A continuous-time Ehrenfest model with catastrophes and its jump-diffusion approximation

TL;DR

This paper introduces a continuous-time Ehrenfest model with catastrophes, analyzes its probabilistic behavior, and develops a jump-diffusion approximation leading to an Ornstein-Uhlenbeck process with catastrophes.

Contribution

It provides a novel analysis of a catastrophe-affected Ehrenfest model and derives a jump-diffusion approximation for better understanding of its dynamics.

Findings

Transient and steady-state probabilities characterized.

First passage time through state 0 analyzed.

Jump-diffusion approximation to Ornstein-Uhlenbeck process derived.

Abstract

We consider a continuous-time Ehrenfest model defined over the integers from -N to N, and subject to catastrophes occurring at constant rate. The effect of each catastrophe instantaneously resets the process to state 0. We investigate both the transient and steady-state probabilities of the above model. Further, the first passage time through state 0 is discussed. We perform a jump-diffusion approximation of the above model, which leads to the Ornstein-Uhlenbeck process with catastrophes. The underlying jump-diffusion process is finally studied, with special attention to the symmetric case arising when the Ehrenfest model has equal upward and downward transition rates.