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.

翻译：Markov-moded L\'evy 过程导致以A_0 + A_1X+A_2 X%2+A_3(X)=0美元等量的矩阵整体方程式,其中A_0美元、A_1美元、A_2美元为基数,而A_3(X)美元为非线性函数,以涉及基数本身指数的基数表示。在本文中,我们提出一些数字方法来解决这一类矩阵方程式,进行理论趋同分析,并通过广泛数字试验显示新方法的有效性。