The filtered martingale problem

Let X be a Markov process characterized as the solution of a martingale problem with generator A, and let Y be a related observation process. The conditional distribution \pi_t of X(t) given observations of Y up to time t satisfies certain martingale properties, and it is shown that any probability-measure-valued process with the appropriate martingale properties can be interpreted as the conditional distribution of X for some observation process. In particular, if Y (t) = \gamma(X(t)) for some measurable mapping \gamma, the conditional distribution of X(t) given observations of Y up to time t is characterized as the solution of a filtered martingale problem. Uniqueness for the original martingale problem implies uniqueness for the filtered martingale problem which in turn implies the Markov property for the conditional distribution considered as a probability-measure-valued process. Other applications include a Markov mapping theorem and uniqueness for filtering equations.