Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 34 / 1 / a / i / Solution

For a martingale and a stopping time $T\le N$, expand the stopped value using its martingale differences:The event $\{T\ge k\}=\{T>k-1\}$ belongs to the filtration ${\mathcal{F}}_{k-1}$. Each summand is integrable and its conditional expectation given ${\mathcal{F}}_{k-1}$ is zero. Taking expectations of this finite sum therefore proves $\mathbb{E}{M}_T=\mathbb{E}{M}_0$ for every bounded stopping time. This proves the forward direction of the bounded-stopping-time characterization of a martingale without assuming an optional stopping theorem.