Moments and ergodicity of the jump-diffusion CIR process

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.