= Solution
The <supermartingale> conclusion gives
$$
\mathbb E\left[Z_tX_t+\int_0^tZ_sc_s\,ds\right]\leq x.
$$
Discard the nonnegative terminal-wealth term and let $t\to\infty$. The cumulative deflated <consumption> increases, so <monotone convergence theorem> yields the <state-price budget constraint>
$$
\boxed{\mathbb E\int_0^\infty Z_tc_t\,dt\leq X_0.}
$$
The normalization is relevant: with $Z_0\ne1$ the right side is $Z_0X_0$; with random integrable initial deflated wealth its <expectation> is the unconditional budget bound.
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