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

We consider the parameter estimation problem for the non-ergodic fractional Ornstein-Uhlenbeck process defined as $dX_t=θX_tdt+dB_t,\ t\geq0$, with a parameter $θ>0$, where $B$ is a fractional Brownian motion of Hurst index $H\in(1/2,1)$. We study the consistency and the asymptotic distributions of the least squares estimator $\hatθ_t$ of $θ$ based on the observation $\{X_s,\ s\in[0,t]\}$ as $t\rightarrow\infty$.