First passage problems for upwards skip-free random walks via the $\Phi,W,Z$ paradigm

TL;DR

This paper develops a unified theory of scale functions for upwards skip-free discrete-time random walks, extending existing spectrally negative Lévy process theory, and applies it to actuarial risk models involving dividends and capital injections.

Contribution

It introduces the novel two-parameter Z scale functions for discrete-time walks, bridging discrete and continuous models, and applies this framework to actuarial risk management problems.

Findings

Introduction of Z scale functions for the first time in this context

Unified framework for discrete and continuous models

Application to dividends and capital injection problems

Abstract

We develop the theory of the $W$ and $Z$ scale functions for right-continuous (upwards skip-free) discrete-time discrete-space random walks, along the lines of the analogue theory for spectrally negative L\'evy processes. Notably, we introduce for the first time in this context the one and two-parameter scale functions $Z$, which appear for example in the joint problem of deficit at ruin and time of ruin, and in problems concerning the walk reflected at an upper barrier. Comparisons are made between the various theories of scale functions as one makes time and/or space continuous. The theory is shown to be fruitful by providing a convenient unified framework for studying dividends-capital injection problems under various objectives, for the so-called compound binomial risk model of actuarial science.