Abstract We prove the existence and pathwise uniqueness of the solution to a stochastic integral equation driven by Poisson random measures based on Kuznetsov measures for a continuous-state branching process. That gives a direct construction of the sample path of a continuous-state branching process with dependent immigration. The immigration rates depend on the population size via some functions satisfying a Yamada–Watanabe type condition. We only assume the existence of the first moment of the process. The existence of excursion law for the continuous-state branching process is not required. By special choices of the ingredients, we can make changes in the branching mechanism or construct models with competition.

中文翻译：

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具有依赖移民的连续状态分支过程的样本路径

摘要 我们证明了基于连续状态分支过程的库兹涅佐夫测度的泊松随机测度驱动的随机积分方程解的存在性和路径唯一性。这给出了具有依赖移民的连续状态分支过程的样本路径的直接构造。通过满足 Yamada-Watanabe 类型条件的一些函数，移民率取决于人口规模。我们只假设过程的第一时刻存在。不需要存在连续状态分支过程的偏移定律。通过对成分的特殊选择，我们可以改变分支机制或构建具有竞争性的模型。