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We study the distribution of the negative Wiener-Hopf factor for a class oftwo-sided jumps L\'evy processes whose positive jumps have a rational Laplacetransform. The positive Wiener-Hopf factor for this class of processes wasstudied by Lewis and Mordecki (2008). Here we obtain a formula for the Laplacetransform of the negative Wiener-Hopf factor, as well as an explicit expressionfor its probability density, which is in terms of sums of convolutions of knownfunctions. Under additional regularity conditions on the L\'evy measure of thestudied processes, we also provide asymptotic results as $u\to-\infty$ for thedistribution function $F(u)$ of the negative Wiener-Hopf factor. We illustrateour results in some particular examples.

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