IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v31y2018i4d10.1007_s10959-017-0765-1.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-017-0765-1
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-017-0765-1?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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. 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.
    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. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. 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).
    4. 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.
    5. Natalia Cardona-Tobón & Juan Carlos Pardo, 2024. "Explosion Rates for Continuous-State Branching Processes in a Lévy Environment," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3395-3425, November.
    6. Xu, Wei, 2023. "Asymptotics for exponential functionals of random walks," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 1-42.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Long, Hongwei & Ma, Chunhua & Shimizu, Yasutaka, 2017. "Least squares estimators for stochastic differential equations driven by small Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1475-1495.
    2. Ying Jiao & Chunhua Ma & Simone Scotti & Chao Zhou, 2018. "The Alpha-Heston Stochastic Volatility Model," Papers 1812.01914, arXiv.org.
    3. Fontana, Claudio & Gnoatto, Alessandro & Szulda, Guillaume, 2023. "CBI-time-changed Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 323-349.
    4. Giorgia Callegaro & Andrea Mazzoran & Carlo Sgarra, 2019. "A Self-Exciting Modelling Framework for Forward Prices in Power Markets," Papers 1910.13286, arXiv.org.
    5. 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).
    6. Yang, Xu, 2017. "Maximum likelihood type estimation for discretely observed CIR model with small α-stable noises," Statistics & Probability Letters, Elsevier, vol. 120(C), pages 18-27.
    7. Ying Jiao & Chunhua Ma & Simone Scotti, 2017. "Alpha-CIR model with branching processes in sovereign interest rate modeling," Finance and Stochastics, Springer, vol. 21(3), pages 789-813, July.
    8. Aur'elien Alfonsi & Guillaume Szulda, 2024. "On non-negative solutions of stochastic Volterra equations with jumps and non-Lipschitz coefficients," Papers 2402.19203, arXiv.org, revised Jul 2024.
    9. Matyas Barczy & Mohamed Ben Alaya & Ahmed Kebaier & Gyula Pap, 2016. "Asymptotic properties of maximum likelihood estimator for the growth rate for a jump-type CIR process based on continuous time observations," Papers 1609.05865, arXiv.org, revised Aug 2017.
    10. Friesen, Martin & Jin, Peng & Rüdiger, Barbara, 2020. "Existence of densities for multi-type continuous-state branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5426-5452.
    11. Grosjean, Nicolas & Huillet, Thierry, 2016. "Deterministic versus stochastic aspects of superexponential population growth models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 455(C), pages 27-37.
    12. Micha{l} Barski & Rafa{l} {L}ochowski, 2024. "Affine term structure models driven by independent L\'evy processes," Papers 2402.07503, arXiv.org.
    13. Ying Jiao & Chunhua Ma & Simone Scotti & Chao Zhou, 2021. "The Alpha‐Heston stochastic volatility model," Mathematical Finance, Wiley Blackwell, vol. 31(3), pages 943-978, July.
    14. Doudou Li & Vladimir Vatutin & Mei Zhang, 2021. "Subcritical Branching Processes in Random Environment with Immigration Stopped at Zero," Journal of Theoretical Probability, Springer, vol. 34(2), pages 874-896, June.
    15. Matyas Barczy & Mohamed Ben Alaya & Ahmed Kebaier & Gyula Pap, 2017. "Asymptotic properties of maximum likelihood estimator for the growth rate of a stable CIR process based on continuous time observations," Papers 1711.02140, arXiv.org, revised Feb 2019.
    16. Ying Jiao & Chunhua Ma & Simone Scotti, 2017. "Alpha-CIR Model with Branching Processes in Sovereign Interest Rate Modelling," Post-Print hal-01275397, HAL.
    17. Ascione, Giacomo & Mehrdoust, Farshid & Orlando, Giuseppe & Samimi, Oldouz, 2023. "Foreign Exchange Options on Heston-CIR Model Under Lévy Process Framework," Applied Mathematics and Computation, Elsevier, vol. 446(C).
    18. Ying Jiao & Chunhua Ma & Simone Scotti, 2016. "Alpha-CIR Model with Branching Processes in Sovereign Interest Rate Modelling," Working Papers hal-01275397, HAL.
    19. Mátyás Barczy & Kristóf Körmendi & Gyula Pap, 2016. "Statistical inference for critical continuous state and continuous time branching processes with immigration," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(7), pages 789-816, October.
    20. Li, Zenghu & Xu, Wei, 2018. "Asymptotic results for exponential functionals of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 108-131.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:31:y:2018:i:4:d:10.1007_s10959-017-0765-1. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.