Let $X_t$ be a generalized Wiener process with drift rate $\mu$ and variance $\sigma^2$, and let $\tau$ be the stopping time
$$\tau:=\inf \left\{ t\geq0: X_t= b\right\}, \quad b\geq0 $$
Can anyone give me some insights on how to compute the generating function:
$$ E[\mathrm{e}^{-\lambda\tau}], \quad \lambda\geq0 $$
Many thanks in advance.