Statistical analysis of the non-ergodic fractional Ornstein-Uhlenbeck process with periodic mean

Abstract: Consider a periodic, mean-reverting Ornstein-Uhlenbeck process $X={X_t,t\geq0}$ of the form $d X_{t}=\left(L(t)+\alpha X_{t}\right) d t+ dB^H_{t}, \quad t \geq 0$, where $L(t)=\sum_{i=1}^{p}\mu_i\phi_i (t)$ is a periodic parametric function, and ${B^H_t,t\geq0}$ is a fractional Brownian motion of Hurst parameter $\frac12\leq H<1$. In the "ergodic" case $\alpha<0$, the parametric estimation of $(\mu_1,\ldots,\mu_p,\alpha)$ based on continuous-time observation of $X$ has been considered in Dehling et al. \cite{DFK}, and in Dehling et al. \cite{DFW} for $H=\frac12$, and $\frac12<H\<1$, respectively. In this paper we consider the "non-ergodic" case $\alpha\>0$, and for all $\frac12\leq H<1$. We analyze the strong consistency and the asymptotic distribution for the estimator of $(\mu_1,\ldots,\mu_p,\alpha)$ when the whole trajectory of $X$ is observed.