Estimating discontinuous periodic signals in a time inhomogeneous diffusion

摘要

We consider a diffusion (ξ_t)_(t≥0) with some T-periodic time dependent input term contained in the drift: under an unknown parameter V ∈ Θ, some discontinuity-an additional periodic signal-occurs at times kT +V,k ∈ IN. Assuming positive Harris recurrence of (ξkT)k∈IN_0 and exploiting the periodicity structure, we prove limit theorems for certain martingales and functionals of the process (ξ_t)_(t≥0). They allow to consider the statistical model parametrized by V ∈ Θ locally in small neighbourhoods of some fixed V, with radius, as n → ∞. We prove convergence of local models to a limit experiment studied by Ibragimov and Khasminskii (Statistical estimation, 1981) and discuss the behaviour of estimators under contiguous alternatives.