Reflected Brownian motion in the quarter plane: An equivalence based on time reversal

Abstract

We consider a semimartingale reflected Brownian motion (SRBM) Z whose state space is the non-negative quarter plane; the apparently more general case of SRBM in a convex wedge can be transformed to the quarter plane by a simple change of variable. The data of the stochastic process Z are a drift vector μ, a covariance matrix Σ, and a 2 × 2 reflection matrix R whose columns are the directions of reflection on the two axes. We consider only the case where R has non-positive off-diagonal elements, that is, the direction of reflection is either normal or toward the origin from each axis.Under that restriction, we define a dual RBM Zˆ that is constructed using the same data (μ,Σ,R) but with certain sign reversals. Using a time reversal argument, we show the following: to find the distribution of the random two-vector Z(t) at an arbitrary time t, assuming that Z(0)=0, it suffices to find the probability that the dual RBM Zˆ, starting from an arbitrary point z in the quarter plane, reaches the origin before time t. Letting t→∞ and assuming that μ and R jointly satisfy the known condition for positive recurrence of Z, we then have the following: to determine the stationary distribution of Z, it suffices to determine the probability that the dual RBM Zˆ, starting from an arbitrary point z, ever reaches the origin.