Uniqueness and robustness of solution of measure-valued equations of nonlinear filtering

We consider the Zakai equation for the unnormalized conditional distribution sigma when the signal process X takes values in a complete separable metric space E and when h is a continuous, possibly unbounded function on E. It is assumed that X is a Markov process which is characterized via a martingale problem for an operator A(0). Uniqueness of solution for the measure-valued Zakai and Fujisaki-Kallianpur-Kunita equations is proved when the test functions belong to the domain of A(0). It is also shown that the conditional distributions are robust.