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

Fractional Ornstein–Uhlenbeck process of the second kind $$(\text {fOU}_{2})$$ ( fOU 2 ) is a solution of the Langevin equation $$\mathrm {d}X_t=-\theta X_t\,\mathrm {d}t+\mathrm {d}Y_t^{(1)}, \ \theta >0$$ d X t = - θ X t d t + d Y t ( 1 ) , θ > 0 with a Gaussian driving noise $$ Y_t^{(1)} := \int ^t_0 e^{-s} \,\mathrm {d}B_{a_s}$$ Y t ( 1 ) : = ∫ 0 t e - s d B a s , where $$ a_t= H e^{\frac{t}{H}}$$ a t = H e t H and $$B$$ B is a fractional Brownian motion with Hurst parameter $$H \in (0,1)$$ H ∈ ( 0 , 1 ) . In this article we consider the case $$H>\frac{1}{2}$$ H > 1 2 , and by using the ergodicity of $$\text {fOU}_{2}$$ fOU 2 process we construct consistent estimators for the drift parameter $$\theta $$ θ based on discrete observations in two possible cases: $$(i)$$ ( i ) the Hurst parameter $$H$$ H is known and $$(ii)$$ ( i i ) the Hurst parameter $$H$$ H is unknown. Moreover, using Malliavin calculus techniques we prove central limit theorems for our estimators which are valid for the whole range $$H \in (\frac{1}{2},1)$$ H ∈ ( 1 2 , 1 ) . Copyright Springer Science+Business Media Dordrecht 2015