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On the limit distributions of continuous-state branching processes with immigration

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  • Keller-Ressel, Martin
  • Mijatović, Aleksandar

Abstract

We consider the class of continuous-state branching processes with immigration (CBI-processes), introduced by Kawazu and Watanabe (1971) [10] and their limit distributions as time tends to infinity. We determine the Lévy–Khintchine triplet of the limit distribution and give an explicit description in terms of the characteristic triplet of the Lévy subordinator and the scale function of the spectrally positive Lévy process, which describe the immigration resp. branching mechanism of the CBI-process. This representation allows us to describe the support of the limit distribution and characterize its absolute continuity and asymptotic behavior at the boundary of the support, generalizing several known results on self-decomposable distributions.

Suggested Citation

  • Keller-Ressel, Martin & Mijatović, Aleksandar, 2012. "On the limit distributions of continuous-state branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 122(6), pages 2329-2345.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:6:p:2329-2345
    DOI: 10.1016/j.spa.2012.03.012
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    References listed on IDEAS

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    1. Masuda, H. & Yoshida, N., 2005. "Asymptotic expansion for Barndorff-Nielsen and Shephard's stochastic volatility model," Stochastic Processes and their Applications, Elsevier, vol. 115(7), pages 1167-1186, July.
    2. Martin Keller-Ressel & Thomas Steiner, 2008. "Yield curve shapes and the asymptotic short rate distribution in affine one-factor models," Finance and Stochastics, Springer, vol. 12(2), pages 149-172, April.
    3. Sato, Ken-iti & Yamazato, Makoto, 1984. "Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type," Stochastic Processes and their Applications, Elsevier, vol. 17(1), pages 73-100, May.
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    Cited by:

    1. Mijatović, Aleksandar & Vidmar, Matija & Jacka, Saul, 2015. "Markov chain approximations to scale functions of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 125(10), pages 3932-3957.
    2. Martin Friesen & Sven Karbach, 2024. "Stationary covariance regime for affine stochastic covariance models in Hilbert spaces," Finance and Stochastics, Springer, vol. 28(4), pages 1077-1116, October.
    3. Likuan Qin & Vadim Linetsky, 2014. "Positive Eigenfunctions of Markovian Pricing Operators: Hansen-Scheinkman Factorization, Ross Recovery and Long-Term Pricing," Papers 1411.3075, arXiv.org, revised Sep 2015.
    4. Matyas Barczy & Leif Doering & Zenghu Li & Gyula Pap, 2013. "Stationarity and ergodicity for an affine two factor model," Papers 1302.2534, arXiv.org, revised Sep 2013.
    5. 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.
    6. 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.
    7. 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.
    8. Micha{l} Barski & Rafa{l} {L}ochowski, 2024. "Affine term structure models driven by independent L\'evy processes," Papers 2402.07503, arXiv.org.
    9. Behme, Anita & Schnurr, Alexander, 2018. "Laplace symbols and invariant distributions," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 217-223.
    10. Barczy, Mátyás & Ben Alaya, Mohamed & Kebaier, Ahmed & Pap, Gyula, 2018. "Asymptotic properties of maximum likelihood estimator for the growth rate for a jump-type CIR process based on continuous time observations," Stochastic Processes and their Applications, Elsevier, vol. 128(4), pages 1135-1164.

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