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

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