Fractional Cox--Ingersoll--Ross process with non-zero <<mean>>

Abstract

In this paper we define the fractional Cox-Ingersoll-Ross process as $X_t:=Y_t^2{1}_{\{t<\inf \{s>0:Y_s=0\}\}}$, where the process $Y=\{Y_t,t\ge 0\}$ satisfies the SDE of the form $d{Y}_t=\frac{1}{2}(\frac{k}{Y_t}-a{Y}_t)dt+\frac{\sigma }{2}d{B}_t^H$, $\{B_t^H,t\ge 0\}$ is a fractional Brownian motion with an arbitrary Hurst parameter $H\in (0,1)$. We prove that $X_t$ satisfies the stochastic differential equation of the form $d{X}_t=(k-a{X}_t)dt+\sigma \sqrt{X_t}\circ d{B}_t^H$, where the integral with respect to fractional Brownian motion is considered as the pathwise Stratonovich integral. We also show that for $k>0$, $H>1/2$ the process is strictly positive and never hits zero, so that actually $X_t=Y_t^2$. Finally, we prove that in the case of $H<1/2$ the probability of not hitting zero on any fixed finite interval by the fractional Cox-Ingersoll-Ross process tends to 1 as $k\to \infty$.

Published at https://doi.org/10.15559/18-VMSTA97 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)