The quadratic covariation for a weighted fractional Brownian motion

Abstract

Let [Formula: see text] be a weighted fractional Brownian motion with indices [Formula: see text] and [Formula: see text] satisfying [Formula: see text] [Formula: see text] [Formula: see text]. In this paper, motivated by the asymptotic property [Formula: see text] for all [Formula: see text], we consider the generalized quadratic covariation [Formula: see text] defined by [Formula: see text] provided the limit exists uniformly in probability. We construct a Banach space [Formula: see text] of measurable functions such that the generalized quadratic covariation exists in [Formula: see text] and the generalized Bouleau–Yor identity [Formula: see text] holds for all [Formula: see text], where [Formula: see text] is the weighted local time of [Formula: see text] and [Formula: see text] is the Beta function.