Mimicking the marginal distributions of a semimartingale

We exhibit conditions under which the flow of marginal distributions of a discontinuous semimartingale $ξ$ can be matched by a Markov process, whose infinitesimal generator is expressed in terms of the local characteristics of $ξ$. 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.

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