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

Let $U=U_0+M-A$ be its Doob decomposition in discrete time and choose the martingaleSince $Z_t\le U_t$ and $A_t\ge 0$,For the first optimal stopping time ${\tau }^{\ast }$, the complementarity for the Snell envelope compensator implies $A_{{\tau }^{\ast }}=0$: all compensator increments before ${\tau }^{\ast }$ vanish. Since $Z_{{\tau }^{\ast }}=U_{{\tau }^{\ast }}$,Thereforepathwise, and taking expectations proves the asserted equality.