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

### The Euler scheme for Lévy driven by stochastic differential equations : Limit theorems

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Résumé: We study the Euler scheme for a stochastic differential equation driven by a L\'evy process $Y$. More precisely we look at the asymptotic behaviour of the normalized error process $u_n(X^n-X)$, where $X$ is the true solution and $X^n$ is its Euler approximation with stepsize $1/n$, and $u_n$ is an appropriate rate going to infinity: if the normalized error processes converge, or are at least tight, we say that the sequence $(u_n)$ is a rate, which in addition is sharp when the limiting process (or processes) is not trivial. We suppose that $Y$ has no Gaussian part (otherwise a 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 $0$, and further we prove the convergence of the sequence $u_n(X^n-X)$ to a non-trivial limit under some further assumptions, which cover all stable processes and a lot of other L\'evy processes whose L\'evy measure behave like a stable L\'evy measure near the origin. 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 again $u_n=(n/\log n)^{1/\al}$ when $\al>1$, but it becomes $u_n=n/(\log n)^2$ if $\al=1$ and $u_n=n$ if $\al<1$.