Parameter estimation for fractional Ornstein-Uhlenbeck processes

Abstract

We study a least squares estimator $\hat {\theta}_T$ for the Ornstein-Uhlenbeck process, $dX_t=\theta X_t dt+\sigma dB^H_t$, driven by fractional Brownian motion $B^H$ with Hurst parameter $H\ge \frac12$. We prove the strong consistence of $\hat {\theta}_T$ (the almost surely convergence of $\hat {\theta}_T$ to the true parameter ${% \theta}$). We also obtain the rate of this convergence when $1/2\le H<3/4$, applying a central limit theorem for multiple Wiener integrals. This least squares estimator can be used to study other more simulation friendly estimators such as the estimator $\tilde \theta_T$ defined by (4.1).