Continuous state branching processes in random environment: The Brownian case
We consider continuous-state branching processes that are perturbed by a Brownian motion. These processes are constructed as the unique strong solution of a stochastic differential equation. The long-term behaviours are studied. In the stable case, the extinction and explosion probabilities are give...
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| Published in | Stochastic processes and their applications Vol. 127; no. 3; pp. 957 - 994 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
01.03.2017
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0304-4149 1879-209X |
| DOI | 10.1016/j.spa.2016.07.006 |
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| Abstract | We consider continuous-state branching processes that are perturbed by a Brownian motion. These processes are constructed as the unique strong solution of a stochastic differential equation. The long-term behaviours are studied. In the stable case, the extinction and explosion probabilities are given explicitly. We find three regimes for the asymptotic behaviour of the explosion probability and five regimes for the asymptotic behaviour of the extinction probability. In the supercritical regime, the process conditioned on eventual extinction has three regimes for the asymptotic behaviour of the extinction probability. Finally, the process conditioned on non-extinction and the process with immigration are given. |
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| AbstractList | We consider continuous-state branching processes that are perturbed by a Brownian motion. These processes are constructed as the unique strong solution of a stochastic differential equation. The long-term behaviours are studied. In the stable case, the extinction and explosion probabilities are given explicitly. We find three regimes for the asymptotic behaviour of the explosion probability and five regimes for the asymptotic behaviour of the extinction probability. In the supercritical regime, the process conditioned on eventual extinction has three regimes for the asymptotic behaviour of the extinction probability. Finally, the process conditioned on non-extinction and the process with immigration are given. |
| Author | Pardo, J.C. Palau, S. |
| Author_xml | – sequence: 1 givenname: S. surname: Palau fullname: Palau, S. email: sandra.palau@cimat.mx – sequence: 2 givenname: J.C. surname: Pardo fullname: Pardo, J.C. email: jcpardo@cimat.mx |
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| Cites_doi | 10.1214/EJP.v12-402 10.1016/0304-4149(76)90011-9 10.1239/jap/1222441825 10.1239/aap/1046366105 10.1090/S0002-9904-1967-11762-2 10.1016/j.spa.2009.11.005 10.1080/03461238.1990.10413872 10.1214/09-PS154 10.1017/S0021900200118108 10.1214/154957805100000122 10.1214/10-AOP629 10.1214/ECP.v16-1685 10.1017/S0021900200005039 10.21136/CMJ.1958.100304 10.1016/j.anihpb.2004.01.006 |
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| Keywords | Brownian motion Supercritical process conditioned on eventual extinction Explosion and extinction probabilities Exponential functional of Brownian motion 60G51 60G80 Q-process 60G17 Continuous state branching processes with immigration in random environment Continuous state branching processes in random environment |
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| SubjectTerms | Brownian motion Continuous state branching processes in random environment Continuous state branching processes with immigration in random environment Explosion and extinction probabilities Exponential functional of Brownian motion Q-process Supercritical process conditioned on eventual extinction |
| Title | Continuous state branching processes in random environment: The Brownian case |
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