Entropies of Cox-Ingersoll-Ross and Bessel processes as functions of time and of related parameters

TL;DR

This paper studies the long-term behavior of various entropy measures related to Cox-Ingersoll-Ross and Bessel processes, revealing their convergence properties and conditions for finiteness as parameters vary.

Contribution

It provides new conditions for the existence of entropy measures for noncentral chi-squared laws and analyzes their asymptotic behavior for CIR and Bessel processes.

Findings

CIR entropies converge to stationary distribution values over time

Squared Bessel process entropies diverge or remain finite depending on parameters

CIR and Bessel process entropies converge as CIR approaches Bessel process

Abstract

We investigate the long-time asymptotic behavior of various entropy measures associated with the Cox-Ingersoll-Ross (CIR) and squared Bessel processes. As the one-dimensional distributions of both processes follow noncentral chi-squared laws, we first derive sufficient conditions for the existence of these entropy measures for a noncentral chi-squared random variable. We then analyze their limiting behavior as the noncentrality parameter approaches zero and apply these results to the CIR and squared Bessel processes. We prove that, as time tends to infinity, the entropies of the CIR process converge to those of its stationary distribution, while for the squared Bessel process, the Shannon, R\'enyi, and generalized R\'enyi entropies diverge, however, the Tsallis and Sharma-Mittal entropies may diverge or remain finite depending on the entropy parameters. Finally, we demonstrate that, as the CIR process converges to the squared Bessel process, the corresponding entropies also converge.