Parameter maximum likelihood estimation problem for time periodic modulated drift Ornstein Uhlenbeck processes

Abstract

In this paper we investigate the large-sample behaviour of the maximum likelihood estimate (MLE) of the unknown parameter $\theta$ for processes following the model

$$\begin{aligned} d\xi _{t}=\theta f(t)\xi _{t}\,dt+d\mathrm {B}_t, \end{aligned}$$

where $f:{\mathbb {R}}\rightarrow {\mathbb {R}}$ is a continuous function with period, say $P>0$. Here the periodic function $f(\cdot )$ is assumed known. We establish the consistency of the MLE and we point out its minimax optimality. These results comply with the well-established case of an Ornstein Uhlenbek process when the function $f(\cdot )$ is constant. However the case when $\int _0^P f(t)dt=0$ and $f(\cdot )$ is not identically null presents some special features. For instance in this case whatever is the value of $\theta$, the rate of convergence of the MLE is $T$ as in the case when $\theta =0$ and $\int _0^Pf(t)dt\ne 0$.