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A continuous-state polynomial branching process

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  • Li, Pei-Sen

Abstract

A continuous-state polynomial branching process is constructed as the pathwise unique solution of a stochastic integral equation with absorbing boundary condition. The process can also be obtained from a spectrally positive Lévy process through Lamperti type transformations. The extinction and explosion probabilities and the mean extinction and explosion times are computed explicitly. Some of those are also new for the classical linear branching process. We present necessary and sufficient conditions for the process to extinguish or explode in finite times. In the critical or subcritical case, we give a construction of the process coming down from infinity. Finally, it is shown that the continuous-state polynomial branching process arises naturally as the rescaled limit of a sequence of discrete-state processes.

Suggested Citation

  • Li, Pei-Sen, 2019. "A continuous-state polynomial branching process," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2941-2967.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:8:p:2941-2967
    DOI: 10.1016/j.spa.2018.08.013
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    References listed on IDEAS

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    1. Berestycki, J. & Döring, L. & Mytnik, L. & Zambotti, L., 2015. "Hitting properties and non-uniqueness for SDEs driven by stable processes," Stochastic Processes and their Applications, Elsevier, vol. 125(3), pages 918-940.
    2. Duhalde, Xan & Foucart, Clément & Ma, Chunhua, 2014. "On the hitting times of continuous-state branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 124(12), pages 4182-4201.
    3. Fu, Zongfei & Li, Zenghu, 2010. "Stochastic equations of non-negative processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 306-330, March.
    4. Chen, Anyue & Li, Junping & Ramesh, N.I., 2008. "Probabilistic approach in weighted Markov branching processes," Statistics & Probability Letters, Elsevier, vol. 78(6), pages 771-779, April.
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    Cited by:

    1. Le, V., 2022. "On the extinction of continuous state branching processes with competition," Statistics & Probability Letters, Elsevier, vol. 185(C).
    2. Pei-Sen Li & Xiaowen Zhou, 2023. "Integral Functionals for Spectrally Positive Lévy Processes," Journal of Theoretical Probability, Springer, vol. 36(1), pages 297-314, March.
    3. Ren, Yan-Xia & Xiong, Jie & Yang, Xu & Zhou, Xiaowen, 2022. "On the extinction-extinguishing dichotomy for a stochastic Lotka–Volterra type population dynamical system," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 50-90.
    4. Foucart, Clément & Li, Pei-Sen & Zhou, Xiaowen, 2020. "On the entrance at infinity of Feller processes with no negative jumps," Statistics & Probability Letters, Elsevier, vol. 165(C).
    5. Vidmar, Matija, 2023. "Complete monotonicity of time-changed Lévy processes at first passage," Statistics & Probability Letters, Elsevier, vol. 193(C).
    6. Foucart, Clément & Vidmar, Matija, 2024. "Continuous-state branching processes with collisions: First passage times and duality," Stochastic Processes and their Applications, Elsevier, vol. 167(C).

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