Stability properties of constrained jump-diffusion processes

Abstract

We consider a class of jump-diffusion processes, constrained to a polyhedral cone G ⊂ ℝn, 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 Γ∅ of a deterministic linear trajectory ∅ remains bounded if and only if ∅ ∈ 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, Lid(x) belongs to a compact subset of Co. © 2002 Applied Probability Trust.