Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 201 / 2

Write $v_m=\mathrm{Var}(X_m)={\sigma }_m^2$. Independence and $\mathbb{E}[X_{n+1}]=0$ giveThus

Put $V=\mathrm{Var}(S_n)={\sum }_{m=1}^n{v}_m$. If ${\max }_{m\le n}{S}_m\ge x$, thenThe Doob maximal inequality for a nonnegative submartingale therefore givesThe right side is minimized at $c=V/x$, and substitution gives

Let $V_n=\mathrm{Var}(S_n)={\sum }_{m=1}^n{v}_m$. ThenConsequently

LetBecause $|X_m|\le K$, one always has $|S_{\tau }|\le x+K$: this is clear if no crossing occurs, and at the first crossing the overshoot is at most one increment. Apply the optional stopping theorem to the martingale from part (c):On the event $\{{\max }_{m\le n}|S_m|\le x\}$ one has $\tau =n$, so $V_{\tau }=V_n$. Since $V_{\tau }\ge 0$ everywhere,Hence