Exponential mean-square stability of the θ-method for neutral stochastic delay differential equations with jumps

ABSTRACT

The exponential mean-square stability of the θ-method for neutral stochastic delay differential equations (NSDDEs) with jumps is considered. With some monotone conditions, the trivial solution of the equation is proved to be exponentially mean-square stable. If the drift coefficient and the parameters satisfy more strengthened conditions, for the constrained stepsize, it is shown that the θ-method can preserve the exponential mean-square stability of the trivial solution for θ ∈ [0, 1]. Since θ-method covers the commonly used Euler–Maruyama (EM) method and the backward Euler–Maruyama (BEM) method, the results are valid for the above two methods. Moreover, they can adapt to the NSDDEs and the stochastic delay differential equations (SDDEs) with jumps. Finally, a numerical example illustrates the effectiveness of the theoretical results.

KEYWORDS:

• Neutral stochastic delay differential equations with jumps
• stochastic θ-method
• exponential mean-square stability

Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The study was supported by the National Natural Science Foundation of China [grant number 61573156], [grant number 61273126], [grant number 61503142]; the Ph.D. Start-up Fund of Natural Science Foundation of Guangdong Province [grant number 2014A030310388]; and the Fundamental Research Funds for the Central Universities [grant number x2zdD2153620].

Notes on contributors

Haoyi Mo

Haoyi Mo was born in 1979. She received her Master degree from School of Mathematics and Information Science, Shaanxi Normal University, China, in 2005. She is currently a Ph.D. candidate at the School of Automation Science and Engineering, South China University of Technology, China. She is also a lecturer of School of Applied Mathematics, Guangdong University of Technology, Guangzhou, China. Her research interests include stability and convergence of the numerical algorithms for stochastic systems.

Xueyan Zhao

Xueyan Zhao was born in 1984. She received her Ph.D. degree in systems engineering from South China University of Technology, Guangzhou, in June 2014. Now she is a teacher at South China University of Technology. Her research interests are stability and stabilization of stochastic systems and nonlinear systems.

Feiqi Deng

Feiqi Deng was born in 1962. He received his Ph.D. degree in control theory and control engineering from South China University of Technology, Guangzhou, in June 1997. Since October 1999, he has been a professor with South China University of Technology and the director of the Systems Engineering Institute of the university. He is currently a member of Technical Committee on Control Theory (TCCT), Chinese Association of Automation, and now he is serving as the chair of the IEEE SMC Guangzhou Chapter, an associate editor-in-chief of Journal of South China University of Technology, and a member of the editorial boards of the following journals: Control Theory and Applications, All about Systems and Control, Journal of Systems Engineering and Electronics, and Journal of Systems Engineering, etc. His main research interests include stability, stabilization, and robust control theory of complex systems, including time-delay systems, non-linear systems and stochastic systems.