Solution
= Solution
The stopping-interval indicator is an <elementary predictable process with stopping-time intervals>. Its <stochastic integral> is
$$
\boxed{(\mathbf1_{(S,T]}\cdot M)_t=M_{t\wedge T}-M_{t\wedge S}.}
$$
It is zero before $S$, equals $M_t-M_S$ between $S$ and $T$, and equals $M_T-M_S$ after $T$. For a <martingale> with jumps, the jump at $S$ is excluded and the jump at $T$ is included, exactly as required by the half-open interval.