Occupation densities in solving exit problems for Markov additive processes and their reflections

Abstract

This paper solves exit problems for spectrally negative Markov additive processes and their reflections. A so-called scale matrix, which is a generalization of the scale function of a spectrally negative Lévy process, plays a central role in the study of exit problems. Existence of the scale matrix was shown in [30, Thm. 3]. We provide a probabilistic construction of the scale matrix, and identify the transform. In addition, we generalize to the MAP setting the relation between the scale function and the excursion (height) measure. The main technique is based on the occupation density formula and even in the context of fluctuations of spectrally negative Lévy processes this idea seems to be new. Our representation of the scale matrix W (x) = e −Λx L(x) in terms of nice probabilistic objects opens up possibilities for further investigation of its properties. IntroductionThis paper solves exit problems for spectrally negative Markov Additive Processes (MAPs) and their reflections. Before entering our discussion on this subject we shall simply begin by defining the class of processes we intend to work with. Let (Ω, F , F, P) be a filtered probability space, with standard (right-continuous and augmented) filtration F = {F t : t ≥ 0}. On this probability space consider a real-valued càdlàg (right-continuous with left limits) process X = {X(t) : t ≥ 0} and a right-continuous jump process J = {J(t) : t ≥ 0} with a finite state space E = {1, . . . , N }, such that (X, J) is adapted to the filtration F. The process (X, J) is a MAP if, given {J(t) = i}, the pair (X(t + s) − X(t), J(t + s)) is independent of F t and has the same law as (X(s) − X(0), J(s)) given {J(0) = i} for all s, t ≥ 0 and i ∈ E. It is common to say that X is an additive component and J is a background process representing the environment. Importantly, a MAP has a very special structure, which we reveal in the following. It is immediate that J is a Markov chain. Furthermore, X evolves as some spectrally negative Lévy process X i while J = i. In addition, a transition of J from i to j = i triggers a jump of X distributed as U ij ≤ 0, where i, j ∈ E. All the above components are independent. This structure explains the other commonly used name for a MAP -'Markov-modulated Lévy process'.The use of MAPs is widespread, making it a classical model in applied probability with a variety of application areas, such as queues, insurance risk, inventories, data communication, finance, environmental problems and so forth, see [4, Ch. XI], [38, Ch. 7], [15,22,21,5,13,17,41,14].Throughout this paper we will consider only spectrally negative MAPs, where the additive component X has no positive jumps. We exclude the trivial cases when the paths of X are a.s. monotone. It is noted that phase-type distributions fit naturally into the framework of MAPs. Positive phase-type jumps can be incorporated into the model using a standard trick, see e.g. [5]. This is achieved by enlarging the number of states of the background process and replacing phasetype jumps by linear stretche...

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