The ultimate boundedness and stability of highly nonlinear neutral stochastic delay differential equations (NSDDEs) are investigated in this paper. Different from many previous works, the highly nonlinear NSDDEs with semi-Markov switching signals are considered for the first time in this paper. Meanwhile, the time delay function in this paper is only required to meet much more relaxed restrictions, which can invalidate plentiful methods with requirements on its derivatives for the stability of NSDDEs. To overcome this difficulty, several novel techniques to tackle NSDDEs with such a delay are developed. Furthermore, a crucial property on the ergodic semi-Markov process is established, which plays a key role in the proof on the existence and boundedness of global solution. The generalized Khasminskii-type theorems are established for the existence and uniqueness of global solution by applying new methods and this property, and the criteria for boundedness and stability are also provided. In particular, feasible measures are provided to deal with the neutral term in our stability criteria. Finally, the effectiveness is verified by a numerical example.

中文翻译：

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半马尔可夫开关信号的高度非线性中性随机时滞微分方程的极限有界性和稳定性


本文研究了高度非线性中性随机时滞微分方程（NSDDE）的极限有界性和稳定性。与许多以前的工作不同，本文首次考虑了具有半马尔可夫开关信号的高度非线性 NSDDE。同时，本文中的时滞函数只需要满足更宽松的限制，这使得许多对其导数有要求的 NSDDE 稳定性方法失效。为了克服这个困难，开发了几种解决 NSDDE 延迟问题的新技术。此外，建立了遍历半马尔可夫过程的一个重要性质，该性质对于证明全局解的存在性和有界性起着关键作用。应用新的方法和这一性质，建立了全局解的存在唯一性的广义Khasminskii型定理，并给出了有界性和稳定性的判据。特别是针对稳定标准中的中性项，提出了可行的措施。最后通过数值算例验证了该方法的有效性。