Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-22T05:29:17.559Z Has data issue: false hasContentIssue false
  • English
  • Français

Distributions of jumps in a continuous-state branching process with immigration

Part of: Stochastic analysis Markov processes

Published online by Cambridge University Press:  09 December 2016

Xin He  and
Zenghu Li
Xin He*
Affiliation:
Beijing Normal University
Zenghu Li*
Affiliation:
Beijing Normal University
*
* Postal address: School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China.
* Postal address: School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the distributional properties of jumps in a continuous-state branching process with immigration. In particular, a representation is given for the distribution of the first jump time of the process with jump size in a given Borel set. From this result we derive a characterization for the distribution of the local maximal jump of the process. The equivalence of this distribution and the total Lévy measure is then studied. For the continuous-state branching process without immigration, we also study similar problems for its global maximal jump.

Keywords

Branching processcontinuous stateimmigrationmaximal jumpjump timejump size

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60H20: Stochastic integral equations
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 1166 - 1177
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bertoin, J. (2011).On the maximal offspring in a critical branching process with infinite variance.J. Appl. Prob. 48,576–582.Google Scholar
Brigo, D. and Mercurio, F. (2006).Interest Rate Models–Theory and Practice,2nd edn.Springer,Berlin.Google Scholar
Chung, K. L. (1982).Lectures from Markov Processes to Brownian Motion.Springer,New York.Google Scholar
Dawson, D. A. and Li, Z. (2006).Skew convolution semigroups and affine Markov processes.Ann. Prob. 34,1103–1142.Google Scholar
Dawson, D. A. and Li, Z. (2012).Stochastic equations, flows and measure-valued processes.Ann. Prob. 40,813–857.Google Scholar
Duquesne, T. and Le Gall, J.-F. (2002).Random trees, Lévy processes and spatial branching processes.Astérisque 281.Google Scholar
Fu, Z. and Li, Z. (2010).Stochastic equations of non-negative processes with jumps.Stoch. Process. Appl. 120,306–330.Google Scholar
Kawazu, K. and Watanabe, S. (1971).Branching processes with immigration and related limit theorems.Theory Prob. Appl. 16,36–54.Google Scholar
Kyprianou, A. E. (2014).Fluctuations of Lévy Processes with Applications,2nd edn.Springer,Heidelberg.Google Scholar
Lamberton, D. and Lapeyre, B. (1996).Introduction to Stochastic Calculus Applied to Finance.Chapman & Hall,London.Google Scholar
Lamperti, J. (1967a).Continuous state branching processes.Bull. Amer. Math. Soc. 73,382–386.Google Scholar
Lamperti, J. (1967b).The limit of a sequence of branching processes.Z. Wahrscheinlichkeitsth. 7,271–288.Google Scholar
Li, Z. (2011).Measure-Valued Branching Markov Processes.Springer,Heidelberg.Google Scholar