The two-sided exit problem for an additive functional of a time-inhomogeneous Markov chain

TL;DR

This paper analyzes the joint distribution of exit times and states for an additive functional driven by a finite state, time-inhomogeneous Markov chain, using operator compositions related to one-sided exit problems.

Contribution

It introduces a novel operator-based framework to characterize the joint distribution of two-sided exit times and states in time-inhomogeneous Markov chains.

Findings

Derived expressions for joint exit time and state distributions.

Connected two-sided exit problems to one-sided first passage operators.

Provided insights into the Markov chain's behavior before exit.

Abstract

We consider an additive functional driven by a time-inhomogeneous Markov chain with a finite state space. Our study focuses on the joint distribution of the two-sided exit time and the state of the driving Markov chain at the time of exit, given in terms of expectation operators. These operators can be expressed as compositions of other operators related to some relevant one-sided exit (or first passage) problems. In addition, we study the law of the driving Markov chain at times prior to the exit time.