It is very well known that expected time for a standard Brownian motion to exit from interval $[a,b]$ (where $a<0$ and $b>0$) is $-ab$. In one of my projects, I wanted to calculate the similar quantity for a compound poisson process. I am not an expert in handling point processes and would be grateful for any help.
Question : $X$ is a compound poisson process starting at position $0$ with arrival rate $\lambda$ and the distribution of jumps as $F(dz)$. What would be the expected time $\tau$ for $X$ to exit $[-a,a]$? Also, what would be the distribution of $X_\tau$?