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

ESAIM: Probability and Statistics (2007)

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

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topLemaire, 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

top- W. Feller, The parabolic differential equations and the associated semi-groups of transformations. Ann. of Math. (2)55 (1952) 468–519. Zbl0047.09303
- W. Feller, Diffusion processes in one dimension. Trans. Amer. Math. Soc.77 (1954) 1–31. Zbl0059.11601
- 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
- S. Karlin and H.M. Taylor, A second course in stochastic processes. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1981). Zbl0469.60001
- D. Lamberton and G. Pagès, Recursive computation of the invariant distribution of a diffusion. Bernoulli8 (2002) 367–405. Zbl1006.60074
- V. Lemaire, Estimation récursive de la mesure invariante d'un processus de diffusion. Ph.D. Thesis, Université de Marne-la-Vallée (2005).
- G. Pagès, Sur quelques algorithmes récursifs pour les probabilités numériques. ESAIM Probab. Statist.5 (2001) 141–170 (electronic). Zbl0998.60073
- 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
- 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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