= Solution
The integral against the local-martingale vector $YP$ is a <local martingale>, provided the <predictable> holdings are stochastically <integrable>. Integrating part (b) gives
$$
M_t=Y_tX_t-Y_0X_0+\int_0^tY_sc_sds.
$$
Both $Y_tX_t$ and the cumulative deflated <consumption> are nonnegative. Hence $M_t\geq-Y_0X_0$. With the usual finite deterministic initial capital, the shifted process
$$
M_t+Y_0X_0=Y_tX_t+\int_0^tY_sc_sds
$$
is a nonnegative <local martingale>, and is therefore a <supermartingale>. For completeness, a <localizing sequence> turns it into true <martingales>; conditional <Fatou lemma> for their nonnegative stopped values gives the <supermartingale> inequality and ordinary Fatou gives <integrability> at each time. Subtracting the initial constant proves
$$
\boxed{M\text{ is a supermartingale with }M_0=0.}
$$
This is <supermartingale control of deflated consumption gains>. It uses $c\geq0$ as well as $X\geq0$; an unrestricted <stochastic integral> is not necessarily a true <supermartingale>.
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