On frequency estimation of periodic ergodic diffusion process

TL;DR

This paper investigates the asymptotic behavior of frequency estimators for periodic ergodic diffusion processes under different smoothness conditions of the trend coefficient, revealing distinct convergence rates and limit distributions.

Contribution

It provides a detailed analysis of the asymptotic properties of maximum likelihood and Bayesian estimators in two scenarios with smooth and discontinuous trend coefficients.

Findings

Estimators are asymptotically normal with rate T^{3/2} in smooth case.

Estimators have different limit distributions with rate T^2 in discontinuous case.

Distinct convergence behaviors depending on the trend coefficient's smoothness.

Abstract

We consider the problem of frequency estimation by observations of the periodic diffusion process possesing ergodic properties in two different situations. The first one corresponds to continuously differentiable with respect to parameter trend coefficient and the second - to discontinuous trend coefficient. It is shown that in the first case the maximum likelihood and bayesian estimators are asymptotically normal with rate $T^{3/2}$ and in the second case these estimators have different limit distributions with the rate $T^2$.