Simulation of exit times and positions for Brownian motions and Diffusions

Madalina Deaconu (1, 2) , Antoine Lejay (1, 2)

1 IECN - Institut Élie Cartan de Nancy
2 TOSCA - Simuler et calibrer des modèles stochastiques

We present in this note some variations of the Monte Carlo method for the random walk 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.

Mots clés: Stochastic Differential Equations; Monte Carlo methods; Random walk on squares; Random walk on rectangles; Variance reduction; Simulation of rare events; Dirichlet/Neumann problems

Domaines: Probabilités [math.PR]

Madalina Deaconu, Antoine Lejay. Simulation of exit times and positions for Brownian motions and Diffusions. ICIAM 2007, 6th International Congress on Industrial and Applied Mathematics, International Council for Industrial and Applied Mathematics (ICIAM), Jul 2007, Zurich, Switzerland. pp.1081401-1081402, ⟨10.1002/pamm.200700564⟩. ⟨inria-00348693⟩