Stability Properties of Constrained Jump-Diffusion Processes

Abstract

We consider a class of jump-diffusion processes, constrained to a polyhedral cone $G\subset\mathbb{R}^n$, where the constraint vector field is constant on each face of the boundary. The constraining mechanism corrects for ``attempts'' of the process to jump outside the domain. Under Lipschitz continuity of the Skorohod map $\Gamma$, it is known that there is a cone ${\cal C}$ such that the image $\Gamma\phi$ of a deterministic linear trajectory $\phi$ remains bounded if and only if $\dot\phi\in{\cal C}$. Denoting the generator of a corresponding unconstrained jump-diffusion by $\cal L$, we show that a key condition for the process to admit an invariant probability measure is that for $x\in G$, ${\cal L}\,{\rm id}(x)$ belongs to a compact subset of ${\cal C}^o$.