Generalisation of fractional Cox-Ingersoll-Ross process

In this paper, we define a generalised fractional Cox-Ingersoll-Ross process (Xt)t >= 0 as a square of singular stochastic differential equation with respect to fractional Brownian motion with Hurst parameter H is an element of (0, 1) taking the form dZt = (f(t, Zt)Z-1 ) t dt + adWtH /2, where f(t, z) is a continuous function on R2+. Firstly, we show that this differential equation has a unique solution (Zt)t >= 0 which is continuous and positive up to the time of the first visit to zero. In addition, we prove that the stochastic process (Xt)t >= 0 satisfies the differential equation dXt = f(t,root Xt)dt + a root Xt degrees dWtH where degrees refers to the Stratonovich integral. Moreover, we prove that the process (Xt) is strictly positive everywhere almost surely for H > 1/2. In the case where H < 1/2, we consider a sequence of increasing functions (fn) and we prove that the probability of hitting zero tends to zero as n -> infinity. These results are illustrated with some simulations using the generalisation of the extended Cox-Ingersoll-Ross process.(c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).