 Brownian running maximum

For standard Brownian motion started at zero, let $M_t={\sup }_{0\le s\le t}{B}_s$. For $t>0$ and $x\ge 0$, the Brownian reflection principle givesThus $M_t$ has the same probability distribution as $|B_t|$, with cumulative distribution function $2\Phi (x/\sqrt{t})-1$ for $x\ge 0$. For $x<0$ the tail probability is one because $M_t\ge 0$; the doubled Gaussian-tail formula is not valid there. Reflecting paths after their first hit of $x>0$ pairs the paths ending below $x$ with those ending above it.