Positive Lamperti drift-implicit scheme for the CIR process

Assume AC and the Brownian conventions of Locally Lipschitz Itô SDE on an open state domain on $[0,T]$ with $U=(0,\mathrm{\infty })$. The input consists of $\kappa ,\theta ,\sigma >0$ satisfying the Feller condition $2\kappa \theta \ge {\sigma }^2$, a step $h>0$, a current transformed value $z_n\ge 0$, and a finite Brownian increment $\mathrm{\Delta }{W}_n$. Set $a=\frac{4\kappa \theta -{\sigma }^2}{8},A=1+\frac{\kappa h}{2},c=z_n+\frac{\sigma }{2}\mathrm{\Delta }{W}_n\mathrm{.}$ For valid parameters, $a>0$ and $A>0$. Compute $z_{n+1}=\frac{c+\sqrt{c^2+4Aah}}{2A},x_{n+1}=z_{n+1}^2\mathrm{.}$ This is the positive-root solution of $A{z}_{n+1}^2-c{z}_{n+1}-ah=0,$ which is the drift-implicit Euler equation for the Lamperti variable $Z=\sqrt{X}$ in the CIR model $d{X}_t=\kappa (\theta -X_t)dt+\sigma \sqrt{X_t}d{W}_t$.

Return invalid-input for parameters outside the stated domain or a non-finite input. Otherwise return success with $(z_{n+1},x_{n+1})$. In exact real arithmetic the discriminant is strictly positive and the displayed root is finite and positive for every $z_n\ge 0$ and $\mathrm{\Delta }{W}_n\in \mathbb{R}$; the algorithm has no rejected root branch. A floating-point implementation that cannot verify a finite positive root returns solve-failure and does not advance the state. This is a specific implicit square-root update, not the ordinary Euler–Maruyama update for $X$.