Drift parameter estimation for fractional Ornstein–Uhlenbeck process of the second kind

Abstract

The fractional Ornstein–Uhlenbeck process of the second kind (fOU₂) is the solution of the Langevin equation $\mathrm{d}{X}_t=-\theta X_t\mathrm{d}t+\mathrm{d}{Y}_t^{(1)},\theta >0$ with driving noise $Y_t^{(1)}:={\int }_0^t{\mathrm{e}}^{-s}\mathrm{d}{B}_{a_s};a_t=H{\mathrm{e}}^{t/H}$ where B is a fractional Brownian motion with Hurst parameter H∈(0, 1). In this article, in the case H>½, we prove that the least-squares estimator ${\hat{\theta }}_T$ introduced in [Hu Y, Nualart D. Parameter estimation for fractional Ornstein–Uhlenbeck processes. Stat. Probab. Lett. 2010;80(11–12):1030–1038], provides a consistent estimator. Moreover, using central limit theorem for multiple Wiener integrals, we prove asymptotic normality of the estimator valid for the whole range H∈(½, 1).

Keywords:

• fractional Ornstein–Uhlenbeck processes
• Malliavin calculus
• Langevin equation
• least-squares estimator

2010 AMS Subject Classifications:

• 60G22
• 60H07
• 62F12

Acknowledgements

Ehsan Azmoodeh thanks the Magnus Ehrnrooth foundation for financial support of the major part of the this work that had been done in Helsinki. José Igor Morlanes thanks Prof. Esko Valkeila for his advice and his financial support via the Academy grant 21245. Both authors thank Tommi Sottinen and Lauri Viitasaari for useful comments and discussions. The authors wish to thank both anonymous referees for careful reading of the previous versions of this paper and also their comments which improved the paper.