= Solution
Write the residual process as
$$
M_t=\sum_{i=1}^n\theta_i\frac{\mathbf1_{\{U_i>t\}}}{1-t}.
$$
For $s<t<1$, the filtration tells us whether each $U_i$ has arrived by time $s$, and its exact arrival time if it has. On $\{U_i>s\}$, its conditional law is uniform on $(s,1)$. <Independence> of the original $U_i$ means that the histories of the other arrivals do not alter this law. Hence
$$
\mathbb E[\mathbf1_{\{U_i>t\}}\mid\mathcal F_s]
=\mathbf1_{\{U_i>s\}}\frac{1-t}{1-s}.
$$
This conditional identity can also be checked against products of bounded functions of the censored arrival times and extended to $\mathcal F_s$ by the <Monotone class theorem>. Summing the identity gives \b[$\mathbb E[M_t\mid\mathcal F_s]=M_s$].
Each $M_t$ is adapted and integrable, with
$$
\mathbb E M_t=\sum_i\theta_i=S.
$$
Its paths are smooth between the finitely many arrival times and have downward jumps of size $\theta_i/(1-U_i)$ at arrivals before time one. The indicators use $U_i>t$, so the value at a jump is the post-jump value: paths are right continuous with finite left limits on $[0,1)$. Thus $M$ is a <càdlàg martingale>, specifically the <uniform-arrival residual mass martingale>.
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