Skip to main content

Continuous Time Mixed State Branching Processes and Stochastic Equations

Abstract

A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes. The process can also be obtained by the pathwise unique solution to a stochastic equation system. From the stochastic equation system we derive the distribution of local jumps and give the exponential ergodicity in Wasserstein-type distances of the transition semigroup. Meanwhile, we study immigration structures associated with the process and prove the existence of the stationary distribution of the process with immigration.

This is a preview of subscription content, access via your institution.

References

  1. Athreya K B, Ney P E. Branching Processes. Berlin: Springer, 1972

    Book  Google Scholar 

  2. Bertoin J, Fontbona J, Martínez S. On prolific indivials in a supercritical continuous-state branching process. J Appl Probab, 2008, 45: 714–726

    MathSciNet  Article  Google Scholar 

  3. Bertoin J, Le Gall J F. Stochastic flows associated to coalescent processes III: Limit theorems. Illinois J Math, 2006, 50: 147–181

    MathSciNet  Article  Google Scholar 

  4. Chen M. From Markov Chains to Non-Equilibrium Particle Systems. 2nd ed. Singapore: World Sci, 2004

    Book  Google Scholar 

  5. Dawson D A, Li Z. Skew convolution semigroups and affine Markov processes. Ann Probab, 2006, 34: 1103–1142

    MathSciNet  Article  Google Scholar 

  6. Dawson D A, Li Z. Stochastic equations, flows and measure-valued processes. Ann Probab, 2012, 40: 813–857

    MathSciNet  Article  Google Scholar 

  7. Dellacherie C, Meyer P A. Probabilites and Potential. Chapters V–VIII. Amsterdam: NorthHolland, 1982

    MATH  Google Scholar 

  8. Ethier S N, Kurtz T G. Markov Processes: Characterization and Convergence. New York: John Wiley and Sons Inc, 1986

    Book  Google Scholar 

  9. Feketa D, Fontbona J, Kyprianou A E. Skeletal stochastic differential equations for continuous-state branching processes. J Appl Probab, 2019, 56: 1122–1150

    MathSciNet  Article  Google Scholar 

  10. Feketa D, Fontbona J, Kyprianou A E. Skeletal stochastic differential equations for superprocesses. J Appl Probab, 2020, 57: 1111–1134

    MathSciNet  Article  Google Scholar 

  11. Feller W. Diffusion processes in genetics//Proceedings 2nd Berkeley Symp Math Statist Probab. Berkeley and Los Angeles: University of California Press, 1950: 227–246

    Google Scholar 

  12. Friesen M, Jin P, Kremer J, Rüdiger B. Exponential ergodicity for stochastic equations of nonnegative processes with jumps. 2019[2019-07-15]. https://arxiv.org/abs/1902.02833

  13. Fu Z, Li Z. Stochastic equations of non-negative processes with jumps. Stochastic Process Appl, 2010, 120: 306–330

    MathSciNet  Article  Google Scholar 

  14. He X, Li Z. Distributions of jumps in a continuous-state branching process with immigration. J Appl Probab, 2016, 53: 1166–1177

    MathSciNet  Article  Google Scholar 

  15. Ikeda N, Watanabe S. Stochastic Differential Equations and Diffusion Processes. Amsterdam/Tokyo: North-Holland/Kodansha, 1989

    MATH  Google Scholar 

  16. Jiřina M. Stochastic branching processes with continuous state space. Czechoslovak Math J, 1958, 8: 292–313

    MathSciNet  Article  Google Scholar 

  17. Ji L, Li Z. Moments of continuous-state branching processes with or without immigration. Acta Math Appl Sin Engl Ser, 2020, (2): 361–373

  18. Ji L, Zheng X. Moments of continuous-state branching processes in Lévy random environments. Acta Math Sci, 2019, 39B(3): 781–796

    Article  Google Scholar 

  19. Jiao Y, Ma C, Scotti S. Alpha-CIR model with branching processes in sovereign interest rate modeling. Finance Stoch, 2017, 21: 789–813

    MathSciNet  Article  Google Scholar 

  20. Jin P, Kremer J, Rüdiger B. Existence of limiting distribution for affine processes. J Math Anal Appl, 2020, 486: 123912, 31 pp

    MathSciNet  Article  Google Scholar 

  21. Kawazu K, Watanabe S. Branching processes with immigration and related limit theorems. Theory Probab Appl, 1971, 16: 36–54

    MathSciNet  Article  Google Scholar 

  22. Li Z. A limit theorem for discrete Galton-Watson branching processes with immigration. J Appl Probab, 2006, 43: 289–295

    MathSciNet  Article  Google Scholar 

  23. Li Z. Measure-Valued Branching Markov Processes. Berlin: Springer, 2011

    Book  Google Scholar 

  24. Li Z, Ma C. Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross model. Stochastic Process Appl, 2015, 125: 3196–3233

    MathSciNet  Article  Google Scholar 

  25. Li Z. Continuous-state branching processes with immigration//Jiao Y. From Probability to Finance, Mathematical Lectures from Peking University. Singapore: Springer, 2020: 1–69

    Google Scholar 

  26. Li Z. Ergodicities and exponential ergodicities of Dawson-Watanabe type processes. 2020[2020-02-22]. https://arxiv.org/abs/2002.09111

  27. Ma C. A limit theorem of two-type Galton-Watson branching processes with immigration. Stat Prob Lett, 2009, 79: 1710–1716

    MathSciNet  Article  Google Scholar 

  28. Ma R. Stochastic equations for two-type continuous-state branching processes with immigration. Acta Math Sinica Engl Ser, 2013, 29: 287–294

    MathSciNet  Article  Google Scholar 

  29. Ma R. Stochastic equations for two-type continuous-state branching processes with immigration and competition. Stat Prob Lett, 2014, 91: 83–89

    MathSciNet  Article  Google Scholar 

  30. Pardoux E. Probabilistic Models of Population Evolution: Scaling Limits, Genealogies and Interactions. Switzerland: Springer, 2016

    Book  Google Scholar 

  31. Pinsky M A. Limit theorems for continuous state branching processes with immigration. Bull Amer Math Soc, 1972, 78: 242–244

    MathSciNet  Article  Google Scholar 

  32. Sato K, Yamazato M. Operator-self-decomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type. Stochastic Process Appl, 1984, 17: 73–100

    MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Shukai Chen.

Additional information

The authors were supported by the National Key R&D Program of China (2020YFA0712900) and the National Natural Science Foundation of China (11531001).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chen, S., Li, Z. Continuous Time Mixed State Branching Processes and Stochastic Equations. Acta Math Sci 41, 1445–1473 (2021). https://doi.org/10.1007/s10473-021-0504-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-021-0504-7

Key words

  • mixed state branching process
  • weak convergence
  • stochastic equation system
  • Wasserstein-type distance
  • stationary distribution

2010 MR Subject Classification

  • 60J80
  • 60H20
  • 60G51