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

The martingale product identity says that $N_t{K}_t-[N,K{]}_t$ is a martingale. Passing to the terminal values of the square-integrable martingales and using $N_0=K_0=0$ givesThe quadratic covariation identity for a stochastic integral isApplying the same product identity to $H\cdot M$ and $K$ therefore gives

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.