Non trivial optimal sampling rate for estimating a Lipschitz-continuous function in presence of mean-reverting Ornstein-Uhlenbeck noise
arxiv(2024)
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.
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