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

TL;DR

This paper develops a method to compute Wiener-Hopf factors for a class of Levy processes with jump densities expressed via theta functions, enabling efficient analysis of their supremum and infimum distributions.

Contribution

It introduces a novel approach to factorize Levy processes with jump densities related to theta functions, providing explicit formulas and series representations for their extremal distributions.

Findings

Wiener-Hopf factors expressed as infinite products over roots.

Series representation for supremum/infimum distributions.

Efficient numerical computation enabled by simple characteristic exponent.

Abstract

In this paper we study the Wiener-Hopf factorization for a class of L\'evy processes with double-sided jumps, characterized by the fact that the density of the L\'evy measure 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 L\'evy processes, defined by the fact that the density of the L\'evy 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.