The Euler Scheme for Lévy Driven Stochastic Differential Equations: Limit Theorems

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 Xnis its Euler approximation with stepsize 1/n, and unis 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/ log n)1/α; when Y is stable but not symmetric, the rate is again un=(n/ log n)1/αwhen α > 1, but it becomes un=n/( log n)2if α = 1 and $u_{n}=n\ \text{if}\ \alpha