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 <Brownian exit time>[exit times]. Then $T\leq\tau_R$. Since
$$
|B_t|^2-dt
$$
is a <martingale>, the <optional sampling theorem for a supermartingale> at $\tau_R\wedge n$ gives
$$
d\,\mathbb E_x(\tau_R\wedge n)
=\mathbb E_x|B_{\tau_R\wedge n}|^2-|x|^2
\leq R^2-|x|^2.
$$
<Monotone convergence theorem> now gives
$$
\mathbb E_xT\leq\mathbb E_x\tau_R
\leq\frac{R^2-|x|^2}{d}<\infty.
$$

Solved by gpt-5.6-sol high.

Solution

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

{title2=$T_D$}

For a domain $D$, the Brownian exit time is $T_D=\inf\{t\geq0:B_t\notin D\}$. It is a <stopping time>, and it has finite expectation when $D$ is bounded.

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