Let \( B^H\) be a fractional Brownian motion with Hurst parameter \(H\in (0,1)\). The fractional Cox-Ingersoll-Ross (CIR) process is defined as the pathwise solution of the stochastic differential equation \[ dX(t)= (k-aX(t)) dt + \sigma \sqrt{ X(t)} \circ dB^H(t), \qquad X(0)>0,\tag{1} \] where \(a,k,\sigma>0\) and the integral is defined as a pathwise Stratonovich integral. It is known that for \(H>1/2\), such process is strictly positive and never hits zero, whereas for \(H<1/2\) there is a positive probability that the process hits zero on a fixed time interval \([0,T]\) and this probability converges to one as \(k\) tends to \(\infty\).
This paper deals with the construction of the fractional CIR process on the whole interval \([0,T]\), when \(H<1/2\). To do this, the authors consider the solution to the stochastic differential equation \[ dY_{\varepsilon} (t)= \frac 12 \left( \frac k { Y_{\varepsilon} (s) \mathbf{1}_{\{Y_{\varepsilon} (s) >0\}} +\varepsilon} -a Y_{\varepsilon} (t) \right) dt +\frac {\sigma}2 dB^H(t),\qquad Y_{\varepsilon}(0)=Y_0>0, \] where \(a,k,\sigma, Y_0, \varepsilon>0\). They prove that the pointwise limit \(Y(t)=\lim_{\varepsilon\rightarrow 0} Y_{\varepsilon}(t)\) exists, and almost surely, it is nonnegative and positive a.e. with respect to the Lebesgue measure. Moreover, \(Y\) is continuous and satisfies the equation \[ Y(t) = Y(\alpha) + \frac 12 \int_{\alpha}^t \left( \frac k { Y(s)} -a Y (s) \right) ds +\frac {\sigma}2 (B^H(t)- B^H(\alpha)), \] for all \(t\in [\alpha, \beta]\) where \((\alpha, \beta)\) is any interval where \(Y\) is positive. Then, it is proved that the process \(X(t)= Y^2(t)\) satisfies the the stochastic differential equation (1) on any interval \((\alpha, \beta)\) where \(Y\) is positive, with initial condition \(X(\alpha)=0\).