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

TL;DR

This paper investigates the asymptotic properties of the maximum likelihood estimator for the growth rate of a jump-type CIR process, revealing different behaviors in subcritical, critical, and supercritical regimes, including new phenomena in the supercritical case.

Contribution

It provides a comprehensive analysis of the MLE's asymptotic behavior for jump-type CIR processes across all regimes, including novel stochastic representations in the supercritical case.

Findings

Weak consistency and asymptotic normality in subcritical case

Strong consistency and mixed normal asymptotics in supercritical case

Non-standard asymptotic behavior in critical case

Abstract

We consider a jump-type Cox--Ingersoll--Ross (CIR) process driven by a standard Wiener process and a subordinator, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate. We distinguish three cases: subcritical, critical and supercritical. In the subcritical case we prove weak consistency and asymptotic normality, and, under an additional moment assumption, strong consistency as well. In the supercritical case, we prove strong consistency and mixed normal (but non-normal) asymptotic behavior, while in the critical case, weak consistency and non-standard asymptotic behavior are described. We specialize our results to so-called basic affine jump-diffusions as well. Concerning the asymptotic behavior of the MLE in the supercritical case, we derive a stochastic representation of the limiting mixed normal distribution, where the almost sure limit of an appropriately scaled jump-type supercritical CIR process comes into play. This is a new phenomenon, compared to the critical case, where a diffusion-type critical CIR process plays a role.