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 $Y{\varepsilon}$, satisfying the SDE of the form $d Y_{\varepsilon}(t)=( \frac{k}{ Y_{\varepsilon}(t)\mathbb{1}{{ Y{\varepsilon}(t)>0}}+\varepsilon}-a Y_{\varepsilon}(t))dt+\sigma dB^H(t)$, as $\varepsilon\downarrow0$. Properties of such limit process are considered. SDE for both the limit process and the fractional Cox-Ingersoll-Ross process are obtained.