Behavior of the Euler scheme with decreasing step in a degenerate situation
ESAIM: Probability and Statistics, Tome 11 (2007), pp. 236-247.

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.

DOI : 10.1051/ps:2007018
Classification : 60H10, 65C30, 37M25
Mots clés : one-dimensional diffusion process, degenerate coefficient, invariant measure, Euler scheme 
@article{PS_2007__11__236_0,
     author = {Lemaire, Vincent},
     title = {Behavior of the {Euler} scheme with decreasing step in a degenerate situation},
     journal = {ESAIM: Probability and Statistics},
     pages = {236--247},
     publisher = {EDP-Sciences},
     volume = {11},
     year = {2007},
     doi = {10.1051/ps:2007018},
     mrnumber = {2320818},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2007018/}
}
TY  - JOUR
AU  - Lemaire, Vincent
TI  - Behavior of the Euler scheme with decreasing step in a degenerate situation
JO  - ESAIM: Probability and Statistics
PY  - 2007
SP  - 236
EP  - 247
VL  - 11
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2007018/
DO  - 10.1051/ps:2007018
LA  - en
ID  - PS_2007__11__236_0
ER  - 
%0 Journal Article
%A Lemaire, Vincent
%T Behavior of the Euler scheme with decreasing step in a degenerate situation
%J ESAIM: Probability and Statistics
%D 2007
%P 236-247
%V 11
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps:2007018/
%R 10.1051/ps:2007018
%G en
%F PS_2007__11__236_0
Lemaire, Vincent. Behavior of the Euler scheme with decreasing step in a degenerate situation. ESAIM: Probability and Statistics, Tome 11 (2007), pp. 236-247. doi : 10.1051/ps:2007018. http://www.numdam.org/articles/10.1051/ps:2007018/

[1] W. Feller, The parabolic differential equations and the associated semi-groups of transformations. Ann. of Math. (2) 55 (1952) 468-519. | Zbl

[2] W. Feller, Diffusion processes in one dimension. Trans. Amer. Math. Soc. 77 (1954) 1-31. | Zbl

[3] I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus. Springer-Verlag, New York, 2nd edition, Graduate Texts in Mathematics 113 (1991). | MR | Zbl

[4] S. Karlin and H.M. Taylor, A second course in stochastic processes. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1981). | MR | Zbl

[5] D. Lamberton and G. Pagès, Recursive computation of the invariant distribution of a diffusion. Bernoulli 8 (2002) 367-405. | Zbl

[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). | Numdam | Zbl

[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). | MR | Zbl

[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). | MR | Zbl

Cité par Sources :