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

TL;DR

This paper introduces a fractional Cox-Ingersoll-Ross process driven by fractional Brownian motion, proving its SDE form, positivity conditions, and asymptotic behavior as the mean parameter grows.

Contribution

It defines a new fractional Cox-Ingersoll-Ross process, derives its stochastic differential equation, and analyzes its positivity and zero-hitting properties for different Hurst parameters.

Findings

The process satisfies a specific SDE with Stratonovich integral.

For H>1/2 and k>0, the process remains positive and does not hit zero.

As k approaches infinity, the probability of not hitting zero tends to 1 for H<1/2.

Abstract

In this paper we define the fractional Cox-Ingersoll-Ross process as $X_t:=Y_t^2\mathbf{1}_{\{t<\inf\{s>0:Y_s=0\}\}}$, where the process $Y=\{Y_t,t\ge0\}$ satisfies the SDE of the form $dY_t=\frac{1}{2}(\frac{k}{Y_t}-aY_t)dt+\frac{\sigma}{2}dB_t^H$, $\{B^H_t,t\ge0\}$ 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 $dX_t=(k-aX_t)dt+\sigma\sqrt{X_t}\circ dB_t^H$, 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 $k\rightarrow\infty$.