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

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 $\Vert B_t\Vert$, 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 $2d{H}^2>1$, 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.

29 pages