On the 1H-variation of the divergence integral with respect to fractional Brownian motion with Hurst parameter H<12

In this paper, we study the 1H-variation of stochastic divergence integrals Xt=∫0tusδBs with respect to a fractional Brownian motion B with Hurst parameter H<12. Under suitable assumptions on the process u, we prove that the 1H-variation of X exists in L1(Ω) and is equal to eH∫0T|us|1Hds, where eH=E[|B1|1H]. In the second part of the paper, we establish an integral representation for the fractional Bessel Process ‖Bt‖, where Bt is a d-dimensional fractional Brownian motion with Hurst parameter H<12. Using a multidimensional version of the result on the 1H-variation of divergence integrals, we prove that if 2dH2>1, then the divergence integral in the integral representation of the fractional Bessel process has a 1H-variation equals to a multiple of the Lebesgue measure.