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Speed of extinction for continuous-state branching processes in a weakly subcritical Lévy environment

Part of: Markov processes Special processes Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  01 December 2023

Natalia Cardona-Tobón  and
Juan Carlos Pardo
Natalia Cardona-Tobón*
Affiliation:
Georg-August-Universität Göttingen
Juan Carlos Pardo*
Affiliation:
Centro de Investigación en Matemáticas
*
*Postal address: Institute for Mathematical Stochastics, Georg-August Universität Göttingen. Goldschmidtstrasse 7 C.P. 37077, Göttingen, Germany. Email: natalia.cardonatobon@uni-goettingen.de
**Postal address: Calle Jalisco s/n. C.P. 36240, Guanajuato, México. Email: jcpardo@cimat.mx
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Abstract

We continue with the systematic study of the speed of extinction of continuous-state branching processes in Lévy environments under more general branching mechanisms. Here, we deal with the weakly subcritical regime under the assumption that the branching mechanism is regularly varying. We extend recent results of Li and Xu (2018) and Palau et al. (2016), where it is assumed that the branching mechanism is stable, and complement the recent articles of Bansaye et al. (2021) and Cardona-Tobón and Pardo (2021), where the critical and the strongly and intermediate subcritical cases were treated, respectively. Our methodology combines a path analysis of the branching process together with its Lévy environment, fluctuation theory for Lévy processes, and the asymptotic behaviour of exponential functionals of Lévy processes. Our approach is inspired by the last two previously cited papers, and by Afanasyev et al. (2012), where the analogue was obtained.

Keywords

Continuous-state branching processesLévy processLévy process conditioned to stay positiverandom environmentlong-term behaviourextinction

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G51: Processes with independent increments; L'evy processes 60H10: Stochastic ordinary differential equations 60K37: Processes in random environments
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 23
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

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