Wiener-Hopf factorization technique for time-inhomogeneous finite Markov chains

Abstract

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., Wiener-Hopf factorization for matrices, Séminaire de Probabilités de Strasbourg 14 (1980), pp. 324–331], which treated the case of finite time-homogeneous Markov chains. Since then, several works extended the results of [Barlow et al., Wiener-Hopf factorization for matrices, Séminaire de Probabilités de Strasbourg 14 (1980), pp. 324–331] 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., Wiener-Hopf factorization for time-inhomogeneous Markov chains and its application, Probab. Math. Stat. 2019, forthcoming], 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.