Stability Properties of Constrained Jump - Diffusion Processes

We consider a class of jump-diffusion processes, constrained to a polyhedral cone G ⊂ IRn, 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 Γ, it is known that there is a cone C such that the image Γφ of a deterministic linear trajectory φ remains bounded if and only if φ̇ ∈ C. Denoting the generator of a corresponding unconstrained jump-diffusion by L, we show that a key condition for the process to admit an invariant probability measure is that for x ∈ G, L id(x) belongs to a compact subset of Co.