Let $G(t)=xe^{\sigma B(t)+ct}$, where $B(t)$ is a standard Brownian motion; i.e. $G(t)$ is a geometric Brownian motion. Given $D<0<U$, let $T=\inf \{t>0:G(t)=U$ or $G(t)=D\}.$ What is the probability that $G(T)=U?$
I know of the analogous result for Brownian motion, but am not sure how to use it here - in particular, how to account for the "drift" of GBM. $G(t)=U \Leftrightarrow B(t)=\frac{\log(x+U)-\log x-ct}{\sigma}$, which is decreasing in $t$.
Any tips or references would be greatly appreciated. Thanks!