Scale Functions of Stochastic Processes without Positive Jumps

Abstract

The study of the behavior of spectrally negative Lévy processes has been a subject of interest for many years. In particular, around 2000, there was a major advancement in the understanding of scale functions for these processes. These scale functions are not only of theoretical interest but are also essential in practical fields such as ruin theory, optimal stopping problems, and optimal control problems. Moreover, recent research has explored the scale functions for other stochastic processes that lack positive jumps, aiming to apply these functions to similar challenges. In this study, we discuss the properties of scale functions for spectrally negative Lévy processes and discuss their applications. In addition, we provide an overview of the scale functions for several other stochastic processes without positive jumps.