Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - University of Rouen Normandy

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.