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

For $Z_t=e^{B_t+at}$, the Itô formula givesThus its local-martingale part is $M_t={\int }_0^t{Z}_s,d{B}_s$, its finite-variation part is $(a+1/2){\int }_0^t{Z}_{s}ds$, and its quadratic variation clock is