Martingale Problems for Conditional Distributions of Markov Processes

Let X be a Markov process with generator A and let Y(t)=γ (X(t)). The conditional distribution πₜ of $X(t)$ given σ (Y(s):s≤ t) is characterized as a solution of a filtered 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 filtering equations, exchangeability of the state distribution of vector-valued processes, verification of quasireversibility, and uniqueness for martingale problems for measure-valued processes. New results on the uniqueness of forward equations, needed in the proof of uniqueness for the filtered martingale problem are also presented.