Consider the following one dimensional SDE.

\begin{align*}

dX_t&= \cos( X_t )^\alpha dW(t)\\

X_0&=0

\end{align*}

Consider the equation for \(\alpha >0\). On what interval do you expect to find the solution at all times ? Classify the behavior at the boundaries.

For what values of \(\alpha < 0\) does it seem reasonable to define the process ? any ? justify your answer.