Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T13:22:42.526Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
∗∗ Postal address: Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong. Email address: [email protected]
∗∗∗ Postal address: School of Mathematics and Computational Science, Xiangtan University, Hunan, 411105, P. R. China Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Anderson, W. J. (1991). Continuous-Time Markov Chains. An Applications-Oriented Approach. Springer, New York.Google Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.Google Scholar
Chen, A. Y. and Renshaw, E. (1990). Markov branching processes with instantaneous immigration. Prob. Theory Relat. Fields 87, 209–240.Google Scholar
Chen, A. Y. and Renshaw, E. (1993). Existence and uniqueness criteria for conservative uni-instantaneous denumerable Markov processes. Prob. Theory Relat. Fields 94, 427–456.Google Scholar
Chen, A., Li, J., Hou, Z. and Ng, K. W. (2010). Decay properties and quasi-stationary distributions for stopped Markovian bulk-arrival and bulk-service queues. Queueing Systems 66, 275–311.Google Scholar
Coolen-Schrijner, P. and Van Doorn, E. A. (2006). Quasi-stationary distributions for birth-death processes with killing. J. Appl. Math. Stoch. Anal. 2006, 15pp.CrossRefGoogle Scholar
Feller, W. (1959). The birth and death processes as diffusion processes. J. Math. Pures Appl. 38, 301–345.Google Scholar
Ferrari, P. A., Kesten, H., Martinez, S. and Picco, P. (1995). Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Prob. 23, 501–521.CrossRefGoogle Scholar
Flaspohler, D. C. (1974). Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains. Ann. Inst. Statist. Math. 26, 351–356.Google Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.Google Scholar
Hou, Z. T. and Guo, Q. F. (1988). Homogeneous Denumerable Markov Processes. Springer, Berlin.Google Scholar
Kaplan, N., Sudbury, A. and Nilsen, T. S. (1975). A branching process with disasters. J. Appl. Prob. 12, 47–59.Google Scholar
Karlin, S. and Tavaré, S. (1982). Linear birth and death processes with killing. J. Appl. Prob. 19, 477–487.Google Scholar
Kingman, J. F. C. (1963). The exponential decay of Markov transition probabilities. Proc. London Math. Soc. (3) 13, 337–358.Google Scholar
Kolb, M. and Steinsaltz, D. (2012). Quasilimiting behavior for one-dimensional diffusions with killing. Ann. Prob. 40, 162–212.Google Scholar
Pakes, A. G. (1995). Quasi-stationary laws for Markov processes: examples of an always proximate absorbing state. Adv. Appl. Prob. 27, 120–145.Google Scholar
Pollett, P. K. (1988). Reversibility, invariance and μ-invariance. Adv. Appl. Prob. 20, 600–621.Google Scholar
Steinsaltz, D. and Evans, S. N. (2007). Quasistationary distributions for one-dimensional diffusions with killing. Trans. Amer. Math. Soc. 359, 1285–1324.Google Scholar
Van Doorn, E. A. (1985). Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process. Adv. Appl. Prob. 17, 514–530.Google Scholar
Van Doorn, E. A. (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. Appl. Prob. 23, 683–700.CrossRefGoogle Scholar
Van Doorn, E. A. (2012). Conditions for the existence of quasi-stationary distributions for birth-death processes with killing. Stoch. Process. Appl. 122, 2400–2410.CrossRefGoogle Scholar
Van Doorn, E. A. and Pollett, P. K. (2011). {Quasi-stationary distributions}. Memorandum 1945, Department of Applied Mathematics, University of Twente, Enschede. Available at http://eprints.eemcs.utwente.nl/20245.Google Scholar
Van Doorn, E. A. and Zeifman, A. I. (2005). Birth-death processes with killing. Statist. Prob. Lett. 72, 33–42.Google Scholar
Van Doorn, E. A. and Zeifman, A. I. (2005). Extinction probability in a birth-death process with killing. J. Appl. Prob. 42, 185–198.Google Scholar
Vere-Jones, D. (1962). Geometric ergodicity in denumerable Markov chains. Quart. J. Math. Oxford (2) 13, 7–28.Google Scholar