  For stochastic processes of non-commuting random variables we formulate a
Cox-Ingersoll-Ross (CIR) stochastic differential equation in the context of
free probability theory which was introduced by Voicelescu. By transforming the
classical CIR equation and the Feller condition, which ensures the existence of
a positive solution, into the free setting (in the sense of having a strictly
positive spectrum), we show the existence of a free CIR equation. The main
challenge lies in the transition from a stochastic differential equation driven
by a classical Brownian motion to a stochastic differential equation driven by
the free analogue to the classical Brownian motion, the so-called free Brownian
motion.
