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

Abstract

We consider a diffusion $(\xi_t)_{t\ge 0}$ with some $T$-periodic time dependent input term contained in the drift: under an unknown parameter $\vth\in\Theta$, some discontinuity - an additional periodic 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 $1/n$ as $\nto$. We prove convergence of local models to a limit experiment studied by Ibragimov and Khasminskii [IH 81] and discuss the behaviour of estimators under contiguous alternatives.