Martingale Problems for Conditional Distributions of Markov Processes
Thomas G. Kurtz, University of Wisconsin, Madison
Abstract
Let $X$ be a Markov process with generator $A$ and let
$Y(t)=gamma (X(t))$. The conditional distribution $pi_t$ of
$X(t)$ given $sigma (Y(s):sleq t)$ 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.
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