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

Published online by Cambridge University Press:  19 June 2007

Vincent Lemaire
Affiliation:
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
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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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2007

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References

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Lamberton, D. and Pagès, G., Recursive computation of the invariant distribution of a diffusion. Bernoulli 8 (2002) 367405.
V. Lemaire, Estimation récursive de la mesure invariante d'un processus de diffusion. Ph.D. Thesis, Université de Marne-la-Vallée (2005).
Pagès, G., Sur quelques algorithmes récursifs pour les probabilités numériques. ESAIM Probab. Statist. 5 (2001) 141170 (electronic). CrossRef
L.C.G. Rogers and D. Williams, Diffusions, Markov processes, and martingales. Vol. 1. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Ltd., Chichester, 2nd edition (1994).
W.F. Stout, Almost sure convergence. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, Probability and Mathematical Statistics 24 (1974).

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