 Solution

A martingale deflator is a strictly positive adapted process $Y$ such that every deflated cum-dividend asset gain has zero conditional drift:Using the definitions of ${\pi }^H$ and ${\xi }^H$,The holdings $H_t$ are ${\mathcal{F}}_{t-1}$-measurable, so the right side is a martingale transform of the deflated asset-gain local martingale. Hence $Z$ is a local martingale.