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 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 $\al<1$.

**Mots Clés:** *Euler scheme ; Stochastic differential equations ; Stability results*

**Date:** 2002-02-28

**Prépublication numéro:** *PMA-711*

**Pdf file : **PMA-711.pdf