions p(x, t) to the pathwise filter equation, and for solutions of the corresponding Zakai equation.