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.
@article{1176991887,
author = {Kurtz, T. G. and Ocone, D. L.},
title = {Unique Characterization of Conditional Distributions in Nonlinear Filtering},
journal = {Ann. Probab.},
volume = {16},
number = {4},
year = {1988},
pages = { 80-107},
language = {en},
url = {http://dml.mathdoc.fr/item/1176991887}
}
Kurtz, T. G.; Ocone, D. L. Unique Characterization of Conditional Distributions in Nonlinear Filtering. Ann. Probab., Tome 16 (1988) no. 4, pp. 80-107. http://gdmltest.u-ga.fr/item/1176991887/