Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant (Q1326346)

This work is concerned with the existence and uniqueness of a class of semimartingale reflecting Brownian motions which live in the nonnegative orthant of \(\mathbb{R}^ d\). Loosely speaking, such a process has a semimartingale decomposition such that in the interior of the orthant the process behaves like a Brownian motion with a constant drift and covariance matrix, and at each of the \((d-1)\)-dimensional faces that form the boundary of the orthant, the bounded variation part of the process increases in a given direction (constant for any particular face) so as to confine the process to the orthant. For historical reasons, this ``pushing'' at the boundary is called instantaneous reflection. \textit{M. I. Reiman} and \textit{R. J. Williams} [ibid. 77, No. 1, 87-97 (1988; Zbl 0617.60081)] proved that a necessary condition for the existence of such a semimartingale reflecting Brownian motion (SRBM) is that the reflection matrix formed by the directions of reflection be completely-\({\mathcal S}\). We prove that this condition is sufficient for the existence of an SRBM and that the SRBM is unique in law. It follows from the uniqueness that an SRBM defines a strong Markov process. Our results have potential application to the study of diffusions arising as approximations to multiclass queueing networks.