Doob L2 maximal inequality
= Doob L2 maximal inequality
{c}
{title2=$\mathbb E\sup_{s\leq t}|M_s|^2\leq4\mathbb E|M_t|^2$}
For a square-integrable martingale starting at zero,
$$
\mathbb E\sup_{s\leq t}|M_s|^2\leq4\mathbb E|M_t|^2.
$$
For a continuous local martingale stopped so that its quadratic variation is integrable, the <Itô isometry> makes the right-hand side $4\mathbb E[M]_t$.
This is the $p=2$ specialization of <Doob's martingale inequality>.