Let $W$ be a Brownian motion started at $a>0$ and let $T_{0}$ be the first time $W$ hits $0$ . The goal is to find the law of $\sup _{t \leq T_{0}} W_{t}$.
My initial attempt is as follows: Reflection principle helps establish the law of $\sup_{s\leq t} W_s$ for fixed $t$. Then we can condition on the $T_0$ and use law of total probability. Is it a valid argument?
I appreciate either hints or solution to this problem. Thank you!
If $y>a$, $$ \mathsf{P}\!\left(\sup_{0\le s\le T_0}W_s<y\right)=\mathsf{P}(T_0<T_y)=1-\frac{a}{y}. $$ This probability can be deduced from the fact that $$ a=\mathsf{E}B_{T_0\wedge T_y}=y\mathsf{P}(T_0\ge T_y). $$