optional stopping

If $x_t$ is a martingale and $\tau$ is a stopping time, then any of the following conditions implies that $\mathbb{E}[x_{\tau}] = \mathbb{E}[x_0]$:

1. The stopping time is bounded (almost surely) by some constant $\tau$.
2. The stopping time has finite expectation, and the expected martingale increments are bounded (there is some $c$ such that $\mathbb{E}\left[\left|x_{t-1} - x_t\right| |x_0, \ldots, x_t \right] \le c$ almost surely for all $t$).
3. The values up to the stopping point are bounded (there is some $c$ such that $|x_{\min(t, \tau)}| \le c$ almost surely for all $t$).

This establishes that there is no successful betting strategy (one that makes money in expectation) for a gambler with a finite lifetime (the first condition) or a house limit on bets (the second condition).