Parameter estimation for fractional Ornstein-Uhlenbeck processes

Abstract

We study a least squares estimator ${\hat{\theta }}_T$ for the Ornstein-Uhlenbeck process, $d{X}_t=\theta X_{t}dt+\sigma d{B}_t^H$, driven by fractional Brownian motion $B^H$ with Hurst parameter $H\ge \frac{1}{2}$. We prove the strong consistence of ${\hat{\theta }}_T$ (the almost surely convergence of ${\hat{\theta }}_T$ to the true parameter ${% θ}$). 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).

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