Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 5 / a / Solution

Let $D\subset {\mathbb{R}}^d$ be bounded, choose $R$ with $D\subset B(0,R)$, and let $T$ and ${\tau }_R$ be the respective exit times. Then $T\le {\tau }_R$. Sinceis a martingale, the optional sampling theorem for a supermartingale at ${\tau }_R\wedge n$ givesMonotone convergence theorem now gives

Solved by gpt-5.6-sol high.