An integral functional driven by fractional Brownian motion

Abstract

Let $B^H$ be a fractional Brownian motion with Hurst index $0<H<1$ and the weighted local time ${\mathcal{L}}^H(\cdot ,t)$. In this paper, we consider the integral functional $${\mathcal{C}}_t^H(a):=\underset{\epsilon \downarrow 0}{\lim }{\int }_0^t{1}_{\{|B_s^H-a|>\epsilon \}}\frac{1}{B_s^H-a}d{s}^{2H}\equiv \frac{1}{\pi }\mathcal{H}{\mathcal{L}}^H(\cdot ,t)(a)$$ in $L^2(\Omega )$ with $a\in \mathbb{R},t\ge 0$ and $\mathcal{H}$ denoting the Hilbert transform. We show that $${\mathcal{C}}_t^H(a)=2\left((B_t^H-a)\log |B_t^H-a|-B_t^H+a\log |a|-{\int }_0^t\log |B_s^H-a|\delta B_s^H\right)$$ for all $a\in \mathbb{R},t\ge 0$ which is the fractional version of Yamada's formula, where the integral is the Skorohod integral. Moreover, we introduce the following {\it occupation type formula}: $${\int }_{\mathbb{R}}{\mathcal{C}}_t^H(a)g(a)da=2H\pi {\int }_0^t(\mathcal{H}g)(B_s^H)s^{2H-1}ds$$ for all continuous functions $g$ with compact support.

30 pages