Exit times and places for a Wiener process with sign-dependent exponential jumps

We consider a one-dimensional jump-diffusion process {X(t), t ≥ 0} whose continuous part is a Wiener process with zero drift. The jumps are exponential and depend on the sign of X(t). Let τ (x) be the first time that the process, starting from X(0) = x, is equal to zero, or |X(t)| = d. We obtain exact analytical expressions for the moment-generating function of τ (x), its mean, and the probability that X(τ (x)) = 0. To do so, we solve integro-differential equations, subject to the appropriate boundary conditions.