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

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.