= Brownian running maximum
{title2=$M_t$}
For standard <Brownian motion> started at zero, let $M_t=\sup_{0\leq s\leq t}B_s$. For $t>0$ and $x\geq0$, the <Brownian reflection principle> gives
$$
\mathbb P(M_t\geq x)=2\mathbb P(B_t\geq x)=\mathbb P(|B_t|\geq x).
$$
Thus $M_t$ has the same <probability distribution> as $|B_t|$, with <cumulative distribution function> $2\Phi(x/\sqrt t)-1$ for $x\geq0$. For $x<0$ the tail probability is one because $M_t\geq0$; 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.
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