Matrix representations of Wiener–Hopf factorisations for Lévy processes

Abstract The Wiener–Hopf factors of a Lévy process are the maximum and the displacement from the maximum at an independent exponential time. The majority of explicit solutions assume the upward jumps to be either phase-type or to have a rational Laplace transform, in which case the traditional expressions are lengthy expansions in terms of roots located by means of Rouché’s theorem. As an alternative, compact matrix formulas are derived, with parameters computable by iteration schemes.