An Exit Probability Approach to Solving High Dimensional Dirichlet Problems

An Efficient Algorithm to Simulate a Brownian Motion Over Irregular Domains

“…From [11] and [12], one can deduce that, when using the coupling algorithm, the numerical integration error along the Brownian path, E D ∆t , is of order one with respect to ∆t. The advantage of this algorithm over the other existing ones described in ( [11], [14], [5]) is that it can handle the simulation of a Brownian motion near an irregular boundary, a boundary with corners while the other existing algorithms impose some smoothness conditions on the boundary. It can also be extended to the case of the reflected and the drifted Brownian motion.…”

“…These results show that the relative error is independent from the Euler time step and it is relatively small (containing one source of error). In the next numerical examples we compare the computational cost of our algorithm with the one proposed in [5] for the same Poisson problems. We recall that in [5] a numerical algorithm of order one is proposed to approximate the exit time and the exit position of the Brownian motion based on the Brownian bridge assumption.…”

“…In this paper, we propose to simulate the exit time τ and the exit position X τ from a bounded domain, by coupling the Euler scheme with the random walk on the hyper-rectangles. The advantage of this algorithm over the ones described in ( [11], [14] and [5]) is to handle domain boundaries with corners and for different types of Brownian motions (natural, reflected, drifted). To show the efficiency of our algorithm, numerical experiments are carried out by comparing the analytical with the Mont Carlo solution of some Poisson problems.…”

“…Its estimation is independent from the time step ∆t. Some algorithms were presented to compute τ c in ( [14], [4]) for the one-dimensional case, in ( [11], [14] and [5]) from a regular domain, ( [15]) for a bi-dimensional cone and more generally in ( [2]) in the case of a reflected Brownian motion. These algorithms are based on considering the Brownian bridge betweeñ X t k andX t k+1 and computing the probability that the Brownian bridge intersects the boundary.…”

Multidimensional hitting time results for Brownian bridges with moving hyperplanar boundaries

“…We take a similar approach in this paper. To the best of our knowledge, the only multidimensional result for the Brownian bridge appears in Buchmann and Petersen (2006). However, they only examine the independent case with one stationary boundary.…”