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Fluctuation theory for upwards skip-free Lévy chains

A fluctuation theory and, in particular, a theory of scale functions is developed for upwards skip-free Lévy chains, i.e. for right-continuous random walks embedded into continuous time as compound Poisson processes. This is done by analogy to the spectrally negative class of Lévy processes -- several results, however, can be made more explicit/exhaustive in our compound Poisson setting. In particular, the scale functions admit a linear recursion, of constant order when the support of the jump measure is bounded, by means of which they can be calculated -- some examples are considered.