Generalisation of Fractional-Cox-Ingersoll-Ross Process

TL;DR

This paper introduces a generalized fractional Cox-Ingersoll-Ross process driven by fractional Brownian motion, establishing existence, positivity, and zero-hitting properties depending on the Hurst parameter, with supporting simulations.

Contribution

It defines a new fractional Cox-Ingersoll-Ross process, proves its existence, uniqueness, and positivity properties, and analyzes zero-hitting probabilities based on the Hurst parameter.

Findings

Unique solution exists and is positive up to first zero visit.

Process is strictly positive for H > 1/2.

Zero-hitting probability tends to zero for H < 1/2.

Abstract

In this paper, we define a generalised fractional Cox-Ingersoll-Ross process as a square of singular stochastic differential equation with respect to fractional Brownian motion with Hurst parameter H in (0,1) and continuous drift function. Firstly, we show that this differential equation has a unique solution which is continuous and positive up to the time of the first visit to zero. In addition, we prove that it is strictly positive everywhere almost surely for H > 1/2. In the case where H < 1/2, we consider a sequence of increasing functions and we prove that the probability of hitting zero tends to zero as n goes to infinity. These results are illustrated with some simulations using the generalisation of the extended Cox-Ingersoll-Ross process.