Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 30K / c / i / Solution

Assume (i), and let $\tau$ be a stopping time. The increments of the stopped process satisfyBecause $\{\tau >n\}\in {\mathcal{F}}_n$, taking conditional expectation given ${\mathcal{F}}_n$ makes the right-hand side zero. Integrability follows froma finite sum of integrable random variables. Hence $M^{\tau }$ is a martingale, proving (ii).