We examine a mean-reverting Ornstein-Uhlenbeck process that perturbs an unknown Lipschitz-continuous drift and aim to estimate the drift's value at a predetermined time horizon by sampling the path of the process. Due to the time varying nature of the drift we propose an estimation procedure that involves an online, time-varying optimization scheme implemented using a stochastic gradient ascent algorithm to maximize the log-likelihood of our observations. The objective of the paper is to investigate the optimal sample size/rate for achieving the minimum mean square distance between our estimator and the true value of the drift. In this setting we uncover a trade-off between the correlation of the observations, which increases with the sample size, and the dynamic nature of the unknown drift, which is weakened by increasing the frequency of observation. The mean square error is shown to be non monotonic in the sample size, attaining a global minimum whose precise description depends on the parameters that govern the model. In the static case, i.e. when the unknown drift is constant, our method outperforms the arithmetic mean of the observations in highly correlated regimes, despite the latter being a natural candidate estimator. We then compare our online estimator with the global maximum likelihood estimator.

翻译：本文研究一个扰动未知Lipschitz连续漂移项的均值回归Ornstein-Uhlenbeck过程，旨在通过采样该过程路径，在预设时间范围内估计漂移项的真实值。考虑到漂移项随时间变化的特性，我们提出一种在线时变优化估计方法，该方法采用随机梯度上升算法最大化观测数据的对数似然函数。本文的目标是探究最优样本量/采样率，以使得估计值与漂移项真实值之间的均方距离最小化。在此框架下，我们发现观测数据相关性（随样本量增加而增强）与未知漂移项的动态特性（可通过提高观测频率削弱）之间存在权衡关系。均方误差随样本量变化呈现非单调性，其全局最小值点的精确描述取决于模型参数。在静态情形（即未知漂移项为常数）下，即使观测数据的算术平均是自然的候选估计量，我们的方法在高相关场景中仍能获得更优性能。最后，我们将所提出的在线估计量与全局最大似然估计量进行了比较。