Berry-Ess\'een bound for drift estimation of fractional Ornstein Uhlenbeck process of second kind

Abstract

In the present paper we consider the Ornstein-Uhlenbeck process of the second kind defined as solution to the equation $dX_{t} = -\alpha X_{t}dt+dY_{t}^{(1)}, \ \ X_{0}=0$, where $Y_{t}^{(1)}:=\int_{0}^{t}e^{-s}dB^H_{a_{s}}$ with $a_{t}=He^{\frac{t}{H}}$, and $B^H$ is a fractional Brownian motion with Hurst parameter $H\in(\frac12,1)$, whereas $\alpha>0$ is unknown parameter to be estimated. We obtain the upper bound $O(1/\sqrt{T})$ in Kolmogorov distance for normal approximation of the least squares estimator of the drift parameter $\alpha$ on the basis of the continuous observation $\{X_t,t\in[0,T]\}$, as $T\rightarrow\infty$. Our method is based on the work of \cite{kp-JVA}, which is proved using a combination of Malliavin calculus and Stein's method for normal approximation.