Wiener-Hopf factorization for a family of Lévy processes related to theta functions

In this paper we study the Wiener–Hopf factorization for a class of Lévy processes with double-sided jumps, characterized by the fact that the density of the Lévymeasure is given by an infinite series of exponential functions with positive coefficients. We express the Wiener–Hopf factors as infinite products over roots of a certain transcendental equation, and provide a series representation for the distribution of the supremum/infimum process evaluated at an independent exponential time. We also introduce five eight-parameter families of Lévy processes, defined by the fact that the density of the Lévy measure is a (fractional) derivative of the theta function, and we show that these processes can have a wide range of behavior of small jumps. These families of processes are of particular interest for applications, since the characteristic exponent has a simple expression, which allows efficient numerical computation of the Wiener–Hopf factors and distributions of various functionals of the process.