Asymptotic properties of maximum likelihood estimator for the growth rate of a stable CIR process based on continuous time observations

TL;DR

This paper investigates the asymptotic behavior of the maximum likelihood estimator for the growth rate of a stable Cox--Ingersoll--Ross process driven by Lévy noise, establishing consistency and asymptotic distributions in various regimes.

Contribution

It provides a comprehensive analysis of the MLE's asymptotic properties for a stable CIR process across subcritical, critical, and supercritical cases, including new results on consistency and normality.

Findings

Strong consistency of the MLE in all cases

Asymptotic normality in the subcritical case

Asymptotic mixed normality in the supercritical case

Abstract

We consider a stable Cox--Ingersoll--Ross process driven by a standard Wiener process and a spectrally positive strictly stable L\'evy process, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate based on continuous time observations. We distinguish three cases: subcritical, critical and supercritical. In all cases we prove strong consistency of the MLE in question, in the subcritical case asymptotic normality, and in the supercritical case asymptotic mixed normality are shown as well. In the critical case the description of the asymptotic behavior of the MLE in question remains open.