Consider a Brownian Motion $B_t$, constants $a \in \mathbb{R},b>0$ and $\tau=\inf\{t\ge 0:B_t=at+b\}$. For $\lambda \ge 0$, show that $$ \mathbb{E}(e^{-\lambda \tau})= \exp(-ba-b\sqrt{a^2+2\lambda})$$ and conclude $P(\tau <\infty) = \min(1,e^{-2ba})$.
Essentially this is like taking the MGF of the random time $-\tau$ which is a function of a brownian motion. Firstly, $$\mathbb{P}(-\tau > t) \leq \mathbb{P}(\tau \le t-1)\mathbb{P}(|B_t - B_{t-1}| \leq 2)$$ $$=\mathbb{P}(\tau \le t-1)\mathbb{P}(|at+b - a(t+1)-b(t+1)| \leq 2)$$ $$ = \mathbb{P}(\tau \le t-1)\mathbb{P}(|a+bt| \leq 2)$$ and hence shows $ \mathbb{E}(e^{-\lambda \tau})$ should be finite, but I'm not sure how to proceed as I'm getting nowhere..
I'm wondering, since It is easy to show that $e^{-\lambda \tau}$ is a martingale, if this helps.
