Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Sharp estimates for the Euler scheme for Lévy driven stochastic differential equations

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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.