Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 6 / b / Solution

For bounded stopping times $S\wedge n\le T\wedge n$, the optional sampling theorem for a supermartingale givesA stopped family drawn from a uniformly integrable martingale is uniformly integrable. Since $X_{T\wedge n}\to X_T$ and $X_{S\wedge n}\to X_S$ almost surely, uniform integrability upgrades both convergences to $L^1$. Passing to the limit yields