Drift parameter estimation for fractional Ornstein-Uhlenbeck process of the Second Kind

TL;DR

This paper establishes the consistency and asymptotic normality of a least squares estimator for the drift parameter in a fractional Ornstein-Uhlenbeck process of the second kind, driven by fractional Brownian motion with Hurst parameter greater than 1/2.

Contribution

It proves the consistency and asymptotic normality of the least squares estimator for the drift parameter in the fractional Ornstein-Uhlenbeck process of the second kind for all Hurst parameters greater than 1/2.

Findings

The estimator is consistent for H > 1/2.

The estimator is asymptotically normal for H in (1/2, 1).

Results extend previous work to the entire range H in (1/2, 1).

Abstract

Fractional Ornstein-Uhlenbeck process of the second kind $(\text{fOU}_{2})$ is solution of the Langevin equation $\mathrm{d}X_t = -\theta X_t\,\mathrm{d}t+\mathrm{d}Y_t^{(1)}, \ \theta >0$ with driving noise $ Y_t^{(1)} := \int^t_0 e^{-s} \,\mathrm{d}B_{a_s}; \ a_t= H e^{\frac{t}{H}}$ where $B$ is a fractional Brownian motion with Hurst parameter $H \in (0,1)$. In this article, in the case $H>1/2$, we prove that the least squares estimator $\hat{\theta}_T$ introduced in [\cite{h-n}, Statist. Probab. Lett. 80, no. 11-12, 1030-1038], provides a consistent estimator. Moreover, using central limit theorem for multiple Wiener integrals, we prove asymptotic normality of the estimator valid for the whole range $H \in(1/2,1)$.