Exact solutions to first-passage problems for jump-diffusion processes

Let $T(x)$ be the first time the time-homogeneous jump-diffusion process $X(t)$, starting from $X(0)=x$, leaves the interval $(a,b)$. The jump size is assumed to have an asymmetric double exponential distribution. The integro-differential equation satisfied by the moment-generating function of $T(x)$ is transformed into an ordinary differential equation and is solved explicitly in particular cases. Explicit and exact results are also obtained for the mean of $T(x)$ as well as the probability $P[X(T(x)) \le a]$.