Stochastic invariance of closed sets for jump-diffusions with non-Lipschitz coefficients

TL;DR

This paper establishes geometric conditions ensuring the stochastic invariance of closed sets for jump-diffusions with non-Lipschitz coefficients, extending previous results to include jump processes under weak regularity assumptions.

Contribution

It provides necessary and sufficient first order geometric conditions for invariance in jump-diffusions, extending prior work to more general non-Lipschitz coefficient settings.

Findings

Characterization of invariance conditions for jump-diffusions.

Extension of previous invariance results to jump processes.

Equivalent semimartingale formulation of the invariance criteria.

Abstract

We provide necessary and sufficient first order geometric conditions for the stochastic invariance of a closed subset of R^d with respect to a jump-diffusion under weak regularity assumptions on the coefficients. Our main result extends the recent characterization proved in Abi Jaber, Bouchard and Illand (2016) to jump-diffusions. We also derive an equivalent formulation in the semimartingale framework.