The distribution and asympotic behaviour of the negative Wiener-Hopf factor for L\'evy processes with rational positive jumps

Abstract

We study the distribution of the negative Wiener-Hopf factor for a class of two-sided jumps L\'evy processes whose positive jumps have a rational Laplace transform. The positive Wiener-Hopf factor for this class of processes was studied by Lewis and Mordecki (2008). Here we obtain a formula for the Laplace transform of the negative Wiener-Hopf factor, as well as an explicit expression for its probability density, which is in terms of sums of convolutions of known functions. Under additional regularity conditions on the L\'evy measure of the studied processes, we also provide asymptotic results as $u\to-\infty$ for the distribution function $F(u)$ of the negative Wiener-Hopf factor. We illustrate our results in some particular examples.