Numerical solution of a matrix integral equation arising in Markov Modulated L\'evy processes

TL;DR

This paper introduces numerical methods for solving complex matrix integral equations from Markov-modulated Lévy processes, providing convergence analysis and demonstrating effectiveness through extensive numerical experiments.

Contribution

It proposes new numerical algorithms for a class of nonlinear matrix integral equations and offers theoretical convergence analysis.

Findings

New numerical methods successfully solve the matrix equations.

The methods show good convergence properties.

Numerical experiments confirm the effectiveness of the approaches.

Abstract

Markov-modulated L\'evy processes lead to matrix integral equations of the kind $ A_0 + A_1X+A_2 X^2+A_3(X)=0$ where $A_0$, $A_1$, $A_2$ are given matrix coefficients, while $A_3(X)$ is a nonlinear function, expressed in terms of integrals involving the exponential of the matrix $X$ itself. In this paper we propose some numerical methods for the solution of this class of matrix equations, perform a theoretical convergence analysis and show the effectiveness of the new methods by means of a wide numerical experimentation.