On Positive Recurrence of Constrained Diffusion Processes

Abstract

Let $G \subset \mathbb{R}^k$ be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces $\{G_i, i =1,\ldots, N\}$, where $n_i$ and $d_i$ denote the inward normal and direction of constraint associated with $G_i$, respectively. Stability properties of a class of diffusion processes, constrained to take values in $G$, are studied under the assumption that the Skorokhod problem defined by the data $\{(n_i,d_i),i = 1,\ldots,N\}$ is well posed and the Skorokhod map is Lipschitz continuous. Explicit conditions on the drift coefficient, $b(\cdot)$, of the diffusion process are given under which the constrained process is positive recurrent and has a unique invariant measure. Define $$\mathscr{C}\doteq \left\{ - \sum^{N}_{i=1} a_i d_i;a_i \ge 0, i \in \{1,\ldots,N\} \right\}.$$ Then the key condition for stability is that there exists $\delta \in (0,\infty)$ and a bounded subset $A$ of $G$ such that for all $x \in G\setminus A, b(x) \in \mathscr{C}$ and $\mathrm{dist} (b(x),\partial\mathscr{C}) \ge \delta$, where $\partial\mathscr{C}$denotes the boundary of $\mathscr{C}$.