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

If $M,N$ are weakly orthogonal continuous martingales, taking the two times equal gives $\mathbb{E}[M_s{N}_s]=0$. Conversely suppose this equal-time condition holds. For $s\le t$, the martingale property and conditional expectation giveFor $t\le s$, condition the other factor at time $t$ instead. Products are integrable by the Cauchy-Schwarz inequality. Thus the equal-time condition implies the full cross-time definition, proving (i) equivalent to (ii).