= Joint distribution of Brownian motion and its running maximum
Let $M_t=\sup_{0\leq s\leq t}B_s$ for a standard <Brownian motion>. For $m\geq0$, the <Brownian reflection principle> gives
$$
\mathbb P(B_t\leq b,M_t\leq m)=
\begin{cases}
\Phi(b/\sqrt t)+\Phi((2m-b)/\sqrt t)-1,&b\leq m,\\
2\Phi(m/\sqrt t)-1,&b>m.
\end{cases}
$$
On $b<m$, the pair $(B_t,M_t)$ therefore has <joint probability density>
$$
f_{B_t,M_t}(b,m)=
\frac{2(2m-b)}{\sqrt{2\pi}\,t^{3/2}}
e^{-(2m-b)^2/(2t)}.
$$
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