The function $z\mapsto\log|z|$ is <harmonic function>[harmonic] on the annulus $\varepsilon<|z|<R$. Hence $\log|B_{t\wedge\tau_\varepsilon\wedge\tau_R}|$ is a bounded martingale by <Itô formula>. Optional stopping gives
$$
\log|x|=p\log\varepsilon+(1-p)\log R,
\qquad p=\mathbb P_x(\tau_\varepsilon<\tau_R).
$$
Solving yields
$$
p=\frac{\log R-\log|x|}{\log R-\log\varepsilon}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 2 / b / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-201/2/b/i/solution