Scale functions of space-time changed processes with no positive jumps

TL;DR

This paper derives the scale functions for space-time changed processes with no positive jumps, linking them to original processes and integral equations, enabling analysis of various Markov processes.

Contribution

It expresses the scale functions of space-time changed processes with no positive jumps in terms of original scale functions and Volterra integral equations, expanding their applicability.

Findings

Expressed scale functions of space-time changed processes using original scale functions.

Connected scale functions of self-similar and branching processes to spectrally negative Lévy processes.

Provided integral equations for calculating scale functions of complex processes.

Abstract

The scale functions were defined for spectrally negative L\'evy processes and other strong Markov processes with no positive jumps, and have been used to characterize their behavior. In particular, I defined the scale functions for standard processes with no positive jumps using the excursion measures in Noba(2020). In this paper, we consider a standard process $X$ with no positive jumps and a standard process $Y$ defined by the space-time change of $X$. We express the scale functions of $Y$ using the scale functions of $X$ defined in Noba(2020) and the Volterra integral equation. From this result, we can express the scale functions of some important processes, such as positive or negative self-similar Markov processes with no positive jumps and continuous-state branching processes, using the scale function of spectrally negative L\'evy processes and the Volterra integral equations.