Generalized covariations, local time and Stratonovich Ito's formula for fractional Brownian motion with Hurst index H ≥ 1/4

• 成果类型: Article
• 署名作者: Gradinaru, M; Russo, F; Vallois, P
• 署名单位: Universite de Lorraine; Universite Paris 13
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2003
• 页码: 1772-1820
• 关键词: stochastic calculus; differential-equations; dirichlet processes; fractal functions; Respect; integration; martingales; BACKWARD

摘要

Given a locally bounded real function g, we examine the existence of a 4-covariation [g(B-H), BH, BH, BH], where BH is a fractional Brownian motion with a Hurst index H greater than or equal to (1)/(4). We provide two essential applications. First, we relate the 4-covariation to one expression involving the derivative of local time, in the case H = (1)/(4), generalizing an identity of Bouleau-Yor type, well known for the classical Brownian motion. A second application is an Ito formula of Stratonovich type for f (B-H). The main difficulty comes from the fact BH has only a finite 4-variation.