Standard and fractional reflected Ornstein-Uhlenbeck processes as the limits of square roots of Cox-Ingersoll-Ross processes

TL;DR

This paper demonstrates that the square root of Cox-Ingersoll-Ross processes converges to reflected Ornstein-Uhlenbeck processes under certain limits, establishing a new connection between these models driven by Wiener or fractional Brownian motion.

Contribution

It introduces a novel limit relationship between CIR and ROU processes driven by standard or fractional Brownian motion, with new representations of reflection functions.

Findings

Square root of CIR converges to ROU as mean reversion parameter varies.

Provides a new representation of the reflection function of ROU.

Simulation results illustrate theoretical convergence.

Abstract

In this paper, we establish a new connection between Cox-Ingersoll-Ross (CIR) and reflected Ornstein-Uhlenbeck (ROU) models driven by either a standard Wiener process or a fractional Brownian motion with $H>\frac{1}{2}$. We prove that, with probability 1, the square root of the CIR process converges uniformly on compacts to the ROU process as the mean reversion parameter tends to either $\sigma^2/4$ (in the standard case) or to $0$ (in the fractional case). This also allows to obtain a new representation of the reflection function of the ROU as the limit of integral functionals of the CIR processes. The results of the paper are illustrated by simulations.