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

ABSTRACT

We consider a stable Cox–Ingersoll–Ross process driven by a standard Wiener process and a spectrally positive strictly stable Lévy 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 behaviour of the MLE in question remains open.

KEYWORDS:

• Stable Cox–Ingersoll–Ross process
• maximum likelihood estimator
• strong consistency
• asymptotic mixed normality

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 60H10
• 91G70
• 60F05
• 62F12

Acknowledgments

We are grateful to Clément Foucart for providing us an idea how to derive (24(25) $\mathrm{E}\left({\mathrm{e}}^{uV}\right)=\exp \{y_0{\psi }_u^{\ast }+{\int }_0^{-{\psi }_u^{\ast }}\frac{F\left(z\right)}{R\left(z\right)}\mathrm{d}z\},u\in {\mathbb{R}}_{-},$(25) ), a formula for the Laplace transform of V in Theorem 7.1. We would like to thank Hatem Zaag for explaining us several methods that may be used for describing the asymptotic behaviour of the ordinary differential equation (17(18) ${\psi }_{u,v}^{\prime }\left(t\right)=\frac{{\sigma }^2}{2}{\psi }_{u,v}(t{)}^2+\frac{{\delta }^{\alpha }}{\alpha }(-{\psi }_{u,v}\left(t\right){)}^{\alpha }-b{\psi }_{u,v}\left(t\right)+v,t\in {\mathbb{R}}_{+},{\psi }_{u,v}\left(0\right)=u.$(18) ). We would like to thank the referees for their comments that helped us to improve the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Gyula Pap  http://orcid.org/0000-0002-1516-5567

Additional information

Funding

This research is supported by Laboratory of Excellence MME-DII [grant number  ANR11-LBX-0023-01] (http://labex-mme-dii.u-cergy.fr/). Mátyás Barczy was supported between September 2016 and January 2017 by the ‘Magyar Állami Eötvös Ösztöndíj 2016’, Tempus Public Foundation [grant number  75141] and from September 2017 by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. Ahmed Kebaier benefited from the support of the Chair Risques Financiers, Fondation du Risque.