Driven by Brownian motion Cox–Ingersoll–Ross and squared Bessel processes : Interaction and phase transition

Abstract

This paper studies two related stochastic processes driven by Brownian motion: the Cox–Ingersoll–Ross (CIR) process and the Bessel process. We investigate their shared and distinct properties, focusing on time-asymptotic growth rates, distance between the processes in integral norms, and parameter estimation. The squared Bessel process is shown to be a phase transition of the CIR process and can be approximated by a sequence of CIR processes. Differences in stochastic stability are also highlighted, with the Bessel process displaying instability while the CIR process remains ergodic and stable.