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Continuous-State Branching Processes in Lévy Random Environments

Author

Listed:
  • Hui He

    (Beijing Normal University)

  • Zenghu Li

    (Beijing Normal University)

  • Wei Xu

    (Beijing Normal University)

Abstract

A general continuous-state branching processes in random environment (CBRE-process) is defined as the strong solution of a stochastic integral equation. The environment is determined by a Lévy process with no jump less than $$-1$$ - 1 . We give characterizations of the quenched and annealed transition semigroups of the process in terms of a backward stochastic integral equation driven by another Lévy process determined by the environment. The process hits zero with strictly positive probability if and only if its branching mechanism satisfies Grey’s condition. In that case, a characterization of the extinction probability is given using a random differential equation with blowup terminal condition. The strong Feller property of the CBRE-process is established by a coupling method. We also prove a necessary and sufficient condition for the ergodicity of the subcritical CBRE-process with immigration.

Suggested Citation

  • Hui He & Zenghu Li & Wei Xu, 2018. "Continuous-State Branching Processes in Lévy Random Environments," Journal of Theoretical Probability, Springer, vol. 31(4), pages 1952-1974, December.
    • Handle: RePEc:spr:jotpro:v:31:y:2018:i:4:d:10.1007_s10959-017-0765-1
    DOI: 10.1007/s10959-017-0765-1
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    References listed on IDEAS

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    1. V. I. Afanasyev & C. Böinghoff & G. Kersting & V. A. Vatutin, 2012. "Limit Theorems for Weakly Subcritical Branching Processes in Random Environment," Journal of Theoretical Probability, Springer, vol. 25(3), pages 703-732, September.
    2. Li, Zenghu & Ma, Chunhua, 2015. "Asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 3196-3233.
    3. Palau, S. & Pardo, J.C., 2017. "Continuous state branching processes in random environment: The Brownian case," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 957-994.
    4. 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.
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    Cited by:

    1. Grégoire Véchambre, 2022. "General Self-Similarity Properties for Markov Processes and Exponential Functionals of Lévy Processes," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2083-2144, December.
    2. 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.
    3. Shukai Chen & Rongjuan Fang & Xiangqi Zheng, 2023. "Wasserstein-Type Distances of Two-Type Continuous-State Branching Processes in Lévy Random Environments," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1572-1590, September.
    4. Xu, Wei, 2023. "Asymptotics for exponential functionals of random walks," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 1-42.

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