Exit problems for additive-increase and multiplicative-decrease Markov-modulated processes

Abstract

We study one-sided, two-sided and reflected exit problems for a finite-state Markov-modulated additive-increase and multiplicative-decrease process. For upward passage, a first-jump decomposition and a spatial Laplace transform lead to a recursion indexed by finite words of phases and to a convergent Green series. For downward passage, the first-jump operator is a strict contraction and the known contribution from $[p_i b,b]$ becomes an explicit boundary forcing term in the same spatial recursion. These allow to derive the corresponding Green representation of the first passage matrix and its finite-dimensional compatibility condition. The derivation of the two-sided exit identity is based on the Markov property, which in this context is equivalent to the cocycle property of the Laplace transform, together with the recursive equation at the upper level. For reflection at the running infimum, a finite $\bar p$-geometric method of steps realizes a certain Picard map that allows to find the exit matrix. As a result, we construct a Doob martingale related to the first passage time of the reflected Markov-modulated AIMD.