We consider a stable Cox-Ingersoll-Ross (α-stable CIR) process defined by $dX_t=(a - b X_t)dt + \sigma X_{t}^{1/2} dW_t + \delta X_{t-}^{1/ \alpha} dL^{\alpha}_t, \quad X_0=x_0 >0,$ where $(L^{\alpha}_t)$ is a stable Lévy process with non-negative jumps and jump activity index $\alpha \in (1,2)$.
We first consider the pure jump case $\sigma=0$. We prove the existence of a joint estimator of $(a, b, \delta, \alpha)$ based on an approximation of the likelihood function, which is consistent and asymptotically conditionally Gaussian. Moreover, uniqueness of the drift estimators is established assuming that $ \delta$ and $\alpha$ are known or consistently estimated. We propose easy-to-implement preliminary estimators of all parameters and improve them by a one-step procedure.
We next consider the case $\sigma >0$. Our aim is to study the joint estimation of $(\sigma, \delta, \alpha)$. Due to the superposition of a Brownian motion and a Lévy process with infinite variation, the usual statistics have an asymptotic bias. We present the problem on a very simple model and explain how to overcome this difficulty.
