On LAN for parametrized continuous periodic signals in a time inhomogeneous diffusion

TL;DR

This paper establishes local asymptotic normality for parameter estimation in a diffusion process with a periodic drift signal, leveraging the process's periodic structure and limit theorems for functionals.

Contribution

It introduces a LAN framework for parametrized periodic signals in diffusions, extending previous methods to more general functionals of the process.

Findings

Proves LAN for the diffusion with periodic drift.

Develops estimators for the unknown parameter.

Utilizes the process's periodic structure for analysis.

Abstract

We consider a diffusion $(\xi_t)_{t\ge 0}$ whose drift involves a $T$-periodic signal. $T$ is fixed and known, whereas the signal depends on an unknown $d$-dimensional parameter $\vartheta\in\Theta$. Assuming positive Harris recurrence of the grid chain $(\xi_{kT})_{k\in\mathbb{N}_0}$ and exploiting the periodic structure of the semigroup, we work with path segments and limit theorems for certain functionals (more general than additive functionals) of the process to prove local asymptotic normality (LAN). Then we consider several estimators for the unknown parameter.