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

Abstract

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