I attended a talk today on BMO martingales. It was my first encounter with the subject, and this may explain my inability to solve this myself.
We take a continuous local martingale $L$, and say that $L\in BMO$ if
$E[\langle L\rangle_{\infty}-\langle L\rangle_T|\mathcal{F}_T]\leq C$
for any stopping time $T$ and an absolute constant $C$. We define then
$G_t^L(\alpha)=\exp\left(\alpha L_t+\left(\frac12-\alpha\right)\langle L\rangle_t\right)$ $\qquad (0\leq t<\infty)$.
Now comes the problem. He wrote down that
$E[G_t^L(\alpha)|\mathcal{F}_T]\geq G_T^L(\alpha)$, for some $\alpha\neq 1$
called it trivial and moved on in his agenda. In other words, we need that
$E\left[ \exp\left( \alpha L_t+\left.\left( \frac12-\alpha\right)\langle L\rangle_t\right)\right| \mathcal{F}_T\right]\geq \exp\left(\alpha L_T+\left(\frac12-\alpha\right)\langle L\rangle_T\right)$.
What immediately comes to my mind with submartingality is to use the Jensen inequality. This gets me next to nowhere. Obviously, I need to apply some property of the BMO martingale to get there, but every result I find on the subject (John-Nirenberg inequality, reverse Hölder inequality, etc.) pulls me in the wrong direction, so to speak. I have also tried seeking help in texts on analysis (Muckenhoupt, Fefferman...) but to no avail. With the lecturer noting that the result is trivial, I suspect it's just me making this problem more complicated than it really is.
Now, corollary 6 in this article combined with theorem 6 of this article leads me to believe that his claim may well be true, but I can neither prove it myself nor find a reference on it. Is it common knowledge that $G$ would be a submartingale in this case, or is it just me overlooking something.