The generalized quadratic covariation for fractional Brownian motion with Hurst index less than 1/2

Abstract

In this paper, we study the generalized quadratic covariation of f(B H ) and B H defined by [Formula: see text] in probability, where f is a Borel function and B H is a fractional Brownian motion with Hurst index 0 < H < 1/2. We construct a Banach space [Formula: see text] of measurable functions such that the generalized quadratic covariation exists in L 2 (Ω) and the Bouleau–Yor identity takes the form [Formula: see text] provided [Formula: see text], where [Formula: see text] is the weighted local time of B H . These are also extended to the time-dependent case, and as an application we give the identity between the generalized quadratic covariation and the 4-covariation [g(B H ), B H , B H , B H ] when [Formula: see text].