Amarjit Budhiraja — Page 6

arXiv2005

Stability Properties of Constrained Jump-Diffusion Processes

We consider a class of jump-diffusion processes, constrained to a polyhedral cone $G\subset\R^n$, 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 \mathcalC such that the image Γϕof a deterministic linear trajectory ϕremains bounded if and only if \dotϕ\in\mathcalC. Denoting the generator of a corresponding unconstrained jump-diffusion by \cll, we show that a key condition for the process to admit an invariant probability measure is that for x\in G, \cll \id(x) belongs to a compact subset of \mathcalC^o.

math.PR↗

arXiv2005

On Positive Recurrence of Constrained Diffusion Processes

Let G \subset \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(\cdot), of the diffusion process are given under which the constrained process is positive recurrent and has a unique invariant measure. Define \C \Df{- \sum_{i=1}^N α_i d_i; α_i \ge 0, i \in \{1, ..., N}}. Then the key condition for stability is that there exists δ\in (0, \infty) and a bounded subset A of G such that for all x \in G\backslash A, b(x) \in \C and \dist(b(x), \partial \C) \ge δ, where \partial \C denotes the boundary of \C.

math.PR↗