The generalized quadratic covariation for fractional Brownian motion with Hurst index less than 1/2 (1106.2302v2)

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.