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Markov branching processes with disasters: Extinction, survival and duality to p-jump processes

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  • Hermann, Felix
  • Pfaffelhuber, Peter

Abstract

A p-jump process is a piecewise deterministic Markov process with multiplicative jumps by a factor of p. We prove a limit theorem for such processes on the unit interval. Via duality with respect to probability generating functions, we deduce limiting results for the survival probabilities of time-homogeneous branching processes with arbitrary offspring distributions, underlying binomial disasters. Extending this method, we obtain corresponding results for time-inhomogeneous birth–death processes underlying time-dependent binomial disasters and continuous state branching processes with p-jumps.

Suggested Citation

  • Hermann, Felix & Pfaffelhuber, Peter, 2020. "Markov branching processes with disasters: Extinction, survival and duality to p-jump processes," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2488-2518.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:4:p:2488-2518
    DOI: 10.1016/j.spa.2019.07.011
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    References listed on IDEAS

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    1. González Casanova, Adrián & Kurt, Noemi & Wakolbinger, Anton & Yuan, Linglong, 2016. "An individual-based model for the Lenski experiment, and the deceleration of the relative fitness," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2211-2252.
    2. Pakes, Anthony G. & Pollett, P. K., 1989. "The supercritical birth, death and catastrophe process: limit theorems on the set of extinction," Stochastic Processes and their Applications, Elsevier, vol. 32(1), pages 161-170, June.
    3. Pakes, Anthony G., 1986. "The Markov branching-castastrophe process," Stochastic Processes and their Applications, Elsevier, vol. 23(1), pages 1-33, October.
    4. Peng, NanFu & Pearl, Dennis K. & Chan, Wenyaw & Bartoszynski, Robert, 1993. "Linear birth and death processes under the influence of disasters with time-dependent killing probabilities," Stochastic Processes and their Applications, Elsevier, vol. 45(2), pages 243-258, April.
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