ESAIM: P&S, August 2007, Vol. 11, pp. 301-326
DOI: 10.1051/ps:2007021

A Donsker theorem to simulate one-dimensional processes with measurable coefficients

Pierre Étoré¹ and Antoine Lejay<sup>2

1</sup>  CERMICS, École Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal Cité Descartes, Champs sur Marne, 77455 Marne la Vallée Cedex 2, France; Pierre.Etore@cermics.enpc.fr
²  Projet OMEGA, Institut Élie Cartan (UMR 7502, Nancy-Université, CNRS, INRIA) and INRIA Lorraine, Campus scientifique, BP 239, 54506 Vandoeuvre-lès-Nancy Cedex, France; Antoine.Lejay@iecn.u-nancy.fr

(Received May 29, 2006. Published online 17 August 2007.)

Abstract
In this paper, we prove a Donsker theorem for one-dimensional processes generated by an operator with measurable coefficients. We construct a random walk on any grid on the state space, using the transition probabilities of the approximated process, and the conditional average times it spends on each cell of the grid. Indeed we can compute these quantities by solving some suitable elliptic PDE problems.

Mathematics Subject Classification. 60J60, 65C

Key words: Monte Carlo methods, Donsker theorem, one-dimensional process, scale function, divergence form operators, Feynman-Kac formula, elliptic PDE problem.


© EDP Sciences, SMAI 2007