# Wiener-Hopf Factorization for Time-Inhomogeneous Markov Chains

## TL;DR

This paper extends Wiener-Hopf factorization theory to finite time-inhomogeneous Markov chains with general time-dependent generator matrices, broadening the scope beyond previous piecewise constant cases.

## Contribution

It develops a Wiener-Hopf type factorization for finite Markov chains with general time-dependent generators, advancing the theoretical framework for inhomogeneous processes.

## Key findings

- Derived Wiener-Hopf factorization for general time-inhomogeneous Markov chains.
- Extended previous results from piecewise constant to general generator functions.
- Provided new mathematical tools for analyzing inhomogeneous Markov processes.

## Abstract

This work contributes to the theory of Wiener-Hopf type factorization for finite Markov chains. This theory originated in the seminal paper Barlow et al. (1980), which treated the case of finite time-homogeneous Markov chains. Since then, several works extended the results of Barlow et al. (1980) in many directions. However, all these extensions were dealing with time-homogeneous Markov case. The first work dealing with the time-inhomogeneous situation was Bielecki et al. (2018), where Wiener-Hopf type factorization for time-inhomogeneous finite Markov chain with piecewise constant generator matrix function was derived. In the present paper we go further: we derive and study Wiener-Hopf type factorization for time-inhomogeneous finite Markov chain with the generator matrix function being a fairly general matrix valued function of time.