A SUFFICIENT CONDITION FOR THE POSITIVE ... - Project Euclid
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A SUFFICIENT CONDITION FOR THE POSITIVE ... - Project Euclid
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A SUFFICIENT CONDITION FOR THE POSITIVE RECURRENCE OF A SEMIMARTINGALE REFLECTING BROWNIAN MOTION IN AN ORTHANT1 By Hong Chen University of British Columbia 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.
1. Introduction. A semimartingale reflecting Brownian motion (SRBM) J evolves in a J-dimensional nonnegative orthant 0;
(12)
u0a Rab + u0b Rb > 0:
Note that any principal submatrices of a nonsingular M-matrix are nonsingular M-matrices. Hence, Rb is a nonsingular M-matrix and Rb−1 exists and is nonnegative. Since R is a non-singular M-matrix, Rba ≤ 0. Then multiplying (12) by Rb−1 Rba from the right, we have (13)
u0b Rba ≤ −u0a Rab Rb−1 Rba :
764
H. CHEN
Combining (11) and (13) yields u0a Ra − Rab R−1 b Rba > 0: Therefore, R is Schur-S. 2 Lemma 3.2. A lower triangular matrix whose inverse exists and is nonnegative is both completely-S and Schur-S. Proof. Let R be a lower triangular matrix whose inverse exists and is nonnegative. Since the inverse of a lower triangular matrix is also a lower triangular matrix, it is clear that the diagonal element of R must be positive. Hence, it is clear that R is completely-S. In order to show that R is Schur-S, it suffices to show that there exists a u > 0 such that u0a Ra − Rab Rb−1 Rba > 0;
(14)
for any partition a; b of J . Recall that R−1 is a lower triangular nonnegative matrix; from its block form (9), it follows that Ra − Rab Rb−1 Rba −1 is a lower triangular nonnegative matrix. This implies that R˜ a x= Ra − Rab Rb−1 Rba is a lower triangular matrix with positive diagonal elements. Then choose u1 = 1; uj = max
