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

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 $d Y_{\varepsilon}(t)=( \frac{k}{ Y_{\varepsilon}(t)\mathbb{1}_{\{ Y_{\varepsilon}(t)>0\}}+\varepsilon}-a Y_{\varepsilon}(t))dt+σdB^H(t)$, 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.

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