Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60H10 Stochastic ordinary differential equations, See Also {
- 65U05 Numerical methods in probability and statistics
- 60J30 Processes with independent increments
- 60F17 Functional limit theorems; invariance principles

**Résumé:** We study the rate of convergence of the Euler scheme, for a stochastic
differential equation driven by a L\'evy process $Y$. A rate is
defined as a sequence such that the normalized error process
$u_n(X^n-X)$ is a uniformly tight sequence of processes, where $X$ is
the true solution and $X^n$ is its Euler approximation with stepsize
$1/n$. This rate is sharp if futher one at least among all limiting
processes of this sequence is not identically $0$.
We suppose that $Y$ has no Gaussian part (otherwise a sharp rate is known
to be $u_n=\rn$). Then rates are given in terms of the concentration
of the L\'evy measure of $Y$ around $0$. For example when $Y$ is a
symmetric stable process with index $\al\in(0,2)$, a sharp rate is
$u_n=(n/\log n)^{1/\al}$; when $Y$ is stable but not symmetric, the
rate is $u_n=n$ for $\al<1$, $u_n=n/(\log n)^2$ if $\al=1$, and
$u_n=n^{\al/(3\al-2)}$ if $\al>1$, but in these cases we do not know
whether these are sharp.

**Mots Clés:** *Euler schemes ; Stochastic differential equations ; Numerical probabilities*

**Date:** 2001-05-11

**Prépublication numéro:** *PMA-656*