HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information
Skip to Main content Skip to Navigation
Reports

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

Vlad Bally 1 Denis Talay 2
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
Complete list of metadata

Cited literature [2 references]  Display  Hide  Download

https://hal.inria.fr/inria-00074427
Contributor : Rapport de Recherche Inria Connect in order to contact the contributor
Submitted on : Wednesday, May 24, 2006 - 3:16:44 PM
Last modification on : Thursday, February 3, 2022 - 11:15:41 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⟩

Share

Metrics

Record views

2675

Files downloads

21998