Let $X_t$ be any nice enough continuous sub-martingale and let $\tau$ be a stopping time which is, let's say bounded.
My question is, are $X_t$ and $1_{\{\tau >t\}}$ positively correlated? That is, is $\mathbb{E}[X_t1_{\{\tau >t\}}] \geq \mathbb{E}[X_t]\mathbb{P}(\tau>t)$?
This seems intuitive enough, since the expectation of $X_t$ increases with time, and it will stop increasing whenever $t$ is bigger than $\tau$. However, I cannot see how to prove it and I'm afraid I'm missing something.