Continuous-State Branching Processes with Immigration

  • Zenghu LiEmail author
Part of the Mathematical Lectures from Peking University book series (MLPKU)


This work provides a brief introduction to continuous-state branching processes with or without immigration. The processes are constructed by taking rescaling limits of classical discrete-state branching models. We give quick developments of the martingale problems and stochastic equations of the continuous-state processes. The proofs here are more elementary than those appearing in the literature before. We have made them readable without requiring too much preliminary knowledge on branching processes and stochastic analysis. Using the stochastic equations, we give characterizations of the local and global maximal jumps of the processes. Under suitable conditions, their strong Feller property and exponential ergodicity are studied by a coupling method based on one of the stochastic equations.


Continuous-state branching process Immigration Rescaling limit Martingale problem Stochastic equation Strong Feller property Exponential ergodicity 

Mathematics Subject Classification (2010)

60J80 60J85 60H10 60H20 


  1. 1.
    Aliev, S.A.: A limit theorem for the Galton–Watson branching processes with immigration. Ukr. Math. J. 37, 535–538 (1985)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aliev, S.A., Shchurenkov, V.M.: Transitional phenomena and the convergence of Galton–Watson processes to Jiřina processes. Theory Probab. Appl. 27, 472–485 (1982)CrossRefGoogle Scholar
  3. 3.
    Athreya, K.B., Ney, P.E.: Branching Processes. Springer, Berlin (1972)Google Scholar
  4. 4.
    Bernis, G., Scotti, S.: Clustering effects through Hawkes processes. From Probability to Finance – Lecture Note of BICMR Summer School on Financial Mathematics. Series of Mathematical Lectures from Peking University. Springer, Berlin (2018+)Google Scholar
  5. 5.
    Bertoin, J., Le Gall, J.-F.: Stochastic flows associated to coalescent processes III: limit theorems. Ill. J. Math. 50, 147–181 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bienaymé, I.J.: De la loi de multiplication et de la durée des families. Soc. Philomat. Paris Extr. 5, 37–39 (1845)Google Scholar
  7. 7.
    Chung, K.L.: Lectures from Markov Processes to Brownian Motion. Springer, Heidelberg (1982)Google Scholar
  8. 8.
    Cox, J., Ingersoll, J., Ross, S.: A theory of the term structure of interest rate. Econometrica 53, 385–408 (1985)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dawson, D.A., Li, Z.: Construction of immigration superprocesses with dependent spatial motion from one-dimensional excursions. Probab. Theory Relat. Fields 127, 37–61 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dawson, D.A., Li, Z.: Skew convolution semigroups and affine Markov processes. Ann. Probab. 34, 1103–1142 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dawson, D.A., Li, Z.: Stochastic equations, flows and measure-valued processes. Ann. Probab. 40, 813–857 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dellacherie, C., Meyer, P.A.: Probabilities and Potential. North-Holland, Amsterdam (1982)Google Scholar
  13. 13.
    Duffie, D., Filipović, D., Schachermayer, W.: Affine processes and applications in finance. Ann. Appl. Probab. 13, 984–1053 (2003)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Duquesne, T., Labbé, C.: On the Eve property for CSBP. Electron. J. Probab. 19, Paper No. 6, 1–31 (2014)Google Scholar
  15. 15.
    El Karoui, N., Méléard, S.: Martingale measures and stochastic calculus. Probab. Theory Relat. Fields 84, 83–101 (1990)MathSciNetCrossRefGoogle Scholar
  16. 16.
    El Karoui, N., Roelly, S.: Propriétés de martingales, explosion et representation de Lévy–Khintchine d’une classe de processus de branchement à valeurs mesures. Stoch. Process. Appl. 38, 239–266 (1991)CrossRefGoogle Scholar
  17. 17.
    Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986)Google Scholar
  18. 18.
    Feller, W.: Diffusion processes in genetics. In: Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability (1951), University of California Press, Berkeley and Los Angeles, pp. 227–246 (1950)Google Scholar
  19. 19.
    Fu, Z., Li, Z.: Stochastic equations of nonnegative processes with jumps. Stoch. Process. Appl. 120, 306–330 (2010)CrossRefGoogle Scholar
  20. 20.
    Galton, F., Watson, H.W.: On the probability of the extinction of families. J. Anthropol. Inst. G. B. Irel. 4, 138–144 (1874)Google Scholar
  21. 21.
    Grey, D.R.: Asymptotic behaviour of continuous time, continuous state-space branching processes. J. Appl. Probab. 11, 669–677 (1974)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Grimvall, A.: On the convergence of sequences of branching processes. Ann. Probab. 2, 1027–1045 (1974)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Harris, T.E.: The Theory of Branching Processes. Springer, Berlin (1963)Google Scholar
  24. 24.
    He, X., Li, Z.: Distributions of jumps in a continuous-state branching process with immigration. J. Appl. Probab. 53, 1166–1177 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer, Heidelberg (1965)Google Scholar
  26. 26.
    Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland, Amsterdam (1989)Google Scholar
  27. 27.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Heidelberg (2003)Google Scholar
  28. 28.
    Jiao, Y., Ma, C., Scotti, S.: Alpha-CIR model with branching processes in sovereign interest rate modeling. Financ. Stoch. 21, 789–813 (2017)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Kawazu, K., Watanabe, S.: Branching processes with immigration and related limit theorems. Theory Probab. Appl. 16, 36–54 (1971)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Keller-Ressel, M., Schachermayer, W., Teichmann, J.: Affine processes are regular. Probab. Theory Relat. Fields 151, 591–611 (2011)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Kyprianou, A.E.: Fluctuations of Lévy Processes with Applications, 2nd edn. Springer, Heidelberg (2014)Google Scholar
  32. 32.
    Lambert, A.: The branching process with logistic growth. Ann. Appl. Probab. 15, 1506–1535 (2005)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Lamberton, D., Lapeyre, B.: Introduction to Stochastic Calculus Applied to Finance. Chapman and Hall, London (1996)Google Scholar
  34. 34.
    Lamperti, J.: The limit of a sequence of branching processes. Z. Wahrsch. verw. Geb. 7, 271–288 (1967)Google Scholar
  35. 35.
    Lamperti, J.: Continuous state branching processes. Bull. Am. Math. Soc. 73, 382–386 (1967)Google Scholar
  36. 36.
    Li, Z.: Immigration structures associated with Dawson–Watanabe superprocesses. Stoch. Process. Appl. 62, 73–86 (1996)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Li, Z.: Asymptotic behavior of continuous time and state branching processes. J. Aust. Math. Soc. Ser. A 68, 68–84 (2000)CrossRefGoogle Scholar
  38. 38.
    Li, Z.: Skew convolution semigroups and related immigration processes. Theory Probab. Appl. 46, 274–296 (2003)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Li, Z.: A limit theorem for discrete Galton–Watson branching processes with immigration. J. Appl. Probab. 43, 289–295 (2006)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Li, Z.: Measure-Valued Branching Markov Processes. Springer, Heidelberg (2011)Google Scholar
  41. 41.
    Li, Z.: Path-valued branching processes and nonlocal branching superprocesses. Ann. Probab. 42, 41–79 (2014)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Li, Z., Ma, C.: Catalytic discrete state branching models and related limit theorems. J. Theor. Probab. 21, 936–965 (2008)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Li, Z., Ma, C.: Asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model. Stoch. Process. Appl. 125, 3196–3233 (2015)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Li, Z., Shiga, T.: Measure-valued branching diffusions: immigrations, excursions and limit theorems. J. Math. Kyoto Univ. 35, 233–274 (1995)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Overbeck, L., Rydén, T.: Estimation in the Cox–Ingersoll–Ross model. Econom. Theory 13, 430–461 (1997)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Pardoux, E.: Probabilistic Models of Population Evolution: Scaling Limits, Genealogies and Interactions. Springer, Switzerland (2016)Google Scholar
  47. 47.
    Parthasarathy, K.R.: Probability Measures on Metric Spaces. Academic, New York (1967)Google Scholar
  48. 48.
    Pitman, J., Yor, M.: A decomposition of Bessel bridges. Z. Wahrsch. verw. Geb. 59, 425–457 (1982)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Situ, R.: Theory of Stochastic Differential Equations with Jumps and Applications. Springer, Heidelberg (2005)Google Scholar
  50. 50.
    Walsh, J.B.: An introduction to stochastic partial differential equations. Ecole d’Eté de Probabilités de Saint-Flour XIV-1984. Lecture Notes in Mathematics, vol. 1180, pp. 265–439. Springer, Berlin (1986)Google Scholar

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© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.School of Mathematical SciencesBeijing Normal UniversityBeijingChina

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