Let $X_t$ be a standard Brownian motion starting at 0 and let
$T=min \{t:|X_t|=1\}$ and $\hat{T}=min \{t:X_t=1\}$
(a) Show that there exist positive constants $c$, $\beta$ such that for all $t>0$,
$$P(T>t)\leq ce^{-\beta t}$$
conclude that $E(T)\leq \infty$
use the reflection principle to fine the density of $\hat{T}$, and show that $E(\hat{T})=\infty$
Please help me to start to solve this problem.
For the second part, let
$ X_t^* = \max_{s\in [0\ t]}X_s$
note that:
$P(\hat T \le t) = P(X_t^*\ge 1)$
and also, by reflection principle, you can show that the pdf of $X_t^*$ is twice the pdf of $X_t$, but defined on $[0\ \infty)$ only.
Put these together, you get the cdf of $\hat T$
$P(\hat T \le t) = 2(1- \phi(1/{\sqrt{t}}))$
where $\phi$ is the cdf of standard normal r.v., and by integration by parts, you can prove that $E(\hat T)=\infty$.