On Time-Changed Birth-Death Processes with Catastrophes

TL;DR

This paper investigates time-changed birth-death processes with catastrophes, deriving fractional differential equations, distributional properties, and explicit formulas, and compares different parameter effects through simulations.

Contribution

It introduces new fractional models of birth-death processes with catastrophes using stable subordinators and provides explicit solutions and simulation algorithms.

Findings

Derived fractional differential equations for state probabilities.

Obtained explicit formulas for distribution of catastrophe times.

Provided simulation algorithms and comparison of expectation plots.

Abstract

We study two time-changed variants of the birth-death process with catastrophe where the time-changing components are the first hitting times of the stable subordinator and the tempered stable subordinator. For both the processes, we derive the governing system of fractional differential equations for their state probabilities. The Laplace transforms of these state probabilities are obtained in terms of those of the corresponding time-changed birth-death processes without catastrophes. We obtain the distribution of catastrophe occurrence times as well as the sojourn times within non-zero states. We study distributional properties of the first visit time to state zero in a particular case. Also, the first occurrence time of an effective catastrophe is studied. Moreover, we study the time-changed linear birth-death processes with catastrophes, derive the explicit expressions for its state probabilities, expectation and variance. For a specific case, we compare the expectation plots across different parameter values and provide an algorithm for simulating sample paths with illustrative plots.