Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 30K / d / Solution

For $n<N$,The recursionshows that if $V_n>Z_n$, then $V_n=\mathbb{E}[V_{n+1}\mid {\mathcal{F}}_n]$ and the increment of $A$ is zero. Conversely, if that increment is positive, the maximum must be attained by $Z_n$, so $V_n=Z_n$. The two nonnegative quantities therefore satisfyFor $n=N$, this also holds because $V_N=Z_N$, regardless of the convention $A_{N+1}=\infty$.