Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 1 / b / ii / Solution

Set $M_t={\int }_0^t{H}_{s}d{B}_s$. Its quadratic variation is $[M{]}_t={\int }_0^t{H}_s^{2}ds$, which is continuous and tends to infinity almost surely by assumption. The stated stopping time is the inverse clock at level one, so $[M{]}_{\tau }=1$. The Dambis-Dubins-Schwarz theorem gives