Let $L=N-H\mathbin\cdot M$. It is a continuous square-integrable <martingale>, and the assumed bracket identity gives
$$
[L,K]_t=[N,K]_t-\int_0^tH_s\,d[M,K]_s=0
$$
for every continuous square-integrable martingale $K$. Choose $K=L$ and use part (i):
$$
\mathbb E[L_\infty^2]=\mathbb E[L]_\infty=0.
$$
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 $N=H\mathbin\cdot M$ up to <indistinguishability of stochastic processes>.

Solution

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

Two stochastic processes $X,Y$ are indistinguishable when
$$
\mathbb P(X_t=Y_t\text{ for every }t)=1.
$$
This is stronger than equality almost surely at each fixed time, though the two notions agree for continuous processes after checking equality on a countable dense set.


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