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

• 成果类型: Article
• 署名作者: Jacod, J
• 署名单位: Universite Paris Cite; Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000667
• 发表日期: 2004
• 页码: 1830-1872
• 关键词: convergence; STABILITY

摘要

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) = rootn). 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 is an element of (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.