Estimation Of all parameters in the Fractional Ornstein-Uhlenbeck model under discrete observations

Abstract

Let the Ornstein-Uhlenbeck process $(X_t)_{t\ge0}$ driven by a fractional Brownian motion $B^{H }$, described by $dX_t = -\theta X_t dt + \sigma dB_t^{H }$ be observed at discrete time instants $t_k=kh$, $k=0, 1, 2, \cdots, 2n+2 $. We propose ergodic type statistical estimators $\hat \theta_n $, $\hat H_n $ and $\hat \sigma_n $ to estimate all the parameters $\theta $, $H $ and $\sigma $ in the above Ornstein-Uhlenbeck model simultaneously. We prove the strong consistence and the rate of convergence of the estimators. The step size $h$ can be arbitrarily fixed and will not be forced to go zero, which is usually a reality. The tools to use are the generalized moment approach (via ergodic theorem) and the Malliavin calculus.