Joint endpoint and maximum law for drifted Brownian motion
= Joint endpoint and maximum law for drifted Brownian motion
{title2=
$\mathbb P(W_t^
u\le x,M_t^
u<a)$
}
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 $\phi_t(y)-\phi_t(2a-y)$ at zero drift. Multiplication by $e^{\nu y-\nu^2t/2}$ and completion of squares gives the formula.