# 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
Document type :
Reports
Domain :
Complete list of metadatas

Cited literature [2 references]

https://hal.inria.fr/inria-00074427
Contributor : Rapport de Recherche Inria <>
Submitted on : Wednesday, May 24, 2006 - 3:16:44 PM
Last modification on : Tuesday, May 14, 2019 - 10:17:42 AM
Long-term archiving on : Sunday, April 4, 2010 - 9:38:27 PM

### Identifiers

• 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⟩

Record views