Explosion, implosion, and moments of passage times for continuous-time Markov chains: a semimartingale approach

TL;DR

This paper develops a semimartingale approach to analyze recurrence, explosion, and implosion phenomena in continuous-time Markov chains, providing sharp conditions and applying results to complex models.

Contribution

It introduces a unified semimartingale framework to quantify recurrence, explosion, and implosion in continuous-time Markov chains, with new conditions and phenomena.

Findings

Established general theorems on recurrence and passage time moments.

Derived sharp conditions for explosion and implosion phenomena.

Applied results to models with complex behaviors.

Abstract

We establish general theorems quantifying the notion of recurrence --- through an estimation of the moments of passage times --- for irreducible continuous-time Markov chains on countably infinite state spaces. Sharp conditions of occurrence of the phenomenon of explosion are also obtained. A new phenomenon of implosion is introduced and sharp conditions for its occurrence are proven. The general results are illustrated by treating models having a difficult behaviour even in discrete time.