Stability Properties of Constrained Jump-Diffusion Processes

Abstract

We consider a class of jump-diffusion processes, constrained to a polyhedral cone G ⊂ IR 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 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 ifφ ∈ 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 C o .

Ergodic Control for Constrained Diffusions: Characterization Using HJB Equations

“…One of the key requirements for the coupling methods used in the above cited works to work, is the existence of a suitable Lyapunov function for the underlying controlled Markov processes. For the processes considered in the present work, the existence of such a Lyapunov function was proved in [1]. Using this Lyapunov function one can show that a Foster type drift criterion is satisfied for an appropriate embedded discrete time controlled Markov chain.…”

Comparison Results for Reflected Jump-diffusions in the Orthant with Variable Reflection Directions and Stability Applications

“…In this section we use the comparison results of Section 3 to establish an ergodicity criterium related to solutions of RSDEs as in equation (2.1), the main idea being to exploit those comparison properties, and the stability results in [1] for the constant reflection directions case, to provide a simple but useful ergodicity condition in the context of state-dependent directions of reflection.…”

“…Stability conditions available in the literature have been established for the constant reflection directions case (see [1]) and critically depend on the Lipschitz continuity of the corre-sponding Skorokhod map, which is not ensured in the state-dependent case.…”

Existence and characterization of product-form invariant distributions for state-dependent stochastic networks in the heavy-traffic diffusion limit

“…In general, sufficient conditions for the existence of invariant distributions for the case of state-dependent directions of reflection remains an open issue. The Lipschitz continuity of the Skorokhod map that is needed for the results in [18] cannot be guaranteed in general for the state-dependent reflection directions case that we consider here. In this paper we show how the cone type of conditions as in [18] play a role in the existence of a product-form stationary distribution in the context of state-dependent directions of reflection.…”

“…An ergodic characterization of the stationary distribution will be also provided. The main stability assumption will correspond to the class of "cone" conditions considered in [18], where sufficient conditions for the existence of a stationary distribution for reflected jump-diffusions with constant directions of reflection are provided (see also [3] in the context of SRBMs). In general, sufficient conditions for the existence of invariant distributions for the case of state-dependent directions of reflection remains an open issue.…”