  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.
