 Solution

The optional stopping theorem says that for a martingale and bounded stopping times $\sigma \le \tau$,For unbounded almost surely finite stopping times, additional hypotheses are essential. Sufficient versions include uniform integrability of the stopped martingale, or an integrable stopping time together with uniformly bounded increments. In the latter case, stop at $\tau \wedge n$ and use $|M_{\tau \wedge n}|\le |M_0|+C\tau$ and the dominated convergence theorem. This is bounded-increment optional stopping with integrable time.