Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 4 / b / ii

Let ${\sigma }_x=\inf \{t\ge 0:{\tilde{B}}_t=x\}$. Continuity gives $\{S\ge x\}=\{{\sigma }_x<\infty \}$. On this event, the Strong Markov property says thatis an independent Brownian motion with drift $-\mu$. It reaches level $y$ with probability $\mathbb{P}(S\ge y)$. Therefore