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

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Fractional Ornstein-Uhlenbeck process of the second kind (fOU₂) is solution of the Langevin equation dXₜ = -θXₜdt+dYₜ⁽¹⁾, θ>0 with Gaussian driving noise Yₜ⁽¹⁾:= ∫ᵗ₀ e⁻ˢ dBₐₛ, where aₜ= H eᵗ/ʰ and B is a fractional Brownian motion with Hurst parameter H ∈ (0,1). In this article, we consider the case H>1/2. Then using the ergodicity of fOU₂ process, we construct consistent estimators of drift parameter θ 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 technique, we prove central limit theorems for our estimators which is valid for the whole range H ∈ (1/2,1).