On the Distribution of Extrema for a Class of L\'evy Processes

TL;DR

This paper derives the characteristic functions of the extrema for certain Levy processes using Riemann-Hilbert techniques, providing new tools for financial and actuarial applications.

Contribution

It introduces a novel application of Riemann-Hilbert methods to obtain extrema distributions for specific Levy processes, including approximation techniques and examples.

Findings

Characteristic functions of extrema derived for exponential type Levy processes

Effective approximation methods demonstrated

Applicable to financial and actuarial modeling

Abstract

Suppose Xt is either a regular exponential type Levy process or a Levy process with a bounded variation jumps measure. The distribution of the extrema of Xt play a crucial role in many financial and actuarial problems. This article employs the well known and powerful Riemann-Hilbert technique to derive the characteristic functions of the extrema for such Levy processes. An approximation technique along with several examples is given.