Probability theory / Partial differential equations Matching marginal distributions of a semimartingale with a Markov process

We give conditions under which the flow of marginal distributions of a discontinuous semimartingale 푋 can be matched by a Markov process whose infinitesimal generator can be expressed in terms of the local characteristics of 푋, generalizing a result of Gyöngy (1986) to the discontinuous case. Our construction preserves the martin-gale property and allows to derive a partial integro-differential equation for the one-dimensional distribution of discontinuous semimartingales, extending the Kolmogorov forward equation to a non-Markovian setting