Multidimensional Exponential Timestepping with Boundary Test

Exponential timestepping algorithms are efficient for exit-time problems because a boundary test can be performed at the end of each timestep, giving high-order convergence in numerical evaluation of mean exit times. Successive time increments are independent random variables with an exponential distribution. We show how to perform exact timestepping for Brownian motion in more than one dimension and consider hitting times of curved surfaces. Taking as examples the exit times from a circle, from a ball, from an ellipse, and from the region bounded by the two lines of a hyperbola, we report the results of numerical experiments performed with the algorithms developed in this work.