Stability Properties of Constrained Jump-Diffusion Processes

We consider a class of jump-diffusion processes, constrained to a polyhedral cone $G\subset\R^n$, 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 \mathcalC such that the image \Gamma\phi of a deterministic linear trajectory \phi remains bounded if and only if \dot\phi\in\mathcalC. Denoting the generator of a corresponding unconstrained jump-diffusion by \cll, we show that a key condition for the process to admit an invariant probability measure is that for x\in G, \cll \id(x) belongs to a compact subset of \mathcalC^o.