A sufficient condition for the positive recurrence of a semimartingale reflecting Brownian motion in an orthant

Dupuis and Williams proved that a sufficient condition for the positive recurrence and the existence of a unique stationary distribution for a semimartingale reflecting Brownian motion in an orthant (SRBM) is that all solutions of an associated deterministic Skorohod problem are attracted to the origin. In this paper, we derive a sufficient condition under which we can construct an explicit linear Lyapunov function for the Skorohod problem. Thus, this implies a sufficient condition for the stability of the deterministic Skorohod problem. The existence of such a linear Lyapunov function is equivalent to the feasibility of a set of linear inequalities. In the two-dimensional case, we recover the necessary and sufficient conditions for the positive recurrence. Some explicit sufficient conditions are derived for the higher-dimensional case.