Markovian projections for functionals of It\^o semimartingales with jumps

TL;DR

This paper extends the theory of Markovian projections to Itô semimartingales with jumps, enabling the construction of Markov processes that match the marginal laws of more complex jump processes.

Contribution

It generalizes Brunick and Shreve's continuous case results to include jump processes, broadening the applicability of Markovian projections.

Findings

Established existence of Markovian projections for jump processes.

Extended the class of processes for which marginal law matching is possible.

Provided theoretical framework for Markovian projections with jumps.

Abstract

Given an It\^o semimartingale $X$, its Markovian projection is an It\^o semimartingale $\widehat{X}$, 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 $\widehat{X}$ 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 It\^o semimartingales with jumps.