= Upper maximal moment bound for a continuous local martingale
{title2=$\mathbb E\sup_{s\leq t}|X_s|^p\leq C_p\mathbb E\langle X\rangle_t^{p/2}$}
For $p\geq2$ and a continuous local martingale starting at zero, the <Itô formula> for $|X|^p$, the <Doob Lp maximal inequality> and <Hölder's inequality> yield $\mathbb E\sup_{s\leq t}|X_s|^p\leq C_p\mathbb E\langle X\rangle_t^{p/2}$. One possible constant is $C_p=[p(p-1)(p/(p-1))^p/2]^{p/2}$. Stop at increasing absolute-value levels to remove an initial boundedness assumption. This proves the upper half of the <Burkholder-Davis-Gundy inequalities> in this range.
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