Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 202 / 2 / b / Solution

Let ${\tau }_m=\inf \{s:|X_s|\ge m\}$. The stopped process $X^{{\tau }_m}$ is bounded, so part (a) gives a continuous adapted quadratic variation $A^{[m]}$. These processes agree before the smaller stopping time, because their dyadic sums agree there and the limits are unique in probability. They therefore paste to a continuous adapted process $A$ with $A_{s\wedge {\tau }_m}=A_s^{[m]}$.

For fixed $t$ and $\epsilon >0$,The second term tends to zero by part (a), while continuity of $X$ on $[0,t]$ makes $\mathbb{P}({\tau }_m\le t)\to 0$. This proves convergence uniformly on compact intervals in probability.