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.
Categories: math.PR math.ST stat.TH
