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

Vincent Lemaire

ESAIM: Probability and Statistics (2007)

  • Volume: 11, page 236-247
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Lemaire, Vincent. "Behavior of the Euler scheme with decreasing step in a degenerate situation." ESAIM: Probability and Statistics 11 (2007): 236-247. <http://eudml.org/doc/250123>.

@article{Lemaire2007,
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. },
author = {Lemaire, Vincent},
journal = {ESAIM: Probability and Statistics},
keywords = {One-dimensional diffusion process; degenerate coefficient; invariant measure; Euler scheme.; one-dimensional diffusion process; Euler scheme},
language = {eng},
month = {6},
pages = {236-247},
publisher = {EDP Sciences},
title = {Behavior of the Euler scheme with decreasing step in a degenerate situation},
url = {http://eudml.org/doc/250123},
volume = {11},
year = {2007},
}

TY - JOUR
AU - Lemaire, Vincent
TI - Behavior of the Euler scheme with decreasing step in a degenerate situation
JO - ESAIM: Probability and Statistics
DA - 2007/6//
PB - EDP Sciences
VL - 11
SP - 236
EP - 247
AB - 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.
LA - eng
KW - One-dimensional diffusion process; degenerate coefficient; invariant measure; Euler scheme.; one-dimensional diffusion process; Euler scheme
UR - http://eudml.org/doc/250123
ER -

References

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  1. W. Feller, The parabolic differential equations and the associated semi-groups of transformations. Ann. of Math. (2)55 (1952) 468–519.  Zbl0047.09303
  2. W. Feller, Diffusion processes in one dimension. Trans. Amer. Math. Soc.77 (1954) 1–31.  Zbl0059.11601
  3. I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus. Springer-Verlag, New York, 2nd edition, Graduate Texts in Mathematics 113 (1991).  Zbl0734.60060
  4. S. Karlin and H.M. Taylor, A second course in stochastic processes. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1981).  Zbl0469.60001
  5. D. Lamberton and G. Pagès, Recursive computation of the invariant distribution of a diffusion. Bernoulli8 (2002) 367–405.  Zbl1006.60074
  6. V. Lemaire, Estimation récursive de la mesure invariante d'un processus de diffusion. Ph.D. Thesis, Université de Marne-la-Vallée (2005).  
  7. G. Pagès, Sur quelques algorithmes récursifs pour les probabilités numériques. ESAIM Probab. Statist.5 (2001) 141–170 (electronic).  Zbl0998.60073
  8. 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).  Zbl0826.60002
  9. W.F. Stout, Almost sure convergence. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, Probability and Mathematical Statistics 24 (1974).  Zbl0321.60022

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