Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 202 / 2 / a / i

The bounded continuous local martingale $X$ is a square-integrable martingale. Since $M^{(n)}$ is a discrete predictable transform of $X$, it has mean zero. The identity $A_t^{(n)}=X_t^2-M_t^{(n)}$ therefore givesuniformly in $t$ and $n$.