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.