  We study the jump-diffusion CIR process, which is an extension of the
Cox-Ingersoll-Ross model and whose jumps are introduced by a subordinator. We
provide sufficient conditions on the L\'evy measure of the subordinator under
which the jump-diffusion CIR process is ergodic and exponentially ergodic,
respectively. Furthermore, we characterize the existence of the $\kappa$-moment
($\kappa>0$) of the jump-diffusion CIR process by an integrability condition on
the L\'evy measure of the subordinator.
