Computation of First Passage of Markov Additive Processes

The computation of the pair of matrices describing the first passage time of a Markov additive process is considered. This pair of matrices is characterized as a solution to an integral matrix equation for which we develop an iterative method. At each step, it requires computing the extremal solution to a mixed linear-quadratic matrix equation, which is accomplished by a quadratically convergent algorithm. When all the jumps are of phase-type distribution, the integral matrix equation can be transformed into a single mixed linear-quadratic matrix equation and thus the pair of matrices can be computed with quadratic convergence.