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

2010/10/25 by Reinhard Hoepfner, Hoepfner, Reinhard, Yury Kutoyants +1
Mathematics · #60J60 #62F12 #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST) #math.PR #math.ST #msc:60J60 #msc:62F12 #stat.TH

paper · pdf · doi:10.48550/arxiv.1010.5105

arxiv created 2010/10/25 · arxiv updated 2010/10/26

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 (1969). 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 (1981), where smoothness of the parametrization (in the sense of Hellinger distance) is Hölder \frac12.

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