If \(r\) is a CIR process and \(Z=r^{\alpha}\), prove that

\[d Z_{t}=\left(\alpha Z_{t}^{1-1 / \alpha}\left(k \theta+(\alpha-1) \sigma^{2} / 2\right)-Z_{t} \alpha k\right) d t+\alpha Z_{t}^{1-1 /(2 \alpha)} \sigma d W_{t}\]

In particular, for \(\alpha=-1, d Z_{t}=Z_{t}\left(k-Z_{t}\left(k \theta-\sigma^{2}\right)\right) d t-Z_{t}^{3 / 2} \sigma d W_{t}\) is the so-called \(3 / 2\) model