Integration with respect to fractional local times with Hurst index $H$ greater than 1/2

Abstract

Let ${\mathcal{L}}^H(x,t)=2H{\int }_0^t\delta (B_s^H-x)s^{2H-1}ds$ be the weighted local time of fractional Brownian motion $B^H$ with Hurst index $1/2<H<1$. In this paper, we use Young integration to study the integral of determinate functions ${\int }_{\mathbb{R}}f(x){\mathcal{L}}^H(dx,t)$. As an application, we investigate the {\it weighted quadratic covariation} $[f(B^H),B^H{]}^{(W)}$ defined by $$[f(B^H),B^H{]}_t^{(W)}:=\underset{n\to \infty }{\lim }2H\sum _{k=0}^{n-1}{k}^{2H-1}\{f(B_{t_{k+1}}^H)-f(B_{t_k}^H)\}(B_{t_{k+1}}^H-B_{t_k}^H),$$ where the limit is uniform in probability and $t_k=kt/n$. We show that it exists and $$[f(B^H),B^H{]}_t^{(W)}=-{\int }_{\mathbb{R}}f(x){\mathcal{L}}^H(dx,t),$$ provided $f$ is of bounded $p$-variation with $1\le p<\frac{2H}{1-H}$. Moreover, we extend this result to the time-dependent case. These allow us to write the fractional Itô formula for new classes of functions.

21 pages

References in corpus (1)

• Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion