Finite-horizon maximum of Brownian motion with drift
= Finite-horizon maximum of Brownian motion with drift
{title2=$M_t=\sup_{s\leq t}(W_s+\nu s)$}
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 $\phi_t(y)-\phi_t(2a-y)$ for $y<a$. Weight it by $e^{\nu y-\nu^2t/2}$ using the <Girsanov theorem> and complete squares in the two Gaussian integrals.