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
$$
\mathbb E Z_\tau\leq\mathbb E U_\tau\leq U_0.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 211 / 4 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-211/4/b/solution