ESAIM: P&S, June 2007, Vol. 11, pp. 236-247
DOI: 10.1051/ps:2007018

Behavior of the Euler scheme with decreasing step in a degenerate situation

Vincent Lemaire

Laboratoire d'Analyse et de Mathématiques Appliquées, UMR 8050, Université de Marne-la-Vallée, 5 boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France; Vincent.Lemaire@univ-mlv.fr

(Received April 3, 2006. Revised October 16, 2006. Published online 19 June 2007.)

Abstract
The aim of this short note is to study the behavior of the weighted empirical measures of the decreasing step Euler scheme of a one-dimensional diffusion process having multiple invariant measures. This situation can occur when the drift and the diffusion coefficient are vanish simultaneously.

Mathematics Subject Classification. 60H10, 65C30, 37M25

Key words: One-dimensional diffusion process, degenerate coefficient, invariant measure, Euler scheme.


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