Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 4 / a / Solution

The strictly increasing continuous clock $[M]$ has a finite continuous inverse ${\tau }_s$. The time change of a continuous process therefore preserves continuity and adaptedness. Moreover,The optional time-change theorem makes $M_{{\tau }_s}$ a continuous local martingale for ${\tilde{\mathcal{F}}}_s={\mathcal{F}}_{{\tau }_s}$, while composition with the increasing map $s\mapsto {\tau }_s$ preserves the finite variation of $A$. Hence $\tilde{X}$ is a continuous semimartingale in the time-changed filtration.