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

Li, Mengling and Deng, Feiqi and Mao, Xuerong (2019) Basic theory and stability analysis for neutral stochastic functional differential equations with pure jumps. Science in China Series F - Information Sciences, 62. 12204. ISSN 1009-2757

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    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.