FIRST-ORDER EULER SCHEME FOR SDES DRIVEN BY FRACTIONAL BROWNIAN MOTIONS: THE ROUGH CASE

• 成果类型: Article
• 署名作者: Liu, Yanghui; Tindel, Samy
• 署名单位: Purdue University System; Purdue University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/17-AAP1374
• 发表日期: 2019
• 页码: 758-826
• 关键词: weighted power variations; LIMIT-THEOREMS; asymptotic-behavior; functionals; CONVERGENCE; paths

摘要

In this article, we consider the so-called modified Euler scheme for stochastic differential equations (SDEs) driven by fractional Brownian motions (fBm) with Hurst parameter 1/3 < H < 1/2. This is a first-order time-discrete numerical approximation scheme, and has been introduced in [Ann. Appl. Probab. 26 (2016) 1147-1207] recently in order to generalize the classical Euler scheme for Ito SDEs to the case H > 1/2. The current contribution generalizes the modified Euler scheme to the rough case 1/3 < H < 1/2. Namely, we show a convergence rate of order n(1/2-2H) for the scheme, and we argue that this rate is exact. We also derive a central limit theorem for the renormalized error of the scheme, thanks to some new techniques for asymptotics of weighted random sums. Our main idea is based on the following observation: the triple of processes obtained by considering the fBm, the scheme process and the normalized error process, can be lifted to a new rough path. In addition, the Holder norm of this new rough path has an estimate which is independent of the step-size of the scheme.