= Terminal scaling inequality for stochastic exponentials
{title2=$\mathbb E\mathcal E(rX)_\infty\geq(\mathbb E\mathcal E(X)_\infty)^{r^2}(\mathbb Ee^{rX_\infty/2})^{-2(r-1)}$}
For a convergent <continuous local martingale> and $r>1$, factor $\mathcal E(X)$ into $\mathcal E(rX)^{1/r^2}$ and an ordinary exponential. Apply the <Holder inequality> and then the concave <Jensen inequality> to the fractional power $2/(r+1)$. The displayed lower bound is informative when the terminal exponential moment is finite.
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