Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 202 / 2 / a / Vanishing quadratic variation / Solution

If $A_t=0$ almost surely, then $X_t^2=M_t$ is a nonnegative martingale starting from zero. A nonnegative random variable of expectation zero vanishes almost surely, so $X_t=0$ almost surely for each $t$. Applying this on the nonnegative rational times and using path continuity shows that $X_t=0$ simultaneously for every $t\ge 0$ almost surely.