Law of Sup of Brownian Motion
We calculate $P(S_t\geq s)$ by introducing a stopping time $\tau=\inf\{t>0\mid B_t>s\}$.
Obviously $P(S_t\geq s)=P(\tau\leq t)$
$P(\tau\leq t)=P(\tau\leq t,B_t<s)+P(\tau\leq t,B_t\geq s)$
$P(\tau\leq t,B_t\geq s)=P(B_t\geq s)$ as if $B_t\geq s$, therefore ,$\tau\leq t$.
We introduce a process $W$ that is equal to $B_t$ if $\tau>t$, and equal to $2s-B_t$ if $\tau\leq t$. W is also a Brownian motion, therfore $P(\tau\leq t,B_t<s)=P(\tau\leq t,W_t<s)=P(\tau\leq t,2s-B_t<s)$ Finally, $P(\tau\leq t)=2P(B_t>s)$