Generalized scale functions of standard processes with no positive jumps

TL;DR

This paper introduces a generalized concept of scale functions for standard processes with no positive jumps, extending the theory beyond spectrally negative Lévy processes, and explores their applications in hitting times, potential measures, and duality.

Contribution

It defines new scale functions for general standard processes with no positive jumps using excursion measures, broadening the scope of existing theories.

Findings

Derived Laplace transforms of hitting times

Analyzed potential measures using new scale functions

Explored duality properties of the processes

Abstract

As a generalization of scale functions of spectrally negative L\'evy processes, we define scale functions of general standard processes with no positive jumps. For this purpose, we utilize excursion measures. Using our new scale functions we study Laplace transforms of hitting times, potential measures and duality.