On the Wiener-Hopf factorization for Levy processes with bounded positive jumps

TL;DR

This paper investigates the Wiener-Hopf factorization for Levy processes with bounded positive jumps, providing explicit product and series representations, asymptotic formulas, and numerical methods for computation.

Contribution

It introduces new explicit formulas and asymptotic expressions for the Wiener-Hopf factors and scale functions in Levy processes with bounded jumps, enhancing numerical computation methods.

Findings

Explicit infinite product representation for the positive Wiener-Hopf factor.

Asymptotic formulas for solutions to the key equation $\, ext{and}\, ext{series representation for scale functions.

Numerical algorithms demonstrated with successful experiments.

Abstract

We study the Wiener-Hopf factorization for Levy processes with bounded positive jumps and arbitrary negative jumps. Using the results from the theory of entire functions of Cartwright class we prove that the positive Wiener-Hopf factor can be expressed as an infinite product in terms of the solutions to the equation $\psi(z)=q$, where $\psi$ is the Laplace exponent of the process. Under some additional regularity assumptions on the Levy measure we obtain an asymptotic expression for these solutions, which is important for numerical computations. In the case when the process is spectrally negative with bounded jumps, we derive a series representation for the scale function in terms of the solutions to the equation $\psi(z)=q$. To illustrate possible applications we discuss the implementation of numerical algorithms and present the results of several numerical experiments.