# Exit and hitting times for the Bessel process $\textrm{d}X_t=\frac{n-1}2\frac{\textrm{d}t}{X_t}+\textrm{d}B_t$

I am trying to analyse the exit time $$T_1:=\inf\{t:X_t\notin[\alpha,2]\}$$ and hitting time $$T_2:=\inf\{t:X_t=0\}$$, where $$\alpha<1$$ is a constant, and $$X_t$$ follows the Bessel process defined by the SDE $$\textrm{d}X_t=\frac{n-1}2\frac{\textrm{d}t}{X_t}+\textrm{d}B_t,$$ with $$n\neq0$$, $$X_0=1$$ a.s. and $$X_t$$ is defined until it hits zero or infinity. I want to find $$\mathbb{P}(X_{T_1}=\alpha)$$ and $$\mathbb{P}(X_{T_2}=0)$$.

Here’s what I've done so far:

a) I am told to show that $$\frac1{X_t^{n-2}}$$ is a local martingale. By Ito's formula, I have $$\textrm{d}\left(\frac1{X_t^{n-2}}\right)=-\frac{n-2}{X_t^{n-1}}\textrm{d}X_t+\frac{(n-1)(n-2)}{2X_t^n}\textrm{d}X_t^2.$$ Since $$\textrm{d}X_t^2=\textrm{d}B_t^2=\textrm{d}t$$, therefore $$\textrm{d}\left(\frac1{X_t^{n-2}}\right)=-\frac{n-2}{X_t^{n-1}}\left(\frac{n-1}2\frac{\textrm{d}t}{X_t}+\textrm{d}B_t\right)+\frac{(n-1)(n-2)}{2X_t^n}\textrm{d}t=-\frac{n-2}{X_t^{n-1}}\textrm{d}B_t.$$ Since the drift term is zero, therefore $$\frac1{X_t^{n-2}}$$ is a local martingale.

b) By the optional stopping theorem, $$\mathbb{E}(X_{T_1})=\mathbb{E}(X_0)=1$$ a.s. Since $$\mathbb{P}(X_{T_1}=\alpha)+\mathbb{P}(X_{T_1}=2)=1$$ and $$\alpha\mathbb{P}(X_{T_1}=\alpha)+2\mathbb{P}(X_{T_1}=2)=1$$, of which I can rearrange to get $$\mathbb{P}(X_T=\alpha)=\frac1{2-\alpha}$$.

My issues now:

a) For the optional stopping theorem, I have implicitly assumed $$X_t$$ is a martingale and that $$T_1$$ is a stopping time with bounded expectation. For the latter I have seen many of those proofs on stackexchange, so my concern lies with the former.  I know that bounded local martingales are martingales, but $$\frac1{X_t^{n-2}}$$ is not bounded and in integrating $$\textrm{d}\left(\frac1{X_t^{n-2}}\right)$$ I get $$\frac1{X_t^{n-2}}=1-(n-2)\int_0^t\frac{\textrm{d}B_s}{X_s^{n-1}}$$, which is not very informative (to me).

b) What are the usual steps in finding $$\mathbb{P}(X_{T_2}=0)$$? Is it a Borel-Cantelli argument? I have seen some for BMs, but I am unclear if they extend to $$X_t$$.

(a) Since we stopped this local martingale by $$T_1$$, it is a bounded martingale.
(b) As you define $$T_2$$ as the hitting time of $$0$$. The real question to ask here is whether $$\mathbb{P}(T_2 < \infty)$$. This is related with recurrence and transience of the process. And in this could be done by, similarly in this question, replacing $$T_1 :=\inf\{t:X_t\notin[\alpha,R]\}$$ and let $$\alpha \to 0$$ and $$R \to \infty$$.
And here is a probably easier way for this problem: note that the Bessel process is the Euclidean norm from $$n$$ dimensional Brownian motion $$B_t$$. The stopping time $$T_1$$ is the same as the first time that $$B_t$$ hits sphere with radius $$\alpha$$ or $$2$$. And the corresponding probability could be computed by optional stopping.