On the Riemann-Hilbert factorization problem for positive definite functions

TL;DR

This paper develops general theorems for positive definite solutions to Riemann-Hilbert problems on the real line and applies these results to Lévy processes to determine the distribution of their extrema at stopping times.

Contribution

It introduces new theorems for solving Riemann-Hilbert problems with positive definite solutions and applies them to analyze Lévy processes.

Findings

Derived general theorems for Riemann-Hilbert problems

Applied theory to Lévy processes' characteristic functions

Determined distributions of extrema at stopping times

Abstract

We give several general theorems concerning positive definite solutions of Riemann-Hilbert problems on the real line. Furthermore, as an example, we apply our theory to the characteristic function of a class of L\'{e}vy processes and we find the distribution of their extrema at a given stopping time.