Continuous-State Branching Processes in Lévy Random Environments

Article

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 \(-1\). 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.

Keywords

Continuous-state branching process Random environment Lévy process Transition semigroup Backward stochastic equation Survival probability Immigration Ergodicity 

Mathematics Subject Classification (2010)

Primary 60J80 60K37 Secondary 60H20 60G51 

References

  1. 1.
    Afanasyev, V.I., Böinghoff, C., Kersting, G., Vatutin, V.A.: Limit theorems for a weakly subcritical branching process in a random environment. J. Theor. Probab. 25, 703–732 (2012)CrossRefMATHGoogle Scholar
  2. 2.
    Afanasy’ev, V.I., Geiger, J., Kersting, G., Vatutin, V.A.: Criticality for branching processes in random environment. Ann. Probab. 33, 645–673 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Aliev, S.A.: A limit theorem for the Galton–Watson branching processes with immigration. Ukrainian Math. J. 37, 535–538 (1985)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bansaye, V., Millan, J.C.P., Smadi, C.: On the extinction of continuous state branching processes with catastrophes. Electron. J. Probab. 18(106), 1–31 (2013)MathSciNetMATHGoogle Scholar
  5. 5.
    Böinghoff, C., Hutzenthaler, M.: Branching diffusions in random environment. Markov Process. Relat. Fields 18, 269–310 (2012)MathSciNetMATHGoogle Scholar
  6. 6.
    Dawson, D.A., Li, Z.: Skew convolution semigroups and affine Markov processes. Ann. Probab. 34, 1103–1142 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dawson, D.A., Li, Z.: Stochastic equations, flows and measure-valued processes. Ann. Probab. 40, 813–857 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    El Karoui, N., Méléard, S.: Martingale measures and stochastic calculus. Probab. Theory Relat. Fields 84, 83–101 (1990)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986)CrossRefMATHGoogle Scholar
  10. 10.
    Feller, W.: Diffusion processes in genetics. In: Proceedings 2nd Berkeley Symp. Math. Statist. Probab., 1950, pp. 227–246. Univ. California Press, Berkeley and Los Angeles (1951)Google Scholar
  11. 11.
    Fu, Z., Li, Z.: Stochastic equations of non-negative processes with jumps. Stoch. Process. Appl. 120, 306–330 (2010)CrossRefMATHGoogle Scholar
  12. 12.
    Guivarch, Y., Liu, Q.S.: Propriétés asymptotiques des processus de branchement en environnement aléatoire. C.R. Acad. Sci. Paris Sér. I Math. 332, 339–344 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hairer, M., Mattingly, J.: The strong Feller property for singular stochastic PDEs. arXiv:1610.03415 (2016)
  14. 14.
    Helland, I.S.: Minimal conditions for weak convergence to a diffusion process on the line. Ann. Probab. 9, 429–452 (1981)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hutzenthaler, M.: Supercritical branching diffusions in random environment. Electron. Commun. Probab. 16, 781–791 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Jiřina, M.: Stochastic branching processes with continuous state space. Czech. Math. J. 8, 292–313 (1958)MathSciNetMATHGoogle Scholar
  17. 17.
    Kawazu, K., Watanabe, S.: Branching processes with immigration and related limit theorems. Theory Probab. Appl. 16, 36–54 (1971)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Keiding, N.: Extinction and exponential growth in random environments. Theor. Popul. Biol. 8, 49–63 (1975)CrossRefMATHGoogle Scholar
  19. 19.
    Kurtz, T.G.: Diffusion approximations for branching processes. In: Branching processes (Conf., Saint Hippolyte, Que., 1976), Vol. 5, 269–292 (1978)Google Scholar
  20. 20.
    Lamperti, J.: The limit of a sequence of branching processes. Z. Wahrsch. verw. Geb. 7, 271–288 (1967a)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lamperti, J.: Continuous state branching processes. Bull. Am. Math. Soc. 73, 382–386 (1967b)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Li, Z.: A limit theorem for discrete Galton–Watson branching processes with immigration. J. Appl. Probab. 43, 289–295 (2006)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Li, Z.: Measure-Valued Branching Markov Processes. Springer, Heidelberg (2011)CrossRefMATHGoogle Scholar
  24. 24.
    Li, Z., Ma, C.: Asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model. Stoch. Process. Appl. 125, 3196–3233 (2015)CrossRefMATHGoogle Scholar
  25. 25.
    Li, Z., Pu, F.: Strong solutions of jump-type stochastic equations. Electron. Commun. Probab. 17(33), 1–13 (2012)CrossRefMATHGoogle Scholar
  26. 26.
    Li, Z., Xu, W.: Asymptotic results for exponential functionals of Lévy processes. arXiv:1601.02363v1 (2016)
  27. 27.
    Palau, S., Pardo, J.C.: Continuous state branching processes in random environment: the Brownian case. Stoch. Process. Appl. 127, 957–994 (2017)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Palau, S., Pardo, J.C.: Branching processes in a Lévy random environment. arXiv:1512.07691v1 (2015)
  29. 29.
    Smith, W.L.: Necessary conditions for almost sure extinction of a branching process with random environment. Ann. Math. Statist. 39, 2136–2140 (1968)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Smith, W.L., Wilkinson, W.E.: On branching processes in random environments. Ann. Math. Statist. 40, 814–827 (1969)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Sharpe, M.: General Theory of Markov Processes. Academic Press, New York (1988)MATHGoogle Scholar
  32. 32.
    Vatutin, V.A.: A limit theorem for an intermediate subcritical branching process in a random environment. Theory Probab. Appl. 48, 481–492 (2004)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Wang, F.: Harnack Inequalities for Stochastic Partial Differential Equations. Springer, Heidelberg (2013)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesBeijing Normal UniversityBeijingPeople’s Republic of China

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