Power variation of some integral fractional processes

Abstract

We consider the asymptotic behaviour of the realized power variation of processes of the form ${\int }_0^t{u}_{s}d{B}_s^H$, where $B^H$ is a fractional Brownian motion with Hurst parameter $H\in (0,1)$, and $u$ is a process with finite $q$-variation, $q<1/(1-H)$. We establish the stable convergence of the corresponding fluctuations. These results provide new statistical tools to study and detect the long-memory effect and the Hurst parameter.