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Uniqueness and Decay Properties of Markov Branching Processes with Disasters

Part of: Markov processes

Published online by Cambridge University Press:  30 January 2018

Anyue Chen ,
Kai Wang Ng  and
Hanjun Zhang
Anyue Chen*
Affiliation:
South University of Science and Technology of China and University of Liverpool
Kai Wang Ng*
Affiliation:
University of Hong Kong
Hanjun Zhang*
Affiliation:
Xiangtan University
*
∗ Postal address: Department of Financial Mathematics and Financial Engineering, South University of Science and Technology of China, Shenzhen, Guangdong, 518055, P. R. China. Email address: chen.ay@sustc.edu.cn
∗∗ Postal address: Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong. Email address: kaing@hku.hk
∗∗∗ Postal address: School of Mathematics and Computational Science, Xiangtan University, Hunan, 411105, P. R. China Email address: zhanghanjunxtu@gmail.com
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Abstract

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In this paper we discuss the decay properties of Markov branching processes with disasters, including the decay parameter, invariant measures, and quasistationary distributions. After showing that the corresponding q-matrix Q is always regular and, thus, that the Feller minimal Q-process is honest, we obtain the exact value of the decay parameter λC. We show that the decay parameter can be easily expressed explicitly. We further show that the Markov branching process with disaster is always λC-positive. The invariant vectors, the invariant measures, and the quasidistributions are given explicitly.

Keywords

Markov branching processdisasterdecay parameterinvariant measureinvariant vectorquasistationary distribution

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J35: Transition functions, generators and resolvents 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 613 - 624
Copyright
© Applied Probability Trust 

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