# The Law of the Euler scheme for stochastic differential equations : I. convergence rate of the distribution function

2 OMEGA - Probabilistic numerical methods
CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, CNRS - Centre National de la Recherche Scientifique : UMR7502
Abstract : We study the approximation problem of $\ee f(X_T)$ by $\ee f(X_T^n)$, where $(X_t)$ is the solution of a stochastic differential equation, $(X^n_t)$ is defined by the Euler discretization scheme with step $\fracTn$,and $f$ is a given function. For smooth $f$'s, Talay and Tubaro have shown that the error $\ee f(X_T)-f(X_T^n)$ can be expanded in powers of $\frac1n$, which permits to construct Romberg extrapolation procedures to accelerate the convergence rate. Here, we prove that the expansion exists also when $f$ is only supposed measurable and bounded, under an additional nondegeneracy condition of Hörmander type for the infinitesimal generator of $(X_t)$): to obtain this result, we use the stochastic variations calculus. In the second part of this work, we will consider the density of the law of $X_T^n$ and compare it to the density of the law of $X_T$. \noindent\bf AMS(MOS) classification: 60H07, 60H10, 60J60, 65C05, 65C20, 65B05
Type de document :
Rapport
[Research Report] RR-2244, INRIA. 1994
Domaine :

Littérature citée [2 références]

https://hal.inria.fr/inria-00074427
Contributeur : Rapport de Recherche Inria <>
Soumis le : mercredi 24 mai 2006 - 15:16:44
Dernière modification le : mercredi 21 mars 2018 - 18:57:30
Document(s) archivé(s) le : dimanche 4 avril 2010 - 21:38:27

### Identifiants

• HAL Id : inria-00074427, version 1

### Citation

Vlad Bally, Denis Talay. The Law of the Euler scheme for stochastic differential equations : I. convergence rate of the distribution function. [Research Report] RR-2244, INRIA. 1994. 〈inria-00074427〉

### Métriques

Consultations de la notice

## 245

Téléchargements de fichiers