= Square root of a CIR diffusion
{c}
{title2=$dZ_t=\left(\frac{4ab-\sigma^2}{8Z_t}-\frac a2Z_t\right)dt+\frac\sigma2dW_t$}
For $dC_t=a(b-C_t)dt+\sigma\sqrt{C_t}\,dW_t$, 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.
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