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

We consider a diffusion (ξt)t≥0 with T -periodic time dependence in its drift; under an unknown parameter θ ∈ Θ, some periodic discontinuity – called signal – occurs at times kT+θ, k ∈ IN . Assuming positive Harris recurrence of (ξkT )k∈IN0 and exploiting the periodicity structure, we prove limit theorems for certain martingales and functionals of the process (ξt)t≥0. They allow to consider the statistical model parametrized by θ ∈ Θ locally in small neighbourhoods of some fixed θ with radius 1 n as n → ∞. 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.