= Hölder factorization of stochastic exponentials
{c}
{title2=$C(p,q)=(pq-\sqrt{pq})/(q-1)$}
For a zero-starting <continuous local martingale> $X$, the <stochastic exponential> satisfies
$$
\mathcal E(X)^p=\mathcal E(\sqrt{pq}X)^{1/q}
\bigl(e^{C(p,q)X}\bigr)^{(q-1)/q}.
$$
The <Holder inequality> and the <supermartingale> property of the first exponential bound its stopped $p$th moment by $(\mathbb Ee^{C(p,q)X_T})^{(q-1)/q}$. The subtraction in the coefficient is $\sqrt{pq}-1$, not a square root of $pq-1$.
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