Commun. Comput. Phys., 8 (2010), pp. 901-916.


An Efficient Algorithm to Simulate a Brownian Motion Over Irregular Domains

S. Zein 1*, A. Lejay 1, M. Deaconu 1

1 INRIA Centre de Recherche Nancy-Grand Est - IECN, France.

Received 24 February 2009; Accepted (in revised version) 3 December 2009
Available online 17 May 2010
doi:10.4208/cicp.240209.031209a

Abstract

In this paper, we present an algorithm to simulate a Brownian motion by coupling two numerical schemes: the Euler scheme with the random walk on the hyper-rectangles. This coupling algorithm has the advantage to be able to compute the exit time and the exit position of a Brownian motion from an irregular bounded domain (with corners at the boundary), and being of order one with respect to the time step of the Euler scheme. The efficiency of the algorithm is studied through some numerical examples by comparing the analytical solution with the Monte Carlo solution of some Poisson problems. The Monte Carlo solution of these PDEs requires simulating Brownian motions of different types (natural, reflected or drifted) over an irregular domain.

AMS subject classifications: 60G50, 60G40, 65C05, 65C30, 65C20

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Key words: Brownian motion, Monte Carlo methods, partial differential equations, Euler scheme, random walk on rectangles.

*Corresponding author.
Email: samih.zein@samtech.com (S. Zein), Antoine.Lejay@iecn.u-nancy.fr (A. Lejay), Madalina.Deaconu@inria.fr (M. Deaconu)
 

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