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

Abstract

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