ON POSITIVE RECURRENCE OF CONSTRAINED DIFFUSION PROCESSES

Let G c R k be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces {G i , i = 1,..., , 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,., , N} is well posed and the Skorokhod map is Lipschitz continuous. Explicit conditions on the drift coefficient, b(.), of the diffusion process are given under which the constrained process is positive recurrent and has a unique invariant measure. Define C =˙ {-Σ α i d i ; α i ≥ 0, i ∈ {1,…,N}}. Then the key condition for stability is that there exists δ ∈ (0, ∞) and a bounded subset A of G such that for all x ∈ G\A, b(x) ∈ C and dist(b(x), ∂C) > δ, where ∂C denotes the boundary of C.