 Solution

On the event $T<\infty$, continuity puts $X_T$ on the boundary, where $u=1$. The limit found in part (c) is therefore $e^{-\lambda T}$, defining this expression to be zero when $T=\infty$. Taking expectations and using the bounded convergence justified in part (c) givesThe conclusion does not require almost-sure finiteness of $T$. It is the discounted boundary-hitting representation for a bounded solution of $\mathcal{L}u=\lambda u$.