Volatility estimation in fractional Ornstein-Uhlenbeck models

Abstract

In this article we study the asymptotic behaviour of the realized quadratic variation of a process ${\int }_0^t{u}_{s}d{Y}_s^{(1)}$% , where $u$ is a $\beta$-Hölder continuous process with $\beta >1-H$ and $Y_t^{(1)}={\int }_0^t{e}^{-s}d{B}_{a_s}^H$, where $a_{t}=He^{\frac{t% }{H}} $ and $B^H$ is a fractional Brownian motion, is connected to the fractional Ornstein-Uhlenbeck process of the second kind. We prove almost sure convergence uniformly in time, and a stable weak convergence for the realized quadratic variation. As an application, we construct strongly consistent estimator for the integrated volatility parameter in a model driven by $Y^{(1)}$.