The Euler scheme for Levy driven stochastic differential equations: limit theorems
| dc.creator | Jacod, Jean | |
| dc.date | 2004-10-05 | |
| dc.date.accessioned | 2026-07-07T05:12:54Z | |
| dc.date.available | 2026-07-07T05:12:54Z | |
| dc.description | 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 α\in(0,2), a sharp rate is u_n=(n/\log n)^{1/α}; when Y is stable but not symmetric, the rate is again u_n=(n/\log n)^{1/α} when α>1, but it becomes u_n=n/(\log n)^2 if α=1 and u_n=n if α<1. | |
| dc.description | 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 | |
| dc.identifier | https://arxiv.org/abs/math/0410118 | |
| dc.identifier | http://arxiv.org/abs/math/0410118 | |
| dc.identifier | Annals of Probability 2004, Vol. 32, No. 3A, 1830-1872 | |
| dc.identifier | doi:10.1214/009117904000000667 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/72754 | |
| dc.subject | Probability | |
| dc.subject | 60J75, 65C30 (Primary) 60J30, 60F17. (Secondary) | |
| dc.title | The Euler scheme for Levy driven stochastic differential equations: limit theorems | |
| dc.type | text |