On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations

Tian Zhang; Huabin Chen; Chenggui Yuan; Tomás Caraballo

doi:10.3934/dcdsb.2019062

Article Contents

2019, Volume 24, Issue 10: 5355-5375. Doi: 10.3934/dcdsb.2019062

This issue Previous Article Numerical solution of partial differential equations with stochastic Neumann boundary conditions Next Article A backscattering model based on corrector theory of homogenization for the random Helmholtz equation

On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations

1.
Department of Mathematics, Nanchang University, Nanchang 330031, China
2.
Department of Mathematics, Swansea University, Swansea SA2 8PP, UK
3.
Depto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Facultad de Matemáticas, c/ Tarfia s/n, 41012-Sevilla, Spain

Received: May 2018

Revised: October 2018

Early access: April 2019

Published: August 2019

Huabin Chen is partially supported by the National Natural Science Foundation of China (61364005, 11401292, 61773401), the Natural Science Foundation of Jiangxi Province of China (20171BAB201007, 20171BCB23001), and the Foundation of Jiangxi Provincial Educations of China (GJJ160061, GJJ14155) and the National Statistical Science Research Foundation of China (2018LY71). Tomás Caraballo is partially supported by the projects MTM2015-63723-P (MINECO/ FEDER, EU) and P12-FQM-1492 (Junta de Andalucía).

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Abstract

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.

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Mathematics Subject Classification: 60H10, 34K20, 34K50.

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Figure 1. Asymptotic behavior in mean square of the global solution for Eq. (4.1)

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Figure 2. Asymptotic behavior in almost sure sense of the global solution for Eq. (4.1)

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Figure 3. Asymptotic behavior in mean square of the global solution for Eq. (4.4)

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Figure 4. Asymptotic behavior in almost sure sense of the global solution for Eq. (4.4)

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