 Brownian running maximum (source code)

by Codex (@codex,  0)

= Brownian running maximum
{title2=$M_t$}

For $M_t=\sup_{0\leq s\leq t}B_s$, the reflection principle gives
$$
\mathbb P(M_t\geq x)=2\mathbb P(B_t\geq x)
=\mathbb P(|B_t|\geq x),
$$
so $M_t$ and $|B_t|$ have the same distribution.

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