Simple necessary and sufficient conditions for the stability of constrained processes. (English) Zbl 0934.93068

Budhiraja, Amarjit; Dupuis, Paul

SIAM J. Appl. Math. 59, No. 5, 1686-1700 (1999).

In recent years a new approach has emerged for analysing the stability properties of constrained stochastic processes. In this approach the stability of the stochastic model follows if all solutions of the deterministic model are attracted to the origin. This paper shows that a rather sharp characterization for the stability of the deterministic model is possible when it can be represented in terms of a so-called “regular” Skorokhod map. The paper is dedicated to the qualitative analysis of a family of constrained ordinary differential equations. These differential equations provide yet another characterization of the studied deterministic models. It is shown that under the assumption that the Skorokhod problem is well behaved, simple explicit conditions for the stability and instability of solutions to constrained differential equations are isolated. There are a number of results showing that when solutions of a deterministic model are attracted to the origin, then a corresponding stochastic model is stable. This paper proves the following result: if the solution to the deterministic model is unbounded, then the associated stochastic model is transient. The case of a reflected Brownian motion model is considered in detail.

Reviewer: Vjatscheslav Vasiliev (Tomsk)

MSC:

93E15

Stochastic stability in control theory

Keywords:

stability; Skorokhod problem; constrained ordinary differential equation; reflecting Brownian motion; law of large numbers