Martingale Problems for Conditional Distributions of Markov Processes

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Abstract
Let X be a Markov process with generator A and let Y (t )= (X(t)). The con- ditional distribution t of X(t )g iven(Y (s ): s t) is characterized as a solution of a ltered martingale problem. As a consequence, we obtain a generator/martingale problem version of a result of Rogers and Pitman on Markov functions. Applications include uniqueness of ltering equations, exchangeability of the state distribution of vector-valued processes, verication of quasireversibility, and uniqueness for martin- gale problems for measure-valued processes. New results on the uniqueness of forward equations, needed in the proof of uniqueness for the ltered martingale problem are also presented.