Basic theory and stability analysis for neutral stochastic functional differential equations with pure jumps

Abstract

This paper investigates the existence and uniqueness of solutions to neutral stochastic functional differential equations with pure jumps (NSFDEwPJs). The boundedness and almost sure exponential stability are also considered. In general, the classical existence and uniqueness theorem of solutions can be obtained under a local Lipschitz condition and linear growth condition. However, there are many equations that do not obey the linear growth condition. Therefore, our first aim is to establish new theorems where the linear growth condition is no longer required whereas the upper bound for the diffusion operator will play a leading role. Moreover, the pth moment boundedness and almost sure exponential stability are also obtained under some loose conditions. Finally, we present two examples to illustrate the effectiveness of our results.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 61573156, 61273126, 61503142), the Ph.D. Start-up Fund of Natural Science Foundation of Guangdong Province (Grant No. 2014A030310388), and Fundamental Research Funds for the Central Universities (Grant No. x2zdD2153620).

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Correspondence to Feiqi Deng.

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Li, M., Deng, F. & Mao, X. Basic theory and stability analysis for neutral stochastic functional differential equations with pure jumps. Sci. China Inf. Sci. 62, 12204 (2019). https://doi.org/10.1007/s11432-017-9302-9

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Keywords

  • neutral term
  • stochastic functional differential equations
  • pure jumps
  • existence and uniqueness theorem
  • stability analysis