Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 201 / 4 / a / Solution

For $0\le s\le t$, write $B_t=B_s+(B_t-B_s)$. The increment is independent of ${\mathcal{F}}_s$ and is normally distributed with variance $t-s$. Its moment generating function givesThe process is integrable for every real $\lambda$, so it is the exponential Brownian martingale.