Free CIR Processes

TL;DR

This paper extends the classical Cox-Ingersoll-Ross (CIR) process to the realm of free probability, formulating a free CIR equation driven by free Brownian motion and establishing conditions for its positivity.

Contribution

It introduces a novel free probability version of the CIR process, adapting classical stochastic differential equations to non-commutative variables.

Findings

Existence of a free CIR process under certain conditions

Transformation of classical Feller condition to free setting

Development of stochastic calculus with free Brownian motion

Abstract

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.