Continuous-State Branching Processes in Lévy Random Environments
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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 \(-1\). 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.
KeywordsContinuous-state branching process Random environment Lévy process Transition semigroup Backward stochastic equation Survival probability Immigration Ergodicity
Mathematics Subject Classification (2010)Primary 60J80 60K37 Secondary 60H20 60G51
We thank Professors Yueyun Hu and Zhan Shi for helpful discussions on branching processes in random environments. We are grateful to Sandra Palau for pointing out a gap in the original proof of Theorem 3.1 and a referee for a list of comments which helped us in improving the presentation of the results.
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