We first study the drift parameter estimation of the fractional Ornstein-Uhlenbeck process (fOU) with periodic mean for every $\frac{ 1}{ 2}<H<1$. More precisely, we extend the consistency proved in \cite{ DFW} for $\frac{ 1}{ 2}<H<\frac{ 3}{ 4}$ to the strong consistency for any $\frac{ 1}{ 2}<H<1$ on the one hand, and on the other, we also discuss the asymptotic normality given in \cite{ DFW}. In the second main part of the paper, we study the strong consistency and the asymptotic normality of the fOU of the second kind with periodic mean for any $\frac{ 1}{ 2}<H<1$.
Categories: math.PR math.ST stat.TH