Joint endpoint and maximum law for drifted Brownian motion ($\mathbb{P}(W_t^u\le x,M_t^u<a)$)

For $W_t^{\nu }=B_t+\nu t$, $M_t^{\nu }={\sup }_{s\le t}{W}_s^{\nu }$, $a>0$ and $x\le a$, the joint distribution function is $\Phi ((x-\nu t)/\sqrt{t})-e^{2a\nu }\Phi ((x-2a-\nu t)/\sqrt{t})$. The Brownian reflection principle gives killed endpoint density ${\varphi }_t(y)-{\varphi }_t(2a-y)$ at zero drift. Multiplication by $e^{\nu y-{\nu }^{2}t/2}$ and completion of squares gives the formula.