Abstract

In this talk I will show how a recent reverse ergodic theorem for inhomogeneous, killed Markov chains can be used to uniquely characterise a semimartingale obliquely reflecting diffusion in some nonsmooth domains. In particular I will discuss two significant cases. In a 2-dimensional piecewise C1 domain, one can prove uniqueness under optimal conditions on the directions of reflection. In fact, in the case of a convex polygon, our conditions reduce to the well known completely-S condition, which is necessary; thus they strictly improve the Dupuis and Ishii (1993) conditions. Moreover our conditions allow for cusps in the boundary of the domain. 2-dimensional piecewise  C1 domains are of interest, for instance, in some singular stochastic control problems. In a piecewise C2  cone in arbitrary dimension, with no cusplike singularities, one can prove uniqueness assuming  conditions analogous to the 2-dimensional case and the existence of certain Lyapunov functions. Piecewise smooth cones are of interest, for instance, in diffusion approximation of some stochastic networks. I will give an example from diffusion approximation of stochastic networks where all the above conditions are met.  Time permitting, some existence results in arbitrary dimension will also be discussed. These are obtained by the constrained martingale problem approach.  This presentation is based on joint works with T.G. Kurtz and a preprint.