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

For $A\in {\mathcal{F}}_n$, define the bounded stopping time $T=n+1$ on $A$ and $T=n$ on $A^c$. Before time $n$ its stopping events are empty, at time $n$ the event $\{T\le n\}=A^c$ belongs to ${\mathcal{F}}_n$, and subsequently it has stopped everywhere. Subtract the assumed mean identity for this $T$ from that for the constant stopping time $n$ to obtainSince $M_n$ is integrable and ${\mathcal{F}}_n$-measurable, this is precisely the defining property of conditional expectation giving $\mathbb{E}[M_{n+1}\mid {\mathcal{F}}_n]=M_n$. Thus the adapted process is a martingale, completing the bounded-stopping-time characterization of a martingale.