Estimating a periodicity parameter in the drift of a time inhomogeneous diffusion

We consider a diffusion $(\xi_t)_{t\ge 0}$ whose drift contains some deterministic periodic signal. Its shape being fixed and known, up to scaling in time, the periodicity of the signal is the unknown parameter $\vartheta$ of interest. We consider sequences of local models at $\vartheta$, corresponding to continuous observation of the process $\xi$ on the time interval $[0,n]$ as $n\to\infty$, with suitable choice of local scale at $\vartheta$. Our tools --under an ergodicity condition-- are path segments of $\xi$ corresponding to the period $\vartheta$, and limit theorems for certain functionals of the process $\xi$ which are not additive functionals. When the signal is smooth, with local scale $n^{-3/2}$ at $\vartheta$, we have local asymptotic normality (LAN) in the sense of Le Cam (1969). When the signal has a finite number of discontinuities, with local scale $n^{-2}$ at $\vartheta$, we obtain a limit experiment of different type, studied by Ibragimov and Khasminskii (1981), where smoothness of the parametrization (in the sense of Hellinger distance) is H\"older $\frac12$.