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 un(Xn−X), where X is the true solution and Xn is its Euler approximation with stepsize 1/n, and un is an appropriate rate going to infinity: if the normalized error processes converge, or are at least tight, we say that the sequence (un) 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 un(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 un=(n/logn)1/α; when Y is stable but not symmetric, the rate is again un=(n/logn)1/α when α>1, but it becomes un=n/(logn)2 if α=1 and un=n if α<1.

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