The function $z\mapsto\log|z|$ is a <harmonic function> on the <annulus> $r_1<|z|<r_2$. The <Itô formula> therefore makes $\log|B_{t\wedge\tau}|$ a bounded <martingale>. The <optional sampling theorem for a supermartingale> gives
$$
\log r=\mathbb E[\log|B_\tau|]=p\log r_1+(1-p)\log r_2,
$$
where $p=\mathbb P(|B_\tau|=r_1)$. Solving this <linear equation> yields the <planar Brownian annulus hitting probability>
$$
p=\frac{\log r_2-\log r}{\log r_2-\log r_1}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 3 / d / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-202/3/d/i/solution