Hölder factorization of stochastic exponentials ($C(p,q)=(pq-\sqrt{pq})/(q-1)$)

For a zero-starting continuous local martingale $X$, the stochastic exponential satisfiesThe Holder inequality and the supermartingale property of the first exponential bound its stopped $p$th moment by $(\mathbb{E}{e}^{C(p,q)X_T}{)}^{(q-1)/q}$. The subtraction in the coefficient is $\sqrt{pq}-1$, not a square root of $pq-1$.

For fixed $q>1$, the coefficient in the Hölder factorization of stochastic exponentials exceeds $\sqrt{q}/(\sqrt{q}+1)>1/2$. Taking $q=1+\epsilon$ and $p=1+{\epsilon }^2$ approaches one-half. Thus a uniform stopped exponential moment at coefficient one-half bounds some $p$th moment of each strict scaling $\mathcal{E}(aM)$, $0<a<1$.