Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 34 / 1 / c / Solution

The stopping-interval indicator is an elementary predictable process with stopping-time intervals. Its stochastic integral isIt 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.