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

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