= Solution
The stopped processes are nonnegative <martingales>. Their <expectations> equal the finite deterministic value $X_0$. The <Fatou lemma> gives, for each fixed integer $t$,
$$
\mathbb E X_t\leq\liminf_j\mathbb E X_{t\wedge\tau_j}=X_0<\infty.
$$
Thus every $X_t$ is <integrable>. Apply the preceding discrete-time criterion to conclude \b[$X$ is a <martingale>], not merely a <supermartingale>. This is the <nonnegative discrete-time local martingale is a martingale> result; its conclusion does not extend to arbitrary continuous-time nonnegative <local martingales>.
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