Let \(d X_{t}=\theta d t+\sigma \sqrt{X_{t}} d W_{t}, X_{0}>0\), where \(\theta>0\) and, for \(a

\[\psi_{a, b}(x)=\frac{x^{1-u}-a^{1-u}}{b^{1-u}-a^{1-u}}\]

where \(u=2 \theta / \sigma^{2}\). Prove also that if \(u>1\), then \(T_{0}\) is infinite and that if \(u<1, \psi_{0, b}(x)=(x / b)^{1-u}\). Thus, the process \(\left(1 / X_{t}, t \geq 0\right)\) explodes in the case \(u<1\).