Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 / 4 / e / Solution

The one-step predictable integrand $\theta$ is bounded by one. If $M$ were a martingale, its transform $\xi =\theta \cdot (M_T-M_{T-1})$ would be an integrable mean-zero random variable. Parts b–d show instead that it is nonnegative almost surely and strictly positive with positive probability, so its expectation is positive. This contradiction proves that $M$ cannot be a martingale.