Simple Necessary and Sufficient Conditions for the Stability of Constrained Processes

In recent years a new approach has emerged for analyzing the stability properties of constrained stochastic processes. In this approach, one associates with the stochastic model a deterministic model (or a family of deterministic models), and, under appropriate conditions, stability of the stochastic model follows if all solutions of the deterministic model are attracted to the origin. In the present work we show that a rather sharp characterization for the stability of the deterministic model is possible when it can be represented in terms of what we call a “regular” Skorokhod Map. Let G be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces {Gi, i = 1, ..., N}, where ni and di denote the inward normal and direction of constraint associated with Gi. Suppose that the Skorokhod Problem defined by the data {(ni, di), i = 1, ..., N} is regular. Under these conditions, the deterministic model mentioned above will correspond to a law of large numbers limit of the original stochastic model with a well defined constant velocity v on the interior of G. Define C . = { − N ∑ i=1 αidi : αi ≥ 0, i ∈ {1, ..., N} } . Then the characterization of stability is as follows. If v is in the interior of C then all solutions of the deterministic model are attracted to the origin. This condition is in a certain sense sharp, and indeed we also prove that if v is outside C then all solutions of the deterministic model diverge to ∞. In the final case, where v is on the boundary of C, we show that all solutions of the deterministic model are bounded, and that for at least one initial condition the corresponding trajectory is not attracted to the origin. Our final contribution is a complement to the existing theory that relates stability of processes and deterministic models. Using reflecting Brownian motion as the canonical example of a constrained process, we prove that if the deterministic model diverges to ∞ for any initial condition, then any associated reflecting Brownian motion must be transient. This result can easily be extended to other classes of constrained processes, and thus the sharp characterization of stability for the deterministic model yields a sharp characterization of stability for related stochastic processes.