Exit Times for a Discrete Markov Additive Process

TL;DR

This paper develops a theory for scale matrices in upward skip-free discrete-time Markov additive chains, enabling the analysis of exit problems and fluctuation identities in a fully discrete setting.

Contribution

It introduces the $ ilde{W}$ and $ ilde{Z}$ scale matrices for discrete Markov additive chains, providing probabilistic constructions, generating functions, and recursion relations.

Findings

Derived explicit forms of scale matrices $ ilde{W}$ and $ ilde{Z}$.

Established connections with occupation mass formulas.

Analyzed distributional properties of reflected processes.

Abstract

In this paper we consider (upward skip-free) discrete-time and discrete-space Markov additive chains (MACs) and develop the theory for the so-called $\tilde{W}$ and $\tilde{Z}$ scale matrices. which are shown to play a vital role in the determination of a number of exit problems and related fluctuation identities. The theory developed in this fully discrete setup follows similar lines of reasoning as the analogous theory for Markov additive processes in continuous-time and is exploited to obtain the probabilistic construction of the scale matrices, identify the form of the generating function and produce a simple recursion relation for $\tilde{W}$, as well as its connection with the so-called occupation mass formula. In addition to the standard one and two-sided exit problems (upwards and downwards), we also derive distributional characteristics for a number of quantities related to the one and two-sided `reflected' processes.