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

Wiener-Hopf Factorization for Time-Inhomogeneous Markov Chains