  We exhibit conditions under which the flow of marginal distributions of a
discontinuous semimartingale $\xi$ can be matched by a Markov process, whose
infinitesimal generator is expressed in terms of the local characteristics of
$\xi$. Our construction applies to a large class of semimartingales, including
smooth functions of a Markov process. We use this result to derive a partial
integro-differential equation for the one-dimensional distributions of a
semimartingale, extending the Kolmogorov forward equation to a non-Markovian
setting.
