Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 2 / b / Solution

Choose stopping times $T_m\uparrow \infty$ such that $X^{T_m}$ is a bounded martingale. Part a constructs $[X^{T_m}]$. Uniqueness in the identityshows consistency on overlapping stopped intervals, so define $[X{]}_{t\wedge T_m}=[X^{T_m}{]}_t$. The stopped dyadic sums converge uniformly on every compact interval in probability, andis a local martingale. This localization constructs the quadratic variation of every continuous local martingale.