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.
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Höpfner, R., Kutoyants, Y. Estimating a periodicity parameter in the drift of a time inhomogeneous diffusion. Math. Meth. Stat. 20, 58–74 (2011). https://doi.org/10.3103/S1066530711010042
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DOI: https://doi.org/10.3103/S1066530711010042
Keywords
- diffusion processes
- inhomogeneity in time
- continuous signals
- discontinuous signals
- periodicity
- localization
- limit experiment