Mimicking the marginal distributions of a semimartingale

TL;DR

This paper establishes conditions for representing the marginal distributions of a discontinuous semimartingale with a Markov process, extending classical equations to non-Markovian cases using local characteristics.

Contribution

It introduces a method to match the marginal distributions of a broad class of semimartingales with Markov processes via their local characteristics.

Findings

Derived a partial integro-differential equation for semimartingale distributions.

Extended the Kolmogorov forward equation to non-Markovian processes.

Applicable to smooth functions of Markov processes.

Abstract

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.