Matrix factorization identity for almost semi-continuous processes on a Markov chain

TL;DR

This paper develops a matrix factorization identity for almost semi-continuous processes on a Markov chain, providing explicit formulas for distributions of extrema and their complements, advancing stochastic process analysis.

Contribution

It introduces a concrete matrix factorization identity for almost semi-continuous processes on Markov chains, enabling explicit distribution calculations.

Findings

Defined matrix factorization components explicitly.

Derived relations for distributions of extrema.

Applied to almost upper semi-continuous processes.

Abstract

In this article almost semi-continuous processes with stationary independent increments on a finite irreducible Markov chain are considered. For these processes the components of matrix factorization identity are concretely defined. On the basis of this concrete definition the relations for the distributions of extrema and distributions of their complements for the almost upper semi-continuous processes are established.