Least squares type estimations for discretely observed nonergodic Gaussian Ornstein-Uhlenbeck processes of the second kind

We consider the nonergodic Gaussian Ornstein-Uhlenbeck processes of the second kind defined by $d{X}_t=\theta X_{t}dt+d{Y}_t^{(1)},t\ge 0,X_0=0$ with an unknown parameter $\theta >0,$ where $d{Y}_t^{(1)}=e^{-t}d{G}_{a_t}$ and $\{G_t,t\ge 0\}$ is a mean zero Gaussian process with the self-similar index $\gamma \in (\frac{1}{2},1)$ and $a_t=\gamma e^{\frac{t}{\gamma }}$. Based on the discrete observations $\{X_{t_i}:t_i=i{\mathrm{\Delta }}_n,i=0,1,\cdots ,n\}$, two least squares type estimators ${\hat{\theta }}_n$ and ${\tilde{\theta }}_n$ of $\theta$ are constructed and proved to be strongly consistent and rate consistent. We apply our results to the cases such as fractional Brownian motion, sub-fractional Brownian motion, bifractional Brownian motion and sub-bifractional Brownian motion. Moreover, the numerical simulations confirm the theoretical results.