Driven by Brownian motion Cox-Ingersoll-Ross and squared Bessel processes: interaction and phase transition

TL;DR

This paper explores the relationship between Cox-Ingersoll-Ross and squared Bessel processes driven by Brownian motion, revealing phase transitions, stability differences, and approximation methods, with implications for stochastic process analysis.

Contribution

It demonstrates that the squared Bessel process is a phase transition of the CIR process and provides insights into their stability and approximation properties.

Findings

Squared Bessel process as phase transition of CIR

Bessel process exhibits instability, CIR remains ergodic

CIR can be approximated by a sequence of CIR processes

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.