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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.

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I don't think this is necessarily true. Let $B_t$ be a standard Brownian motion, $X_t := B_t + t$, $\tau := \inf\{t : X_t > \varepsilon\}\wedge T$ for some $T > \varepsilon > 0$. Then for $\varepsilon < t < T$ we have $\mathbb{E}[X_t 1_{\tau > t}] \le \varepsilon \mathbb{P}(\tau > t)$, but $\mathbb{E}[X_t] = t > \varepsilon$ so $\mathbb{E}[X_t 1_{\tau > t}] \le \varepsilon \mathbb{P}(\tau > t) < \mathbb{E}[X_t] \mathbb{P}(\tau > t)$

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