First-Passage Time Problems for Jump-Diffusion Processes with Positive and Negative Jumps

Let τ(x) be the first time that a jump-diffusion process with positive and negative jumps leaves the interval (a,b). The moment-generating function M(x) of τ(x) satisfies an integro-differential equation (IDE). In the case of exponential jumps, we transform this IDE into an ordinary differential equation (ODE). This is similar to the IDE satisfied by m(x):=E[τ(x)]. Explicit solutions to particular problems are obtained for both M(x) and m(x) by solving the appropriate ODEs.