I am trying to solve the following problem but I got stuck and would appreciate some tips and/or references on how to approach it.
$X_t$ is a BM with drift $\mu$ and variance $\sigma^2$ which starts at $x$ (i.e. $X_0 = x$). I define the following stopping times $\tau_U = \inf\{t:X_t = x_U\}$, $\tau_D = \inf\{t:X_t = x_D\}$ and $T > 0$ for $x_D<x<x_U$. $E\left( \cdot \right)$ defines the expectation operator and $1_{\{\cdot\}}$ defines the indicator function. I am interested in finding:
$E\left( e^{-r \tau_U}1_{\{\tau_U < \tau_D \land T \}}\right)$ and $E\left( e^{-r \tau_D} 1_{\{\tau_D < \tau_U \land T \}}\right)$
For $r=0$, the two expressions above become simply the probabilities of hitting one barrier before touching the other barrier and before $T$ periods have passed.
I was able to derive the expressions without the excursion time $T$ (i.e $E\left( e^{-r \tau_U}1_{\{\tau_U < \tau_D\}}\right)$ and $E\left( e^{-r \tau_D} 1_{\{\tau_D < \tau_U \}}\right)$). However, I don't know how to approach the problem when the excursion time $T$ is introduced.
Any references/tips on how to solve this problem? Thanks!