Consider the following ODE and SDE:
\[\dot x_t = x^2_t \qquad x_0 >0\]
\[d X_t = X^2_t dt + \sigma |X_t|^\alpha dW_t\qquad X_0 >0\]
where \(\alpha >0\) and \(\sigma >0\).
- Show that \(x_t\) blows up in finite time.
- Find the values of \(\sigma\) and \(\alpha\) so that \(X_t\) does not explode (off to infinity).
[ From Klebaner, ex 6.12]