For $Z_t=e^{B_t+at}$, the <Itô formula> gives
$$
dZ_t=Z_t\,dB_t+\left(a+\frac12\right)Z_t\,dt.
$$
Thus its local-martingale part is $M_t=\int_0^tZ_s,dB_s$, its finite-variation part is $(a+1/2)\int_0^tZ_sds$, and its <quadratic variation> clock is
$$
[M]_t=\int_0^tZ_s^2ds.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-202/4/b/solution