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

Ehsan Azmoodeh () and Lauri Viitasaari ()

Statistical Inference for Stochastic Processes, 2015, vol. 18, issue 3, 205-227

Abstract: 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

Keywords: Fractional Ornstein–Uhlenbeck processes; Malliavin calculus; Multiple Wiener integrals; Central limit theorem (CLT); Parameter estimation; 60G22; 60H07; 62F99 (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (4)

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DOI: 10.1007/s11203-014-9111-8

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