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{c}
{title2=$L^p$}

For $p>1$ and a martingale $(M_k)_{k\leq n}$,
$$
\left\|\max_{k\leq n}|M_k|\right\|_p
\leq\frac p{p-1}\|M_n\|_p.
$$

This strong $L^p$ bound is one form of <Doob's martingale inequality>.

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Upper maximal moment bound for a continuous local martingale

Doob Lp maximal inequality ( $L^{p}$ )

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Doob Lp maximal inequality (Lp)

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Doob Lp maximal inequality ( $L^{p}$ )

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