Parameter estimation for fractional Ornstein-Uhlenbeck processes

TL;DR

This paper investigates the properties of a least squares estimator for the fractional Ornstein-Uhlenbeck process, proving its strong consistency and convergence rate for certain Hurst parameters, and discusses related estimators.

Contribution

It establishes the strong consistency and convergence rate of the least squares estimator for fractional Ornstein-Uhlenbeck processes with Hurst parameter H ≥ 1/2, including H in [1/2, 3/4).

Findings

Proves almost sure convergence of the estimator.

Derives the convergence rate for H in [1/2, 3/4).

Discusses alternative estimators like equate for simulation.

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).