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

Take the first entry into the stopping region:This set is nonempty because $U_T=Z_T$. Before ${\tau }^{\ast }$, the recursion hasso the stopped process $U_{t\wedge {\tau }^{\ast }}$ is a martingale. Optional sampling and $U_{{\tau }^{\ast }}=Z_{{\tau }^{\ast }}$ giveThus ${\tau }^{\ast }$ is an optimal stopping time.