Wasserstein bounds in CLT of approximative MCE and MLE of the drift parameter for Ornstein-Uhlenbeck processes observed at high frequency

Abstract

AbstractThis paper deals with the rate of convergence for the central limit theorem of estimators of the drift coefficient, denotedθ, for the Ornstein-Uhlenbeck process$X := \{X_{t},t\geq 0\}$X:={Xt,t≥0}observed at high frequency. We provide an approximate minimum contrast estimator and an approximate maximum likelihood estimator ofθ, namely$\widetilde{\theta}_{n}:= {1}/{ (\frac{2}{n} \sum_{i=1}^{n}X_{t_{i}}^{2} )}$θ˜n:=1/(2n∑i=1nXti2), and$\widehat{\theta}_{n}:= -{\sum_{i=1}^{n} X_{t_{i-1}} (X_{t_{i}}-X_{t_{i-1}} )}/{ (\Delta _{n} \sum_{i=1}^{n} X_{t_{i-1}}^{2} )}$θˆn:=−∑i=1nXti−1(Xti−Xti−1)/(Δn∑i=1nXti−12), respectively, where$t_{i} = i \Delta _{n}$ti=iΔn,$i=0,1,\ldots , n $i=0,1,…,n,$\Delta _{n}\rightarrow 0$Δn→0. We provide Wasserstein bounds in the central limit theorem for$\widetilde{\theta}_{n}$θ˜nand$\widehat{\theta}_{n}$θˆn.