Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 37 / 4 / b / Solution

Translate the horizontal coordinate by $x$. Then $X_t+x$ is a standard Brownian motion starting at zero, and continuity identifies $\tau$ with its first-passage time to $x$. The vertical coordinate remains an independent standard Brownian motion starting at zero. Part (a) therefore gives the Cauchy exit law from a Brownian half-plane:The TeX transcription's initial value $-z$ and later $Y_T$ are both corrected here to the PDF's $-x$ and $Y_{\tau }$.