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

In this paper, we establish a new connection between Cox–Ingersoll–Ross (CIR) and reflected Ornstein–Uhlenbeck (ROU) models driven by either standard Wiener process or fractional Brownian motion with H>12. We prove that, probability 1, the square root of CIR converges uniformly on compacts to ROU as mean reversion parameter tends σ2/4 (in case) 0 case). This also allows obtain representation reflection function limit integral functionals processes. The results paper are illustrated simulations.