On the nonexplosion and explosion for nonhomogeneous Markov pure jump processes

TL;DR

This paper develops new, sharp criteria for determining when nonhomogeneous Markov pure jump processes in Borel spaces will either explode or not, with conditions that are necessary under certain compactness and boundedness assumptions.

Contribution

It introduces novel drift-type conditions for explosion and nonexplosion, improving upon existing criteria and applicable to a broad class of processes in Borel spaces.

Findings

New necessary and sufficient conditions for nonexplosion.

Conditions are sharp and relate to weak Feller and local boundedness.

Applications demonstrated in various process settings.

Abstract

In this paper, we obtain new drift-type conditions for nonexplosion and explosion for nonhomogeneous Markov pure jump processes in Borel state spaces. The conditions are sharp; e.g., the one for nonexplosion is necessary if the state space is in addition locally compact and the $Q$-function satisfies weak Feller-type and local boundedness conditions. We comment on the relations of our conditions with the existing ones in the literature, and demonstrate some possible applications.