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Fractional Ornstein–Uhlenbeck process of the second kind \((\text {fOU}_{2})\) is a solution of the Langevin equation \(\mathrm {d}X_t = -\theta X_t\,\mathrm {d}t+\mathrm {d}Y_t^{(1)}, \ \theta >0\) with a Gaussian driving noise \( Y_t^{(1)} := \int ^t_0 e^{-s} \,\mathrm {d}B_{a_s}\), where \( a_t= H e^{\frac{t}{H}}\) and \(B\) is a fractional Brownian motion with Hurst parameter \(H \in (0,1)\). In this article we consider the case \(H>\frac{1}{2}\), and by using the ergodicity of \(\text {fOU}_{2}\) process we construct consistent estimators for the drift parameter \(\theta \) based on discrete observations in two possible cases: \((i)\) the Hurst parameter \(H\) is known and \((ii)\) the Hurst parameter \(H\) is unknown. Moreover, using Malliavin calculus techniques we prove central limit theorems for our estimators which are valid for the whole range \(H \in (\frac{1}{2},1)\).

Parameter estimation based on discrete observations of fractional Ornstein–Uhlenbeck process of the second kind