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

We consider a mean-reverting Ornstein–Uhlenbeck process that perturbs an unknown Lipschitz-continuous drift and want to estimate the value of that drift at some fixed time horizon. The estimation is based on a finite number of observations obtained by sampling the path of the process at equally spaced instant of times between the time origin and prescribed time horizon. Aim of the study is to find the optimal sample size/rate for achieving the minimum mean square distance between our estimator and the true value of the drift at the time horizon. The estimation procedure we utilize consists of an online time-varying optimization scheme employed through a stochastic gradient ascent algorithm for the log-likelihood of our observations. We discover 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 rule the model. In the static case, i.e. when the unknown drift is constant, our method performs in highly correlated regimes even better than the arithmetic mean of the observations, which represents in such particular instance a natural candidate estimator. A comparison with the global maximum likelihood estimator is also presented.