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.This paper provides some sufficient conditions for the existence and uniqueness and the stochastic stability of the global solution for nonlinear neutral stochastic functional differential equations. When the drift term and the diffusion term satisfy a locally Lipschitz condition, and the Lyapunov monotonicity condition has a sign-changed time-varying coefficient, the existence and uniqueness of the global solution for such equations will be studied by using the Lyapunov–Krasovskii function approach and the theory of stochastic analysis. The stability in \(p\)th-moment, the asymptotical stability in \(p\)th-moment, and the exponential stability in \(p\)th-moment will be investigated. Different characterizations for these three kinds of stochastic stability in moment will be established, which are presented with respect to integration conditions. These results have seldom been reported in the existing literature. The almost surely exponential stability for the global solution of such equations is also discussed. Some discussions and comparisons are provided. Two examples are given to check the effectiveness of the theoretical results obtained.Keywordsnonlinear neutral stochastic functional differential equationtime-varying equationglobal solutionexistence and uniquenessstochastic stabilityMSC codes34D2034K2093E1534F0560H10
Chen et al. (2024) studied this question.