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

For every stopping time $\tau$, the martingale case of the optional sampling theorem for a supermartingale gives $\mathbb{E}{X}_{\tau }=X_0=0$. HenceUse the optimal stopping time from part c and the pathwise inequalityTaking expectations gives