Abstract. Numerical methods for stochastic differential equations are relatively inefficient when used to approximate mean exit times. In particular, although the basic Euler-Maruyama method has weak order equal to one for approximating the expected value of the solution, the order reduces to one half when it is used in a straightforward manner to approximate the mean value of a (stopped) exit time. Consequently, the widely used standard approach of combining an Euler-Maruyama discretization with a Monte Carlo simulation leads to a computationally expensive procedure. 1. Background and notation. We begin with the system of stochastic differential equations (SDEs),Brownian motion, and we let (Ω, F, P, F t ) be a complete, filtered probability space satisfying the usual conditions. For a specified open set O ⊂ R d , the stopped exit time is the first time at which X(s) leaves the open set O, or T if this is smaller. Our quantity of interest is the expected value of this random variable. *

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