On a finite quasi birth-death process with catastrophes and its diffusion approximation

TL;DR

This paper analyzes a finite quasi-birth-death process with catastrophes, exploring its transient and long-term behavior, and introduces a diffusion approximation converging to a reflected Ornstein-Uhlenbeck jump diffusion.

Contribution

It provides a detailed analysis of a QBD process with catastrophes and establishes a new diffusion approximation for its behavior.

Findings

The process exhibits specific transient and steady-state characteristics.

The diffusion approximation converges to a reflected Ornstein-Uhlenbeck jump diffusion.

The phase switching mechanism influences the process dynamics significantly.

Abstract

We study a multi-type Ehrenfest process modeled as a finite quasi-birth-death (QBD) process. We assume that the transitions are allowed only to the two adjacent levels of the same phase and are characterized by linear rates. The crucial element lies in the phase switching mechanism at the origin, which is governed by an irreducible stochastic matrix. The process evolution is interrupted by catastrophic events, whose occurrences are controlled by a Poisson process. Each catastrophe resets the system state to zero, initiating a new cycle of evolution until the next resetting event. We conduct a comprehensive analysis, addressing both the transient and long-term behavior of this process. Furthermore, we derive a diffusive approximation, by proving its convergence to a reflected Ornstein-Uhlenbeck jump diffusion process.