On Finite Time Span Estimators of Parameters for Ornstein-Uhlenbeck Processes

Abstract

We study the bias and the mean-squared error of the maximum likelihood estimators (MLE) of parameters associated with a two-parameter mean-reverting process for a finite time $T$. Using the likelihood ratio process, we derive the expressions for MLEs, then compute the bias and the MSE via the change of measure and Ito's formula. We apply the derived expressions to the general Ornstein-Uhlenbeck process, where the bias and the MSE are numerically computed through a joint moment-generating function of key functionals of the O-U process. A numerical study is provided to illustrate the behaviour of bias and the MSE for the MLE of the mean-reverting speed parameter.