Here is the question:
$(B_t,t\ge 0)$ is a standard brwonian motion, starting at $0$. $S_t=\sup_{0\le s\le t} B_s$. $T=\inf\{t\ge 0: B_t=S_1\}$. Show that $T$ follows the arcsinus law with density $g(t)=\frac{1}{\pi\sqrt{t(1-t)}}1_{]0,1[}(t)$.
I used Markov property to get the following equality:
$P(T
where $\hat{S}_{1-t}$ is defined for the brownian motion $\hat{B}_s=B_{t+s}-B_t$, which is independant of $F_t$.
However the reflexion principle tells us that $S_t-B_t$ has the same law as $S_t$, so we can also write that $P(T
To this point, we can calculate $P(T
Do you know how to get the arcsinus law? Thank you.