We show that the positive Wiener-Hopf factor of a Lévy process with positive jumps having a rational Fourier transform is a rational function itself, expressed in terms of the parameters of the jump distribution and the roots of an associated equation. Based on this, we give the closed form of the ruin probability for a Lévy process, with completely arbitrary negatively distributed jumps, and finite intensity positive jumps with a distribution characterized by a rational Fourier transform. We also obtain results for the ladder process and its Laplace exponent. A key role is played by the analytic properties of the characteristic exponent of the process and by a Baxter-Donsker-type formula for the positive factor that we derive.

In this paper, we present an iterative procedure to calculate explicitly the Laplace transform of the distribution of the maximum for a Lévy process with positive jumps of phase type. We derive error estimates showing that this iteration converges geometrically fast. Subsequently, we determine the Laplace transform of the law of the upcrossing ladder process and give an explicit pathwise construction of this process.