= Solution
For a <stopping time> $T\ge s$, the <optional sampling theorem> for the <L2-bounded continuous martingale> $M$ gives $\mathbb E[M_T\mid\mathcal F_s]=M_s$. Consequently
$$
\mathbb E[M_TN_s]=\mathbb E[M_sN_s].
$$
If $N_s$ 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.
Back to article page
