On the $\frac{1}{H}$-variation of the divergence integral with respect to fractional Brownian motion with Hurst parameter $H < 1/2$

TL;DR

This paper investigates the $rac{1}{H}$-variation of divergence integrals with respect to fractional Brownian motion for $H<1/2$, establishing existence, explicit formulas, and applications to fractional Bessel processes.

Contribution

It provides the first explicit characterization of the $rac{1}{H}$-variation of divergence integrals for $H<1/2$, including an integral representation for fractional Bessel processes.

Findings

The $rac{1}{H}$-variation of $X_t$ exists in $L^1$ and equals $e_H imes$ the integral of $|u_s|^H$.

An integral representation for the fractional Bessel process is established.

Under $2dH^2 > 1$, the divergence integral in the Bessel process has a $rac{1}{H}$-variation proportional to Lebesgue measure.

Abstract

In this paper, we study the $\frac{1}{H}$-variation of stochastic divergence integrals $X_t = \int_0^t u_s {\delta}B_s$ with respect to a fractional Brownian motion $B$ with Hurst parameter $H < \frac{1}{2}$. Under suitable assumptions on the process u, we prove that the $\frac{1}{H}$-variation of $X$ exists in $L^1({\Omega})$ and is equal to $e_H \int_0^T|u_s|^H ds$, where $e_H = \mathbb{E}|B_1|^H$. In the second part of the paper, we establish an integral representation for the fractional Bessel Process $\|B_t\|$, where $B_t$ is a $d$-dimensional fractional Brownian motion with Hurst parameter $H < \frac{1}{2}$. Using a multidimensional version of the result on the $\frac{1}{H}$-variation of divergence integrals, we prove that if $2dH^2 > 1$, then the divergence integral in the integral representation of the fractional Bessel process has a $\frac{1}{H}$-variation equals to a multiple of the Lebesgue measure.