I’m asked to calculate
$$ P\left(\sup_{t\geq 0} W_t-t-1\geq 0\right)$$
Where $W_t$ is the Brownian motion.
I tried to calculate it as the way I calculated the distribution of $\sup_{0\leq s\leq t} W_s$ by considering the stopping time $\tau_b=\inf\{t>0,W_t-t=b\}$ for $b>0$.
But, here I can’t say $\tau_b$ is a stopping time with finite expection, which is important in calculating the distribution of $\sup_{0\leq s\leq t} W_s$. So, I don’t know how to do it now.
