Non trivial optimal sampling rate for estimating a Lipschitz-continuous function in presence of mean-reverting Ornstein-Uhlenbeck noise

TL;DR

This paper investigates the optimal sampling rate for estimating a Lipschitz-continuous drift in a mean-reverting Ornstein-Uhlenbeck process, revealing a non-monotonic mean square error that depends on the trade-off between observation correlation and drift dynamics.

Contribution

It introduces an online, time-varying estimation method using stochastic gradient ascent and characterizes the optimal sampling rate for minimizing estimation error.

Findings

Mean square error is non-monotonic in sample size.

Optimal sample size balances correlation and drift variability.

Method outperforms simple averaging in highly correlated regimes.

Abstract

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.