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

## TL;DR

This paper derives explicit formulas and asymptotic behaviors for the distribution of the negative Wiener-Hopf factor in a class of Lévy processes with rational positive jumps, extending previous work on the positive factor.

## Contribution

It provides a new explicit formula for the Laplace transform and density of the negative Wiener-Hopf factor for Lévy processes with rational positive jumps, along with asymptotic results.

## Key findings

- Explicit Laplace transform of the negative Wiener-Hopf factor.
- Closed-form expression for its probability density.
- Asymptotic behavior of the distribution as u approaches -infinity.

## 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.