Modern Stochastics: Theory and Applications

In this paper, we deal with an Ornstein–Uhlenbeck process driven by sub-fractional Brownian motion of the second kind with Hurst index $H\in (\frac{1}{2},1)$. We provide a least squares estimator (LSE) of the drift parameter based on continuous-time observations. The strong consistency and the upper bound $O(1/\sqrt{n})$ in Kolmogorov distance for central limit theorem of the LSE are obtained. We use a Malliavin–Stein approach for normal approximations.

A problem of drift parameter estimation is studied for a nonergodic weighted fractional Vasicek model defined as $d{X_{t}}=\theta (\mu +{X_{t}})dt+d{B_{t}^{a,b}}$, $t\ge 0$, with unknown parameters $\theta >0$, $\mu \in \mathbb{R}$ and $\alpha :=\theta \mu $, whereas ${B^{a,b}}:=\{{B_{t}^{a,b}},t\ge 0\}$ is a weighted fractional Brownian motion with parameters $a>-1$, $|b|<1$, $|b|<a+1$. Least square-type estimators $({\widetilde{\theta }_{T}},{\widetilde{\mu }_{T}})$ and $({\widetilde{\theta }_{T}},{\widetilde{\alpha }_{T}})$ are provided, respectively, for $(\theta ,\mu )$ and $(\theta ,\alpha )$ based on a continuous-time observation of $\{{X_{t}},\hspace{2.5pt}t\in [0,T]\}$ as $T\to \infty $. The strong consistency and the joint asymptotic distribution of $({\widetilde{\theta }_{T}},{\widetilde{\mu }_{T}})$ and $({\widetilde{\theta }_{T}},{\widetilde{\alpha }_{T}})$ are studied. Moreover, it is obtained that the limit distribution of ${\widetilde{\theta }_{T}}$ is a Cauchy-type distribution, and ${\widetilde{\mu }_{T}}$ and ${\widetilde{\alpha }_{T}}$ are asymptotically normal.

Stationary processes have been extensively studied in the literature. Their applications include modeling and forecasting numerous real life phenomena such as natural disasters, sales and market movements. When stationary processes are considered, modeling is traditionally based on fitting an autoregressive moving average (ARMA) process. However, we challenge this conventional approach. Instead of fitting an ARMA model, we apply an AR(1) characterization in modeling any strictly stationary processes. Moreover, we derive consistent and asymptotically normal estimators of the corresponding model parameter.