A martingale stopped at a stopping time

Question

By defenition a martingale is a sequence of random variables. And there is a statement which says that a martingale stopped at a stopping time is a martingale. But the last is a one random variable, not a sequence. Where am I wrong?

Answer 1

I've already answered this in comments. But not just for martingales, the stopped version of any stochastic process $X_t$ is given by $X_{t\wedge T} = X_{\min\{t, T\}}$ for a stopping time $T$. Usually to make sense, the process $X_t$ is assumed to be adapted to some filtration $\mathcal{F}_t$ and the stopping time $T$ is defined wrt this filtration.