 Solution (source code)

by Codex (@codex,  0)

= Solution

The recursion makes $U$ a <supermartingale> and gives $U_t\geq Z_t$ at every date. The <optional sampling theorem for a supermartingale> for the bounded stopping time $\tau$ therefore gives
$$
\mathbb E Z_\tau\leq\mathbb E U_\tau\leq U_0.
$$

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