an:07098993
Zbl 1420.60077
Zhang, Tian; Chen, Huabin; Yuan, Chenggui; Tom??s Caraballo, author.givenNames
On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations
EN
Discrete Contin. Dyn. Syst., Ser. B 24, No. 10, 5355-5375 (2019).
00437553
2019
j
60H10 34K20 34K50
stochastic differential equations; time-varying delay; asymptotic behavior; stability; Markov switching
Summary: In this paper, the existence and uniqueness, the stability analysis for the global solution of highly nonlinear stochastic differential equations with time-varying delay and Markovian switching are analyzed under a locally Lipschitz condition and a monotonicity condition. In order to overcome a difficulty stemming from the existence of the time-varying delay, one integral lemma is established. It should be mentioned that the time-varying delay is a bounded measurable function. By utilizing the integral inequality, the Lyapunov function and some stochastic analysis techniques, some sufficient conditions are proposed to guarantee the stability in both moment and almost sure senses for such equations, which can be also used to yield the almost surely asymptotic behavior. As a by-product, the exponential stability in \( p \)th (\(p\geq 1\))-moment and the almost sure exponential stability are analyzed. Finally, two examples are given to show the usefulness of the results obtained.