The Euler scheme for Lévy driven stochastic differential equations: limit theorems

Abstract

We study the Euler scheme for a stochastic differential equation driven by a Lévy process Y. More precisely, we look at the asymptotic behavior of the normalized error process u_n(Xⁿ−X), where X is the true solution and Xⁿ 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 $u_{n}=\sqrt {n}$ ). Then rates are given in terms of the concentration of the Lévy measure of Y around 0 and, further, we prove the convergence of the sequence u_n(Xⁿ−X) to a nontrivial limit under some further assumptions, which cover all stable processes and a lot of other Lévy processes whose Lévy measure behave like a stable Lévy measure near the origin. For example, when Y is a symmetric stable process with index α∈(0,2), a sharp rate is u_n=(n/logn)^1/α; when Y is stable but not symmetric, the rate is again u_n=(n/logn)^1/α when α>1, but it becomes u_n=n/(logn)² if α=1 and u_n=n if α<1.