Let $Z_t$ be a standard Brownian motion, and define its first-hitting time to an exponential time-varying boundary as follows: $$ \tau=\inf\{t\geq0:Z_t=e^{at}\} $$ where $a\in\mathbb{R}$ is given. This link has discussed some property of the distribution of the stopping time $\tau$ when $a<0$. But I wonder how to further compute the following expectation regarding $\tau$ for general $a\in\mathbb{R}$ or for $a>0$: $$ \mathbb{E}[e^{-r\tau}] $$ where $r>0$ is given and is known as, say, discount factor.
Thank you very much for your assistances. Best wishes.
