Let $W=(W_t:t\ge0)$ denote a standard Brownian motion, and let $W^*_t:=\sup_{0\le s\le t}W_s$ be its running maximum. For constant $\lambda>0$ define the stopping time $$ \tau:=\inf\{t\ge0:W_t\le W^*_t-\lambda\}. $$ Identify the distribution of $W_\tau^*$.
It is suggested that the solution can be found by appealing to the strong Markov property of $W$, but I'm stumped as to how to do this. By other means I found that $W_\tau^*\sim\text{Exp}(\lambda^{-1})$, which seems at least intuitively plausible, but it would be useful to verify this answer using the method suggested. Any suggestions would be greatly appreciated.