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# The Law of the Euler Scheme for Stochastic Differential Equations : II. Convergence Rate of the Density

1 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 : In the first part of this work~\cite{Bally-Talay-94-1} we have studied 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 $f(\cdot)$ is a given function, only supposed measurable and bounded. We have proven that the discretization error can be expanded in terms of powers of $\frac1n$ under a nondegeneracy condition of Hörmander type for the infinitesimal generator of $(X_t)$. In this second part, we consider the density of the law of a small perturbation of $X_T^n$ and we compare it to the density of the law of $X_T$: we prove that the difference between the densities can also be expanded in terms of $\frac1n$. \noindent{\bf AMS(MOS) classification}: 60H07, 60H10, 60J60, 65C05, 65C20, 65B05.
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Submitted on : Wednesday, May 24, 2006 - 2:18:12 PM
Last modification on : Friday, February 4, 2022 - 3:18:41 AM
Long-term archiving on: : Thursday, March 24, 2011 - 12:11:30 PM

### Identifiers

• HAL Id : inria-00074016, version 1

### Citation

Vlad Bally, Denis Talay. The Law of the Euler Scheme for Stochastic Differential Equations : II. Convergence Rate of the Density. [Research Report] RR-2675, INRIA. 1995, pp.30. ⟨inria-00074016⟩

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