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

TL;DR

This paper investigates the estimation of the unknown periodicity parameter in the drift of a time-inhomogeneous diffusion process, establishing asymptotic properties under different smoothness conditions of the signal.

Contribution

It introduces a novel approach to analyze the local asymptotic behavior of the periodicity parameter estimation in diffusions with periodic signals, including cases with discontinuities.

Findings

LAN holds for smooth signals with local scale n^{-3/2}

Different limit experiment for signals with finite discontinuities

Hölder 1/2 smoothness in the non-smooth case

Abstract

We consider a diffusion $(\xi_t)_{t\ge 0}$ 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 $\vartheta$ of interest. We consider sequences of local models at $\vartheta$, corresponding to continuous observation of the process $\xi$ on the time interval $[0,n]$ as $n\to\infty$, with suitable choice of local scale at $\vartheta$. Our tools --under an ergodicity condition-- are path segments of $\xi$ corresponding to the period $\vartheta$, and limit theorems for certain functionals of the process $\xi$ which are not additive functionals. When the signal is smooth, with local scale $n^{-3/2}$ at $\vartheta$, 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 $\vartheta$, 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 H\"older $\frac12$.