Fluctuation theory of continuous-time skip-free downward Markov chains with applications to branching processes with immigration

TL;DR

This paper introduces a new fluctuation theory for continuous-time skip-free Markov chains, extending existing discrete-time results and applying it to branching processes with immigration to derive new fluctuation identities.

Contribution

It develops a novel methodology for continuous-time skip-free Markov chains and applies it to derive comprehensive fluctuation identities for branching processes with immigration.

Findings

Derived new fluctuation identities for Markov branching processes with immigration

Unified approach for fluctuation identities in continuous-time skip-free Markov chains

Recovered identities for skip-free downward compound Poisson processes

Abstract

We develop a new methodology for the fluctuation theory of continuous-time skip-free Markov chains, extending the recent work of Choi and Patie [5] for discrete-time skip-free Markov chains. As the main application we use it to derive a full set of fluctuation identities regarding exiting a finite or infinite interval for Markov branching processes with immigration, thereby uncovering many new results for this classical family of continuous-time Markov chains. The theory also allows us to recover in a simple manner fluctuation identities for skip-free downward compound Poisson processes.