Skip to main content Accessibility help
×
Home

Ornstein-Uhlenbeck type processes and branching processes with immigration

  • Zeng-Hu Li (a1)

Abstract

It is shown that an Ornstein-Uhlenbeck type process associated with a spectrally positive Lévy process can be obtained as the fluctuation limits of both discrete state and continuous state branching processes with immigration.

Copyright

Corresponding author

Postal address: Department of Mathematics, Beijing Normal University, Beijing 100875, P. R. China. Email address: lizh@email.bnu.edu.cn

Footnotes

Hide All

Research supported by the National Natural Science Foundation of China (Grant No. 19361060).

Footnotes

References

Hide All
Athreya, K. B., and Ney, P. E. (1972). Branching Processes. Springer, Berlin.
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.
Bingham, N. H. (1976). Continuous branching processes and spectral positivity. Stoch. Proc. Appl. 4, 217242.
Ethier, S. N., and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.
Gorostiza, L. G. (1996). Fluctuation theorem for a superprocess with small branching rate. Sobretiro de Aportaciones Matematicas, IV Simposio de Probabilidad y Procesos Estocasticos 12, Sociedad Matematica Mexicana, pp. 119127.
Hadjiev, D. I. (1985). The first passage problem for generalized Ornstein–Uhlenbeck processes with non-positive jump. Séminaire de Probabilités XIX (Lecture Notes in Mathematics, 1123). Springer, Berlin, pp. 80-90.
Holley, H., and Stroock, D. W. (1978). Generalized Ornstein–Uhlenbeck processes and infinite particle branching Brownian motions. Publ. RIMS Kyoto Univ. 14, 741788.
Kawazu, K., and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Theoret. Prob. Appl. 16, 3451.
Lamperti, J. (1967). Continuous-state branching processes. Bull. Amer. Math. Soc. 73, 382386.
Le Gall, J. F., and Le Jan, Y. (1998a). Branching processes in Lévy processes: the exploration process. Ann. Prob. 26, 213252.
Le Gall, J. F., and Le Jan, Y. (1998b). Branching processes in Lévy processes: Laplace functionals of snakes and superprocess. Ann. Prob. 26, 14071432.
Li, Z.-H. (1991). Integral representations of continuous functions. Chinese Sci. Bull. 36, 979983.
Li, Z.-H. (1998). Immigration processes associated with branching particle systems. Adv. Appl. Prob. 30, 657675.
Li, Z.-H. (1999). Measure-valued immigration diffusions and generalized Ornstein–Uhlenbeck diffusions. Acta Math. Appl. Sinica 15, 310320.
Samorodnitsky, G., and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.
Sato, K. (1990). Processes with Independent Increments. Kinokuniya, Tokyo (in Japanese). English translation: (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
Sato, K., and Yamazato, M. (1984). Operator-selfdecomposable distributions as limit distributions of processes of Ornstein–Uhlenbeck type. Stoch. Proc. Appl. 17, 73100.
Shiga, T. (1990). A recurrence criterion for Markov processes of Ornstein–Uhlenbeck type. Prob. Theory Rel. Fields 85, 425447.
Wolfe, S. J. (1982). On a continuous analogue of the stochastic difference equation X n = ρX n − 1 + B n . Stoch. Proc. Appl. 12, 301312.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed