On the Asymptotic Behavior for Neutral Stochastic Differential Delay Equations

This note investigates the existence and uniqueness as well as the stability of the general decay rate of the global solution for neutral stochastic differential equations with time-varying delay under a locally Lipschitz condition, a contractive condition, and a monotonicity condition. The stability results are derived by using the Lyapunov function approach and some stochastic analysis techniques, which not only cover the exponential stability in the <inline-formula><tex-math notation="LaTeX">$p$</tex-math></inline-formula>th<inline-formula><tex-math notation="LaTeX">$(p>0)$</tex-math></inline-formula>-moment and the almost sure exponential stability, but also the polynomial stability in the <inline-formula><tex-math notation="LaTeX">$p$</tex-math></inline-formula>th<inline-formula><tex-math notation="LaTeX">$(p>0)$</tex-math></inline-formula>-moment and the almost sure polynomial stability. Two examples including one coupled system consisting of a mass–spring–damper connected to a pendulum and the nonlinear external random force are given to illustrate the effectiveness of the obtained results.

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