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

32 pagesInternational audienceIn this paper we investigate the large-sample behaviour of the maximum likelihood estimate (MLE) of the unknown parameter $\theta$ for processes following the model $d\xi_t = \theta f(t)\xi_t dt + dB_t$, where $f : R \rightarrow R$ 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 $\int^P_0 f(t) dt = 0$ 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 $\int^P_0 f(t)dt \neq 0$