Continuous-State Branching Processes in Lévy Random Environments

Hui He (), Zenghu Li () and Wei Xu ()
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Hui He: Beijing Normal University
Zenghu Li: Beijing Normal University
Wei Xu: Beijing Normal University

Journal of Theoretical Probability, 2018, vol. 31, issue 4, 1952-1974

Abstract: Abstract A general continuous-state branching processes in random environment (CBRE-process) is defined as the strong solution of a stochastic integral equation. The environment is determined by a Lévy process with no jump less than $$-1$$ - 1 . We give characterizations of the quenched and annealed transition semigroups of the process in terms of a backward stochastic integral equation driven by another Lévy process determined by the environment. The process hits zero with strictly positive probability if and only if its branching mechanism satisfies Grey’s condition. In that case, a characterization of the extinction probability is given using a random differential equation with blowup terminal condition. The strong Feller property of the CBRE-process is established by a coupling method. We also prove a necessary and sufficient condition for the ergodicity of the subcritical CBRE-process with immigration.

Keywords: Continuous-state branching process; Random environment; Lévy process; Transition semigroup; Backward stochastic equation; Survival probability; Immigration; Ergodicity; Primary 60J80; 60K37; Secondary 60H20; 60G51 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (6)

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DOI: 10.1007/s10959-017-0765-1

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