Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 2 / c / ii / Solution

Let $L=N-H\cdot M$. It is a continuous square-integrable martingale, and the assumed bracket identity givesfor every continuous square-integrable martingale $K$. Choose $K=L$ and use part (i):Thus $L_{\infty }=0$ almost surely, and the conditional expectation property gives $L_t=\mathbb{E}[L_{\infty }\mid {\mathcal{F}}_t]=0$ for every $t$. Hence $N=H\cdot M$ up to indistinguishability of stochastic processes.