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Behavior of the Euler scheme with decreasing stepin a degenerate situation

Published online by Cambridge University Press:  19 June 2007

Vincent Lemaire
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; [email protected]
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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.

Keywords

One-dimensional diffusion processdegenerate coefficientinvariant measureEuler scheme.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 11: Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthdaySpecial Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday , June 2007 , pp. 236 - 247
Copyright
© EDP Sciences, SMAI, 2007

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References

Feller, W., The parabolic differential equations and the associated semi-groups of transformations. Ann. of Math. (2) 55 (1952) 468–519. CrossRef
Feller, W., Diffusion processes in one dimension. Trans. Amer. Math. Soc. 77 (1954) 1–31. CrossRef
I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus. Springer-Verlag, New York, 2nd edition, Graduate Texts in Mathematics 113 (1991).
S. Karlin and H.M. Taylor, A second course in stochastic processes. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1981).
Lamberton, D. and Pagès, G., Recursive computation of the invariant distribution of a diffusion. Bernoulli 8 (2002) 367–405.
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) 141–170 (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).