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(X −X), where X is the true solution and X 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 un = √ 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(X n −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); 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.