Abstract
The distributional properties of a multi-dimensional continuous-state branching process are determined by its cumulant semigroup, which is defined by the backward differential equation. We provide a proof of the assertion of Rhyzhov and Skorokhod (Theory Probab. Appl., 1970) on the uniqueness of the solutions to the equation, which is based on a characterization of the process as the pathwise unique solution to a system of stochastic equations.
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Acknowledgements
We would like to thank the two anonymous referees for their valuable comments and suggestions, which have led to a number of improvements in the presentation of the work.
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Conflict of Interest Zeng Hu LI is an editorial board member for Acta Mathematica Sinica English Series and was not involved in the editorial review or the decision to publish this article. All authors declare that there are no competing interests.
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This work was supported by the National Key R&D Program of China (Grant No. 2020YFA0712900) and the National Natural Science Foundation of China (Grant No. 12271029)
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Li, P.S., Li, Z.H. Uniqueness Problem for the Backward Differential Equation of a Continuous-State Branching Process. Acta. Math. Sin.-English Ser. 40, 1825–1836 (2024). https://doi.org/10.1007/s10114-024-3107-0
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DOI: https://doi.org/10.1007/s10114-024-3107-0
Keywords
- Continuous-state branching process
- multi-dimensional
- backward differential equation
- stochastic equation
- generator