Past exam of the stochastic differential equations course of the University of Oxford / 2011 / 3 / c / Solution

Boundedness makes each $N_u$ Q-integrable. For $s\le u\le t$, the density formula and conditioning first on ${\mathcal{F}}_u$ giveThe middle equality uses that $N_u$ is ${\mathcal{F}}_u$-measurable and $Z$ is a martingale; the next uses the assumed martingale property of $NZ$. Thus $N$ is a Q-martingale.