Estimation of all parameters in the fractional Ornstein–Uhlenbeck model under discrete observations

Abstract

Let the Ornstein–Uhlenbeck process \((X_t)_{t\ge 0}\) 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 \(t_k=kh\), \(k=0, 1, 2, \ldots , 2n+2 \). We propose an ergodic type statistical estimator \({\hat{\theta }}_n \), \({\hat{H}}_n \) 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 estimator. 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.