Markovian projections for functionals of Itô semimartingales with jumps

Abstract

Given an Itô semimartingale $X$, its Markovian projection is an Itô semimartingale $\hat{X}$, with Markovian differential characteristics, that matches the one-dimensional marginal laws of $X$. One may even require certain functionals of the two processes to have the same fixed-time marginals, at the cost of enhancing the differential characteristics of $\hat{X}$ but still in a Markovian sense. In the continuous case, the definitive result on existence of Markovian projections was obtained by Brunick and Shreve~\cite{MR3098443}. In this paper, we extend their result to the fully general setting of Itô semimartingales with jumps.