An integral functional driven by fractional Brownian motion

Let $B^H$ be a fractional Brownian motion with Hurst index $0<H<1$ and the weighted local time ${\mathscr L}^H(\cdot,t)$. In this paper, we consider the integral functional ${\mathcal C}^H_t(a):=\lim_{\varepsilon\downarrow 0}\int_0^t1_{\{|B^H_s-a|>\varepsilon\}}\frac1{B^H_s-a}ds^{2H}\equiv \frac1π{\mathscr H}{\mathscr L}^H(\cdot,t)(a)$ in $L^2(Ω)$ with $a\in {\mathbb R}, t\geq 0$ and ${\mathscr H}$ denoting the Hilbert transform. We show that ${\mathcal C}^H_t(a)=2\left((B^H_t-a)\log|B^H_t-a|-B^H_t+a\log|a| -\int_0^t\log|B^H_s-a|δB^H_s\right)$ for all $a\in {\mathbb R}, t\geq 0$ 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}: $\int_{\mathbb R}{\mathcal C}^H_t(a)g(a)da=2Hπ\int_0^t({\mathscr H}g)(B^H_s)s^{2H-1}ds$ for all continuous functions $g$ with compact support.

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