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

In this paper we investigate the large-sample behaviour of the maximum likelihood estimate (MLE) of the unknown parameter θ for processes following the model dξt = θf(t)ξt dt+ dBt, where f : R → R is a continuous function with period, say P > 0. Here the periodic function f(·) 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(·) is constant. However the case when ∫ P 0 f(t)dt = 0 and f(·) is not identically null presents some special features. For instance in this case whatever is the value of θ, the rate of convergence of the MLE is T as in the case when θ = 0 and ∫ P 0 f(t)dt 6= 0.