Fluctuation theory for upwards skip-free L\'evy chains

TL;DR

This paper develops a fluctuation theory and explicit scale functions for upwards skip-free Lévy chains, a class of compound Poisson processes, enabling easier calculation and analysis of their probabilistic behavior.

Contribution

It introduces a new fluctuation theory for upwards skip-free Lévy chains, including explicit recursive formulas for scale functions, enhancing analytical tractability.

Findings

Scale functions admit a linear recursion for bounded support.

Explicit calculations of scale functions are possible in this setting.

Several examples demonstrate the applicability of the theory.

Abstract

A fluctuation theory and, in particular, a theory of scale functions is developed for upwards skip-free L\'evy 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 L\'evy 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.