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.
Citation
T. G. Kurtz. D. L. Ocone. "Unique Characterization of Conditional Distributions in Nonlinear Filtering." Ann. Probab. 16 (1) 80 - 107, January, 1988. https://doi.org/10.1214/aop/1176991887
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