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

Abstract

Let $B^H$ be a fractional Brownian motion with Hurst index $0<H<1/2$. In this paper we study the {\it generalized quadratic covariation} $[f(B^H),B^H]^{(W)}$ defined by $$ [f(B^H),B^H]^{(W)}_t=\lim_{\epsilon\downarrow 0}\frac{2H}{\epsilon^{2H}}\int_0^t\{f(B^{H}_{s+\epsilon})-f(B^{H}_s)\}(B^{H}_{s+\epsilon}- B^{H}_s)s^{2H-1}ds, $$ where the limit is uniform in probability and $x\mapsto f(x)$ is a deterministic function. We construct a Banach space ${\mathscr H}$ of measurable functions such that the generalized quadratic covariation exists in $L^2$ and the Bouleau-Yor identity takes the form $$ [f(B^H),B^H]_t^{(W)}=-\int_{\mathbb {R}}f(x){\mathscr L}^{H}(dx,t) $$ provided $f\in {\mathscr H}$, where ${\mathscr L}^{H}(x,t)$ is the weighted local time of $B^H$. This allows us to write the fractional It\^{o} formula for absolutely continuous functions with derivative belonging to ${\mathscr H}$. These are also extended to the time-dependent case.