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

Abstract

We consider a diffusion (ξ t ) t≥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 ϑ of interest. We consider sequences of local models at ϑ corresponding to continuous observation of the process ξ on the time interval [0, n] as n → ∞, with suitable choice of local scale at ϑ. Our tools - under an ergodicity condition — are path segments of ξ corresponding to the period ϑ, and limit theorems for certain functionals of the process ξ, which are not additive functionals. When the signal is smooth, with local scale n −3/2 at ϑ, we have local asymptotic normality (LAN) in the sense of Le Cam [21]. When the signal has a finite number of discontinuities, with local scale n −2 at ϑ, we obtain a limit experiment of different type, studied by Ibragimov and Khasminskii [14], where smoothness of the parametrization (in the sense of Hellinger distance) is Hölder 1/2.