Least squares estimation for non-ergodic weighted fractional Ornstein-Uhlenbeck process of general parameters

Let $B^{a,b}:=\{B_t^{a,b},t\ge 0\}$ be a weighted fractional Brownian motion of parameters $a>-1$, $\mathrm{\mid }b\mathrm{\mid }<1$, $\mathrm{\mid }b\mathrm{\mid }<a+1$. We consider a least square-type method to estimate the drift parameter $\theta >0$ of the weighted fractional Ornstein-Uhlenbeck process $X:=\{X_t,t\ge 0\}$ defined by $X_0=0; \ dX_t=\theta X_tdt+dB_t^{a,b}$. In this work, we provide least squares-type estimators for $\theta$ based continuous-time and discrete-time observations of $X$. The strong consistency and the asymptotic behavior in distribution of the estimators are studied for all $(a,b)$ such that $a>-1$, $\mathrm{\mid }b\mathrm{\mid }<1$, $\mathrm{\mid }b\mathrm{\mid }<a+1$. Here we extend the results of \cite{SYY2,SYY} (resp. \cite{CSC}), where the strong consistency and the asymptotic distribution of the estimators are proved for $-\frac{1}{2}<a<0$, $-a<b<a+1$ (resp. $-1<a<0$, $-a<b<a+1$).