Stochastic derivatives for fractional diffusions

• 成果类型: Article
• 署名作者: Darses, Sebastien; Nourdin, Ivan
• 署名单位: Universite Marie et Louis Pasteur; Universite Paris Cite; Sorbonne Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000001169
• 发表日期: 2007
• 页码: 1998-2020
• 关键词: ordinary differential-equations; brownian-motion; time-reversal; integration; Respect

摘要

In this paper, we introduce some fundamental notions related to the so-called stochastic derivatives with respect to a given a-field Q. In our framework, we recall well-known results about Markov-Wiener diffusions. We then focus mainly on the case where X is a fractional diffusion and where 62 is the past, the future or the present of X. We treat some crucial examples and our main result is the existence of stochastic derivatives with respect to the present of X when X solves a stochastic differential equation driven by a fractional Brownian motion with Hurst index H > 1/2. We give explicit formulas.