Unique characterization of conditional distributions in nonlinear filtering

A 'filtered' martingale problem is defined for the problem of estimating a process X from observations of Y, where (X,Y) is Markov. We give conditions on the generator of (X,Y) that imply that the conditional distribution is the unique solution to this filtered martingale problem. We apply this result to prove uniqueness of solutions of the Kushner-Stratonovich and Zakai equations of non-linear filtering.