Rogers functions and fluctuation theory

Abstract

Extending earlier work by Rogers, Wiener-Hopf factorisation is studied for a class of functions closely related to Nevanlinna-Pick functions and complete Bernstein functions. The name 'Rogers functions' is proposed for this class. Under mild additional condition, for a Rogers function f, the Wiener--Hopf factors of f(z)+q, as well as their ratios, are proved to be complete Bernstein functions in both z and q. This result has a natural interpretation in fluctuation theory of L\'evy processes: for a L\'evy process X_t with completely monotone jumps, under mild additional condition, the Laplace exponents kappa(q;z), kappa*(q;z) of ladder processes are complete Bernstein functions of both z and q. Integral representation for these Wiener--Hopf factors is studied, and a semi-explicit expression for the space-only Laplace transform of the supremum and the infimum of X_t follows.