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

TL;DR

This paper establishes Wasserstein bounds for the rate of convergence in the central limit theorem of drift estimators for Ornstein-Uhlenbeck processes observed at high frequency, using approximate MLE and MCE methods.

Contribution

It introduces new Wasserstein bounds for the CLT of drift estimators in high-frequency Ornstein-Uhlenbeck processes, based on approximate maximum likelihood and contrast estimators.

Findings

Wasserstein bounds are derived for the CLT of the estimators.

Results apply to high-frequency observations as elta_n approaches zero.

The estimators' convergence rates are quantitatively characterized.

Abstract

This paper deals with the rate of convergence for the central limit theorem of estimators of the drift coefficient, denoted $\theta$, for a Ornstein-Uhlenbeck process $X \coloneqq \{X_t,t\geq0\}$ observed at high frequency. We provide an Approximate minimum contrast estimator and an approximate maximum likelihood estimator of $\theta$, namely $\widetilde{\theta}_{n}\coloneqq {1}/{\left(\frac{2}{n} \sum_{i=1}^{n}X_{t_{i}}^{2}\right)}$, and $\widehat{\theta}_{n}\coloneqq -{\sum_{i=1}^{n} X_{t_{i-1}}\left(X_{t_{i}}-X_{t_{i-1}}\right)}/{\left(\Delta_{n} \sum_{i=1}^{n} X_{t_{i-1}}^{2}\right)}$, respectively, where $ t_{i} = i \Delta_{n}$, $ i=0,1,\ldots, n $, $\Delta_{n}\rightarrow 0$. We provide Wasserstein bounds in central limit theorem for $\widetilde{\theta}_{n}$ and $\widehat{\theta}_{n}$.