Cox–Ingersoll–Ross model ($d{r}_t=\sigma \sqrt{r_t}d{B}_t+(a-b{r}_t)dt$)

A nonnegative mean-reverting short-rate diffusion with positive parameters. Its mean-reversion level is $a/b$ and its diffusion size is proportional to $\sqrt{r}$. The Feller positivity condition for the CIR model controls whether zero is reachable. Its stationary law is a gamma distribution and its zero-coupon bond prices are exponential-affine.

For $d{C}_t=a(b-C_t)dt+\sigma \sqrt{C_t}d{W}_t$, the Itô formula applied to $Z=\sqrt{C}$ gives the displayed dynamics before hitting zero. In particular $2ab={\sigma }^2$ does not eliminate the reciprocal drift. The cancellation condition is $4ab={\sigma }^2$. Under that condition $C$ can be represented as the square of a signed Ornstein-Uhlenbeck process with drift $-aY/2$ and volatility $\sigma /2$; its principal square root is the reflected process $|Y|$, not an unrestricted signed OU process. The mean of $C$ is $b+(C_0-b)e^{-at}$ for all admissible positive parameters.

For the CIR model under its money-market risk-neutral measure, the Feynman-Kac formula gives $B^{\prime }=1-bB-{\sigma }^2{B}^2/2$ and $A^{\prime }/A=-aB$, with $B(0)=0$, $A(0)=1$. Solving this Riccati equation gives explicit exponential-affine zero-coupon bond prices. The positive root parameter is $\sqrt{b^2+2{\sigma }^2}$; the squared diffusion coefficient, not the volatility itself, enters this expression.

For a CIR model started at a positive rate, this condition makes the zero boundary inaccessible. Below the threshold, zero can be reached, but the usual nonnegative solution does not cross into negative rates. Thus nonnegativity and strict positivity are different properties.