= Equivalent terminal and maximal norms for L2-bounded continuous martingales
For a <L2-bounded continuous martingale>, the terminal and maximal <norms> satisfy
$$
\|M_\infty\|_2\leq\left\|\sup_{t\geq0}|M_t|\right\|_2\leq2\|M_\infty\|_2.
$$
The first inequality follows from <almost sure convergence>. Apply the <Doob L2 maximal inequality> on $[0,T]$ and then the <monotone convergence theorem> as $T\to\infty$ for the second. These are <equivalent norms> on the space modulo <indistinguishability of stochastic processes>; neither inequality needs a zero initial value.
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