Simulation of exit times and positions for Brownian motions and diffusions

We present in this note some variations of the Monte Carlo method for the randomwalk on spheres which allow to solve many elliptic and parabolic problems involving the Laplace operator or second-order differential operators. In these methods, the spheres are replaced by rectangles or parallelepipeds. Our first method constructs the exit time and the exit position of a rectangle for a Brownian motion. The second method exhibits a variance reduction technique. The main point is to reduce the problem only to the use of some distributions related to the standard one-dimensional Brownian motion.