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

TL;DR

This paper develops a Young integral approach to analyze weighted local times of fractional Brownian motion with Hurst index greater than 1/2, leading to a fractional Itô formula for new function classes.

Contribution

It introduces a novel Young integration method for weighted local times of fractional Brownian motion with H>1/2, extending the fractional Itô formula to broader function classes.

Findings

Existence of weighted quadratic covariation for fractional Brownian motion.

Representation of covariation as an integral of weighted local time.

Extension of results to time-dependent functions.

Abstract

Let ${\mathscr L}^H(x,t)=2H\int_0^t\delta(B^H_s-x)s^{2H-1}ds$ 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 $\int_{\mathbb R}f(x){\mathscr L}^H(dx,t)$. As an application, we investigate the {\it weighted quadratic covariation} $[f(B^H),B^H]^{(W)}$ defined by $$ [f(B^H),B^H]^{(W)}_t:=\lim_{n\to \infty}2H\sum_{k=0}^{n-1} k^{2H-1}\{f(B^H_{t_{k+1}})-f(B^H_{t_{k}})\}(B^H_{t_{k+1}}-B^H_{t_{k}}), $$ where the limit is uniform in probability and $t_k=kt/n$. We show that it exists and $$ [f(B^H),B^H]^{(W)}_t=-\int_{\mathbb R}f(x){\mathscr L}^H(dx,t), $$ provided $f$ is of bounded $p$-variation with $1\leq p<\frac{2H}{1-H}$. Moreover, we extend this result to the time-dependent case. These allow us to write the fractional It\^{o} formula for new classes of functions.