Unique Characterization of Conditional Distributions in Nonlinear Filtering

Abstract

Let $(X, Y)$ solve the martingale problem for a given generator $A$. This paper studies the problem of uniquely characterizing the conditional distribution of $X(t)$ given observations $\{Y(s)\mid 0 \leq s \leq t\}$. We define a filtered martingale problem for $A$ and we show, given appropriate hypotheses on $A$, that the conditional distribution is the unique solution to the filtered martingale problem for $A$. Using these results, we then prove that the solutions to the Kushner-Stratonovich and Zakai equations for filtering Markov processes in additive white noise are unique under fairly general circumstances.