**Matija Vidmar**

Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia; matija.vidmar@fmf.uni-lj.si

Received: 20 July 2018; Accepted: 16 September 2018; Published: 18 September 2018

**Abstract:** 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 the compound Poisson setting. Importantly, 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 presented. An application to the modeling of an insurance company's aggregate capital process is briefly considered.

**Keywords:** Lévy processes; non-random overshoots; skip-free random walks; fluctuation theory; scale functions; capital surplus process
