Clarification of basic factorization identity is for the almost semi-continuous latticed Poisson processes on the Markov chain

Abstract

Let ${xi(t), x(t)}$ be a homogeneous semi-continuous lattice Poisson process on the Markov chain.The jumps of one sign are geometrically distributed, and jumps of the opposite sign are arbitrary latticed distribution. For a suchprocesses the relations for the components of two-sided matrix factorization are established.This relations define the moment genereting functions for extremumf of the process and their complements.