MULTILEVEL MONTE CARLO ALGORITHMS FOR LEVY-DRIVEN SDES WITH GAUSSIAN CORRECTION

• 成果类型: Article
• 署名作者: Dereich, Steffen
• 署名单位: Philipps University Marburg
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/10-AAP695
• 发表日期: 2011
• 页码: 283-311
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; euler scheme; partial sums; simulation; approximation

摘要

We introduce and analyze multilevel Monte Carlo algorithms for the, computation of E(f) (Y), where Y = (Y(t))t is an element of[0, 1] is the solution of a multidimensional Levy-driven stochastic differential equation and f is a real-valued function on the path space. The algorithm relies on approximations obtained by simulating large jumps of the Levy process individually and applying a Gaussian approximation for the small jump part. Upper bounds are provided for the worst case error over the class of all measurable real functions f that are Lipschitz continuous with respect to the supremum norm. These upper bounds are easily tractable once one knows the behavior of the Levy measure around zero. In particular, one can derive upper bounds from the Blumenthal-Getoor index of the Levy process. In the case where the Blumenthal-Getoor index is larger than one, this approach is superior to algorithms that do not apply a Gaussian approximation. If the Levy process does not incorporate a Wiener process or if the Blumenthal-Getoor index beta is larger than 4/3, then the upper bound is of order tau(-(4-beta)/(6 beta)) when the runtime tau tends to infinity. Whereas in the case, where 13 is in [I, 4, and the Levy process has a Gaussian component, we obtain bounds of order tau(-beta/(6 beta-4)). In particular, the error is at most of order tau(-1/6).