A mixed fractional CIR model: positivity and an implicit Euler scheme

We consider a Cox--Ingersoll--Ross (CIR) type short rate model driven by a mixed fractional Brownian motion. Let $M=B+B^H$ be a one-dimensional mixed fractional Brownian motion with Hurst index $H>1/2$, and let $\mathbf{M}=(M,\mathbb{M}^{\mathrm{It\hat{o}}})$ denote its canonical It\^o rough path lift. We study the rough differential equation \begin{equation}\label{eqn1} \dd r_t = k(\theta-r_t)\,\dd t + \sigma\sqrt{r_t}\,\dd\mathbf{M}_t,\qquad r_0>0, \end{equation} and prove that, under the Feller condition $2k\theta>\sigma^2$, the unique rough path solution is almost surely strictly positive for all times. The proof relies on an It\^o type formula for rough paths, together with refined pathwise estimates for the mixed fractional Brownian motion, including L\'evy's modulus of continuity for the Brownian part and a law of the iterated logarithm for the fractional component. As a consequence, the positivity property of the classical CIR model extends to this non-Markovian rough path setting. We also establish the convergence of an implicit Euler scheme for the associated singular equation obtained by a square-root transformation.