A stable Cox–Ingersoll–Ross model with restart

We consider a stable Cox–Ingersoll–Ross model in a domain D=[0,∞)D=[0,∞) which is killed at the boundary and which, while alive, jumps instantaneously at an exponentially distributed random time with intensity λ>0λ>0 to a new point, according to a distribution ν. From this new point, it repeats the above behavior independently of what has transpired previously. We give the infinitesimal generator of this new process and prove that the corresponding semigroup is compact and thus the spectrum of the generator consists exclusively of eigenvalues. We further show that the principal eigenvalue gives the exponential rate of decay of not exiting the domain by time t. Finally, we study some properties of the principal eigenvalue.