2004/10/05 by Jean Jacod
Mathematics · #math.PR #msc:60J75 #msc:65C30 #msc:60J30 #msc:60F17.
paper · pdf · doi:10.1214/009117904000000667
published as Annals of Probability 2004, Vol. 32, No. 3A, 1830-1872 · Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000667
arxiv created 2004/10/05 · arxiv updated 2009/12/01
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 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 un=√ 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 un(Xn-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 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/α when α>1, but it becomes un=n/(log n)2 if α=1 and un=n if α<1.