Last passage time above a level for Brownian motion with negative drift
= Last passage time above a level for Brownian motion with negative drift
{title2=$T=\sup\{t\geq0:W_t-at>b\}$}
For $a,b>0$ and the convention $\sup\varnothing=0$,
$$
\mathbb P(T\leq t)
=\Phi\!\left(a\sqrt t+\frac b{\sqrt t}\right)
-e^{-2ab}\Phi\!\left(\frac b{\sqrt t}-a\sqrt t\right),
\qquad t>0.
$$