Let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion starting at zero and denote by $S=(S_t)_{t\ge 0}$ its running maximum, i.e. $S_t=\sup_{0\le s\le t}B_s$. Given a fixed number $p>1$, define the stopping time
$$\tau~~:=~~\inf\Big\{t>0: S_t>\frac{p}{p-1}|B_t|\Big\}.$$
My question is whether we may show that $|B_{\tau}|^p$ is integrable. Thanks a lot for the reply!