Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 4 / b / Solution

The Dambis-Dubins-Schwarz theorem says that there is Brownian motion $W$ such thatWhen $[M]$ is strictly increasing, define its inverseand set $W_s=M_{T_s}$. Optional sampling shows that $W$ is a continuous local martingale, while time change gives $[W{]}_s=s$. The Lévy characterization of Brownian motion makes $W$ Brownian, and inverse time change gives the displayed representation.