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 $$[f(B^H),B^H{]}_t^{(W)}=\underset{\epsilon \downarrow 0}{\lim }\frac{2H}{{\epsilon }^{2H}}{\int }_0^t\{f(B_{s+\epsilon }^H)-f(B_s^H)\}(B_{s+\epsilon }^H-B_s^H)s^{2H-1}ds,$$ where the limit is uniform in probability and $x\mapsto f(x)$ is a deterministic function. We construct a Banach space $\mathcal{H}$ of measurable functions such that the generalized quadratic covariation exists in $L^2$ and the Bouleau-Yor identity takes the form $$[f(B^H),B^H{]}_t^{(W)}=-{\int }_{\mathbb{R}}f(x){\mathcal{L}}^H(dx,t)$$ provided $f\in \mathcal{H}$, where ${\mathcal{L}}^H(x,t)$ is the weighted local time of $B^H$. This allows us to write the fractional Itô formula for absolutely continuous functions with derivative belonging to $\mathcal{H}$. These are also extended to the time-dependent case.

22 pages