Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 35 / 2 / b / Solution

The Dambis-Dubins-Schwarz theorem states: let $M$ be a continuous local martingale with $M_0=0$ and $[M{]}_{\infty }=\infty$ almost surely. SetThen $W_u=M_{{\tau }_u}$ is standard Brownian motion relative to $({\mathcal{G}}_u)$ andStrict increase of the bracket is not required in the general theorem: $M$ is constant on intervals on which its bracket is constant. Continuity of $M$ is required. A nonzero initial value is handled by applying the theorem to $M-M_0$ and adding it back. If the terminal bracket is finite, the finite-lifetime extension of the Dambis-Dubins-Schwarz theorem supplies the Brownian motion up to that bracket time and, if necessary, an independent continuation on an enlarged probability space.