Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 201 / 1 / b / Solution

On upper exit, $b\le S_T\le b+c$; on lower exit, $-a-c\le S_T\le -a$. With $p=\mathbb{P}(S_T\ge b)$ and $\mathbb{E}{S}_T=0$,Solving gives

Moreover $(S_T+a)(S_T-b)\ge 0$, so $\mathbb{E}{S}_T^2\ge ab$, which is stronger than the requested lower bound. Also $S_T\in [-a-c,b+c]$, and henceTaking expectations and using $\mathbb{E}{S}_T=0$ gives $\mathbb{E}{S}_T^2\le (a+c)(b+c)$, stronger than the requested upper bound. Sincethe two stated estimates follow from part a.