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

Abstract

Given a locally bounded real function g, we examine the existence of a 4-covariation [g(BH), BH,BH, BH], where BH is a fractional Brownian motion with a Hurst index H ≥ 1/4. We provide two essential applications. First, we relate the 4-covarialion 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 Itô formula of Stratonovich type for f(BH). The main difficulty comes from the fact B H has only a finite 4-variation.