Reflecting Brownian motion (RBM) is a stochastic process that behaves like a Brownian motion in the interior of its domain and is pushed into the interior whenever it reaches the boundary of its domain. RBM’s in the positive orthant were first introduced by Harrison and Reiman 1981, and arise naturally in a wide variety of settings, e.g., heavy traffic queueing networks. A difficult question that has received a significant amount of attention is identifying the asymptotics for tail probabilities associated with RBM in the positive orthant. In this work we focus on the specific tail probability that a stable RBM started near the origin exits a large box before returning to the origin. We develop particle based algorithms to estimate this probability. Using results of Dean and Dupuis 2008 we are able to develop algorithms that efficiently estimate this tail probability in two dimensions. In three and higher dimensions, we are not able to construct an efficient estimator, but we do construct estimators that are provably superior (in an asymptotic sense) to standard Monte Carlo. Numerical results show the benefits of our algorithm to standard Monte Carlo. This is based on joint work with Xin Liu and Zicheng Wang.