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 $$\theta $$ θ for processes following the model $$\begin{aligned} d\xi _{t}=\theta f(t)\xi _{t}\,dt+d\mathrm {B}_t, \end{aligned}$$ d ξ t = θ f ( t ) ξ t d t + d B t , where $$f:{\mathbb {R}}\rightarrow {\mathbb {R}}$$ f : R → R is a continuous function with period, say $$P>0$$ P > 0 . Here the periodic function $$f(\cdot )$$ 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 $$f(\cdot )$$ f ( · ) is constant. However the case when $$\int _0^P f(t)dt=0$$ ∫ 0 P f ( t ) d t = 0 and $$f(\cdot )$$ f ( · ) is not identically null presents some special features. For instance in this case whatever is the value of $$\theta $$ θ , the rate of convergence of the MLE is $$T$$ T as in the case when $$\theta =0$$ θ = 0 and $$\int _0^Pf(t)dt\ne 0$$ ∫ 0 P f ( t ) d t ≠ 0 . Copyright Springer Science+Business Media Dordrecht 2015