Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 38 / 2 / a / iv / Solution

For a stopping time $T\ge s$, the optional sampling theorem for the L2-bounded continuous martingale $M$ gives $\mathbb{E}[M_T\mid {\mathcal{F}}_s]=M_s$. ConsequentlyIf $N_s$ is unbounded, first multiply by a bounded truncation and then remove it using $L^2$ bounds and the Cauchy-Schwarz inequality. Thus (ii) implies (iv). Taking $T=s$ proves the converse. For $T=\infty$, the same calculation uses the $L^2$ terminal value.