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.
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