Analytic property of generalized scale functions for standard processes with no negative jumps and its application to quasi-stationary distributions

TL;DR

This paper studies the mathematical properties of generalized scale functions for certain stochastic processes, extending their analytic features and applying these results to determine conditions for quasi-stationary distributions.

Contribution

It characterizes generalized scale functions as solutions to integral equations, extends them analytically, and introduces a boundary classification to analyze quasi-stationary distributions.

Findings

Generalized scale functions are solutions to Volterra integral equations.

The paper derives a resolvent identity for these functions.

A new boundary classification extends Feller's theory for process analysis.

Abstract

For a generalized scale function of standard processes, we characterize it as a unique solution to a Volterra type integral equation. This allows us to extend it to an entire function and to derive a useful identity that we call the resolvent identity. We apply this result to study the existence of a quasi-stationary distribution for the processes killed at hitting boundaries. A new classification of the boundary, which is a natural extension of Feller's for one-dimensional diffusions, is introduced and plays a central role to characterize the existence.