Estimating discontinuous periodic signals in a non-time homogeneous diffusion process

We consider a diffusion $(\xi_t)_{t\ge 0}$ with $T$-periodic time dependence in its drift; under an unknown parameter $\vth\in\Theta$, some periodic discontinuity -- called signal -- occurs at times $kT{+}\vth$, $k\in\bbn$. Assuming positive Harris recurrence of $(\xi_{kT})_{k\in\bbn_0}$ and exploiting the periodicity structure, we prove limit theorems for certain martingales and functionals of the process $(\xi_t)_{t\ge 0}$. They allow to consider the statistical model parametrized by $\vth\in\Theta$ locally in small neighbourhoods of some fixed $\vth$ with radius $\frac{1}{n}$ as $\nto$. We prove convergence of local models to a limit experiment studied first by Ibragimov and Khasminskii [IH 81] which is not quadratic in its parameter. We discuss the behaviour of estimators under contiguous alternatives, and prove a local asymptotic minimax bound under quadratic loss which is attained by the corresponding Bayes estimator.