---
author:
- 'Tomasz R. Bielecki[^1] ,'
- 'Igor Cialenco,'
- 'Ruoting Gong,'
- Yicong Huang
date: ' [First Circulated: January 14, 2018 ]{}'
title: 'Wiener-Hopf factorization for time-inhomogeneous Markov chains and its application'
---

[l@ p[350pt]{}]{}\
<span style="font-variant:small-caps;">Abstract</span>:  & In this paper we derive the Wiener-Hopf factorization for a finite-state time-inhomogeneous Markov chain. To the best of our knowledge, this study is the first attempt to investigate the Wiener-Hopf factorization for time-inhomogeneous Markov chains. In this work we only deal with a special class of time-inhomogeneous Markovian generators, namely piece-wise constant, which allows to use an appropriately tailored randomization technique. Besides the mathematical importance of the Wiener-Hopf factorization methodology, there is also an important computational aspect: it allows for efficient computation of important functionals of Markov chains.\
<span style="font-variant:small-caps;">Keywords:</span>  & Wiener-Hopf factorization, inhomogeneous Markov chain, fluctuation theory, randomization method, additive functional.\
<span style="font-variant:small-caps;">MSC2010:</span>  & 60J27, 60J28, 60K25\

Introduction
============

In this paper we derive the Wiener-Hopf factorization (WHf) for a finite-state time-inhomogeneous Markov chain. As far as we know, our study is the first attempt to investigate the Wiener-Hopf factorization for time-inhomogeneous Markov chains. In this pioneering study we only deal with a special class of time-inhomogeneous Markovian generators, 