I'm studying the exit of $X_t$, characterized by 1D SDEs, from the interval using this concrete example: $\begin{cases}dX_t=\mu dt+\sin(X_t)^{\alpha}dW_t\\ X_0=\pi/2\end{cases}$, where $\mu\in\mathbb{R}$ constant and $\alpha>0$. I'm interested in studying these questions:
On which interval does the solution exist all the time?
Classify the behavior at the boundaries in terms of the parameters.
For what values of $\alpha<0$ does it seem reasonable to define the process?
For the first question, I answered that by using the theorem on the existence and uniqueness of solutions of SDEs: For an end time $T>0$, if $\sigma(\cdot)$ and $\mu(\cdot)$ are globally Lipschitz, ($\exists K>0$ so that for any $t\in[0,T]$ and any $x$ and $y$, $|\mu(t,x)-\mu(t,y)|+|\sigma(t,x)-\sigma(t,y)|\leq K|x-y|$), the SDE with initial condition $X_0=x$ has a strongly uniquely solution. Here I argue that $\sigma(t,x)=\sin(x)^\alpha$ is Lipschitz for $\alpha>1$ and not Lipschitz for $0<\alpha\leq1$.
For the second and the third question, I have trouble answering them. For the second one, I plan to use this result from Feller's theory to argue: Let $[\alpha,\beta]$ be the interval. of interest, and fix $\zeta\in(\alpha,\beta)$, and define the integrals
$$I_\alpha=\int_{\alpha}^{\zeta}[S(z)-S(\alpha)]m(z)dz,\quad J_\alpha=\int_{\alpha}^{\zeta}[M(z)-M(\alpha)]S'(z)dz.$$
where $m(z)$ satisfies $m(x)dx=\frac{1}{(S'\sigma^2)(x)}dx$, $S(x)=\int_{\alpha}^x\exp\left(-\int_{\beta}^y \frac{2\mu(z)}{2\sigma^2(z)}dz\right)dy$, and $M(x)=\int_{-\infty}^x m(y)dy$. This is theorem that I'll reference:
I'd like to classify the behavior at $x=0$. But are there any boundary points that I need to consider?
I appreciate if anyone can give their approaches to 2 and 3.