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.

摘要

这项工作为有限马尔可夫链的Wiener-Hopf类型分解理论做出了贡献。该理论起源于开创性的论文[Barlow等人，矩阵的Wiener-Hopf因式分解，斯特拉斯堡工业学院（SéminairedeProbabilitésde Strasbourg）14（1980），第324-331页]，该论文处理了有限时均质马尔可夫链的情况。从那以后，几项工作从多个方向扩展了[Barlow等人，Wiener-Hopf矩阵分解，斯特拉斯堡工业大学学报（14）（1980），第324-331页]的结果。但是，所有这些扩展都涉及时间均匀的马尔可夫问题。处理时间不均匀情况的第一项工作是[Bielecki等人，时间不均匀马尔可夫链的Wiener-Hopf因式分解及其应用，Probab。数学。统计 [2019，即将推出]，其中推导了具有分段常数生成器矩阵函数的时间不均匀有限马尔可夫链的Wiener-Hopf型分解。在本文中，我们走得更远：我们推导并研究了时间非均质有限Markov链的Wiener-Hopf型分解，其中生成器矩阵函数是时间的相当通用的矩阵值函数。