= Solution
A <martingale deflator> is a strictly positive adapted process $Y$ such that every deflated cum-dividend asset gain has zero conditional drift:
$$
\mathbb E\!\left[Y_t(P_t+\delta_t)\mid\mathcal F_{t-1}\right]
=Y_{t-1}P_{t-1}.
$$
Using the definitions of $\pi^H$ and $\xi^H$,
$$
Z_t-Z_{t-1}
=H_t\mathbin\cdot
\left[Y_t(P_t+\delta_t)-Y_{t-1}P_{t-1}\right].
$$
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>.
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