# are sub-martingales correlated with the indicator function of stopping times?

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.

## 1 Answer

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 $$\mathbb{E}[X_t 1_{\tau > t}] \le \varepsilon \mathbb{P}(\tau > t) < \mathbb{E}[X_t] \mathbb{P}(\tau > t)$$