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

Abstract

Fractional Ornstein-Uhlenbeck process of the second kind $({\text{fOU}}_2)$ is solution of the Langevin equation $\mathrm{d}{X}_t=-\theta X_t\mathrm{d}t+\mathrm{d}{Y}_t^{(1)},\theta >0$ with Gaussian driving noise $Y_t^{(1)}:={\int }_0^t{e}^{-s}\mathrm{d}{B}_{a_s}$, 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 $H>\frac{1}{2}$. Then using the ergodicity of ${\text{fOU}}_2$ process, we construct consistent estimators of 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 technique, we prove central limit theorems for our estimators which is valid for the whole range $H\in (\frac{1}{2},1)$.

Modified version. arXiv admin note: text overlap with arXiv:1302.6047

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