Lévy identity for Brownian motion
= Lévy identity for Brownian motion
{c}
For a standard <Brownian motion> $B$, its <Brownian running maximum> $M$, and its <local time of a semimartingale> $L^0$ at zero, the process pairs satisfy
$$
(M_t-B_t,M_t)_{t\geq0}
\overset d=
(|B_t|,L_t^0)_{t\geq0}.
$$
In particular, $M-B$ and $|B|$ are both <Reflected Brownian motion> and have the same law as processes.