Fractional Cox--Ingersoll--Ross process with small Hurst indices

Abstract

In this paper the fractional Cox-Ingersoll-Ross process on ${\mathbb{R}}_{+}$ for $H<1/2$ is defined as a square of a pointwise limit of the processes $Y_{\epsilon }$, satisfying the SDE of the form $d{Y}_{\epsilon }(t)=(\frac{k}{Y_{\epsilon }(t)1_{\{Y_{\epsilon }(t)>0\}}+\epsilon }-a{Y}_{\epsilon }(t))dt+\sigma d{B}^H(t)$, as $\epsilon \downarrow 0$. Properties of such limit process are considered. SDE for both the limit process and the fractional Cox-Ingersoll-Ross process are obtained.

Published at https://doi.org/10.15559/18-VMSTA126 in the Modern Stochastics: Theory and Applications (https://vmsta.org/) by VTeX (http://www.vtex.lt/)