Finite-horizon maximum of Brownian motion with drift ($M_t={\sup }_{s\le t}(W_s+\nu s)$)

For $a>0$ and $t>0$, the distribution function is $\Phi ((a-\nu t)/\sqrt{t})-e^{2a\nu }\Phi ((-a-\nu t)/\sqrt{t})$. The Brownian reflection principle gives the zero-drift terminal density below the barrier as ${\varphi }_t(y)-{\varphi }_t(2a-y)$ for $y<a$. Weight it by $e^{\nu y-{\nu }^{2}t/2}$ using the Girsanov theorem and complete squares in the two Gaussian integrals.

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.