Skip to main content
SpringerLink
Log in

Catalytic Discrete State Branching Models and Related Limit Theorems

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

Catalytic discrete state branching processes with immigration are defined as strong solutions of stochastic integral equations. We provide main limit theorems of those processes using different scalings. The class of limit processes of the theorems includes essentially all continuous state catalytic branching processes and spectrally positive regular affine processes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aldous, D.: Stopping times and tightness. Ann. Probab. 6, 335–340 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andersen, T.G., Lund, J.: Estimating continuous-time stochastic volatility models of the short-term interest rate. J. Econom. 72, 343–377 (1997)

    Article  Google Scholar 

  3. Athreya, K.B., Ney, P.E.: Branching Processes. Spring, New York (1972)

    MATH  Google Scholar 

  4. Bingham, N.H.: Continuous branching processes and spectral positivity. Stoch. Process. Their Appl. 4, 217–242 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  5. Çinlar, E., Jacod, J.: Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures. In: Seminar on Stochastic Processes (1981)

  6. Cox, J., Ingersoll, J., Ross, S.: A theory of the term structure of interest rate. Econometrica 53, 385–408 (1985)

    Article  MathSciNet  Google Scholar 

  7. Dawson, D.A., Fleischmann, K.: A continuous super-Brownian motion in a super-Brownian medium. J. Theor. Probab. 10, 213–276 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dawson, D.A., Fleischmann, K.: Catalytic and mutually catalytic branching. In: Infinite Dimensional Stochastic Analysis, pp. 145–170. Royal Netherlands Academy of Arts and Sciences, Amsterdam (2000)

    Google Scholar 

  9. Dawson, D.A., Fleischmann, K.: Catalytic and mutually catalytic super-Brownian motions. In: Progr. Probab., vol. 52, pp. 89–110. Birkhäuser, Basel (2002)

    Google Scholar 

  10. Dawson, D.A., Li, Z.H.: Construction of immigration superprocesses with dependent spatial motion from one-dimensional excursions. Probab. Theory Relat. Fields. 127, 37–61 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Duffie, D., Filipović, D., Schachermayer, W.: Affine processes and applications in finance. Ann. Appl. Probab. 13, 984–1053 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. El Karoui, N., Méléard, S.: Martingale measures and stochastic calculus. Probab. Theory Relat. Fields 84, 83–101 (1990)

    Article  MATH  Google Scholar 

  14. Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986)

    MATH  Google Scholar 

  15. Heston, S.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6, 327–343 (1993)

    Article  Google Scholar 

  16. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland Mathematical Library, vol. 24. North-Holland/Kodansha, Amsterdam/Tokyo (1989)

    MATH  Google Scholar 

  17. Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften, vol. 288. Springer, Berlin (1987)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Lamperti, J.: Continuous-state branching processes. Bull. Am. Math. Soc. 73, 382–386 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  20. Le Gall, J.F., Le Jan, Y.: Branching processes in Lévy processes: The exploration process. Ann. Probab. 26, 213–252 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  21. Li, Z.H.: Measure-valued branching processes with immigration. Stoch. Process. Their Appl. 43, 249–264 (1992)

    Article  MATH  Google Scholar 

  22. Li, Z.H.: Ornstein–Uhlenbeck type processes and branching processes with immigration. J. Appl. Probab. 37, 627–634 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Li, Z.H.: A limit theorem of discrete Galton–Watson branching processes with immigration. J. Appl. Probab. 43, 289–295 (2005)

    Article  Google Scholar 

  24. Mamon, R.S.: A time-varying Markov chain model of term structure. Stat. Probab. Lett. 60, 309–312 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics, vol. 68. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  26. Taira, K.: Diffusion Processes and Partial Differential Equations. Academic Press, London (1988)

    MATH  Google Scholar 

  27. Venttsel’, A.D.: On boundary conditions for multi-dimensional diffusion processes. Theory Probab. Appl. 4, 164–177 (1959)

    Article  Google Scholar 

  28. Wang, H.: A class of measure-valued branching diffusions in a random medium. Stoch. Anal. Appl. 16, 753–786 (1998)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Zenghu Li

  2. School of Mathematical Sciences, Nankai University, Tianjin, 300071, People’s Republic of China

    Chunhua Ma

Authors
  1. Zenghu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chunhua Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chunhua Ma.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, Z., Ma, C. Catalytic Discrete State Branching Models and Related Limit Theorems. J Theor Probab 21, 936–965 (2008). https://doi.org/10.1007/s10959-008-0161-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-008-0161-y

Keywords

AMS Subject Classifications

Search

Navigation