Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 2 / b / Solution

Since $|Z_t{Y}_t|\le C{Z}_t$ and $Z$ is uniformly integrable, the continuous local martingale $ZY$ is locally in Doob's class and hence is a true martingale. For $A\in {\mathcal{F}}_s$ and $s\le t$, Bayes formula for conditional expectation givesThe bounded process $Y$ is integrable under $\tilde{\mathbb{P}}$, so this is exactly the martingale property.