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...

For a spectrally one-sided Lévy process, we extend various two-sided exit identities to the situation when the process is only observed at arrival epochs of an independent Poisson process. In addition, we consider exit problems of this type for processes reflected either from above or from below. The resulting Laplace transforms of the main quantities of interest are in terms of scale functions and turn out to be simple analogues of the classical formulas.

Let M and τ be the supremum and its time of a Lévy process X on some finite time interval. It is shown that zooming in on X at its supremum, that is, considering ((X τ +tε − M )/a ε ) t∈R as ε ↓ 0, results in (ξ t ) t∈R constructed from two independent processes having the laws of some self-similar Lévy process X conditioned to stay positive and negative. This holds when X is in the domain of attraction of X under the zoomingin procedure as opposed to the classical zooming out of Lamperti (1962). As an application of this result we establish a limit theorem for the discretization errors in simulation of supremum and its time, which extends the result of Asmussen, Glynn, and Pitman (1995) for the Brownian motion. Additionally, complete characterization of the domains of attraction when zooming in on a Lévy process at 0 is provided.2010 Mathematics Subject Classification. Primary 60G51, 60F17; secondary 60G18, 60G52.

In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions. This result provides us with a technique that can be used to derive various further identities.