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Continuous state branching processes in random environment: The Brownian case

Author

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  • Palau, S.
  • Pardo, J.C.

Abstract

We consider continuous-state branching processes that are perturbed by a Brownian motion. These processes are constructed as the unique strong solution of a stochastic differential equation. The long-term behaviours are studied. In the stable case, the extinction and explosion probabilities are given explicitly. We find three regimes for the asymptotic behaviour of the explosion probability and five regimes for the asymptotic behaviour of the extinction probability. In the supercritical regime, the process conditioned on eventual extinction has three regimes for the asymptotic behaviour of the extinction probability. Finally, the process conditioned on non-extinction and the process with immigration are given.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:3:p:957-994
    DOI: 10.1016/j.spa.2016.07.006
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    References listed on IDEAS

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    1. Bingham, N. H., 1976. "Continuous branching processes and spectral positivity," Stochastic Processes and their Applications, Elsevier, vol. 4(3), pages 217-242, August.
    2. 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. 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.
    2. 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).
    3. Marguet, Aline & Smadi, Charline, 2024. "Spread of parasites affecting death and division rates in a cell population," Stochastic Processes and their Applications, Elsevier, vol. 168(C).

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