Local asymptotic normality for shape and periodicity in the drift of a time inhomogeneous diffusion

We consider a one-dimensional diffusion whose drift contains a deterministic periodic signal with unknown periodicity T and carrying some unknown d-dimensional shape parameter \(\vartheta \). We prove local asymptotic normality (LAN) jointly in \(\vartheta \) and T for the statistical experiment arising from continuous observation of this diffusion. The local scale turns out to be \(n^{-1/2}\) for the shape parameter and \(n^{-3/2}\) for the periodicity which generalizes known results about LAN when either \(\vartheta \) or T is assumed to be known.