Numerical solution of a matrix integral equation arising in Markov Modulated Lévy processes

Markov-modulated Lévy processes lead to matrix integral equations of the kind A0 + A1X + A2X + A3(X) = 0 where A0, A1, A2 are given matrix coefficients, while A3(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.