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

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 $L^1(\mathrm{\Omega })$ and is equal to $e_H{\int }_0^T\mathrm{\mid }{u}_s{\mathrm{\mid }}^{H}ds$, 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 $\mathrm{\parallel }{B}_t\mathrm{\parallel }$, where $B_t$ is a $d$-dimensional fractional Brownian motion with Hurst parameter $H < \frac{1}{2}$\mathrm{.}Usingamultidimensionalversionoftheresultonthe$\frac{1}{H}$-variationofdivergenceintegrals,weprovethatif$2dH^2 > 1$, 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.