Parameter maximum likelihood estimation problem for time-periodic-drift Langevin type stochastic differential equations

In this paper we investigate the large-sample behavior 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, and which is observed through continuous time interval [0, T ] as T → ∞. Here the periodic function f(·) is assumed known. We establish the consistency of the MLE and we point out its minimax efficiency. These results comply with the well-established case when the function f(·) is constant non null. However the case when ∫ P 0 f(t)dt = 0 and f(·) is not identically null presents some particularities. 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. Futhermore when ∫ P 0 f(t)dt = 0, the MLE is locally efficient for the quadratic risk.