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

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 $Y_t^{(1)}:={\int }_0^t{e}^{-s}\mathrm{d}{B}_{a_s};a_t=H{e}^{\frac{t}{H}}$ 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 $H\in (1/2,1)$.

18 pages

References in corpus (1)

• Parameter Estimation for Fractional Ornstein-Uhlenbeck Processes: Non-ergodic Case