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The moment exponential stability criterion of nonlinear hybrid stochastic differential equations and its discrete approximations

Part of: Functional-differential and differential-difference equations Stochastic analysis Stability

Published online by Cambridge University Press:  19 July 2016

Xiaofeng Zong ,
Fuke Wu  and
Chengming Huang
Xiaofeng Zong
Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People's Republic of China (xfzong87816@gmail.com)
Fuke Wu
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, People's Republic of China (wufuke@mail.hust.edu.cn; chengming_huang@hotmail.com)
Chengming Huang
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, People's Republic of China (wufuke@mail.hust.edu.cn; chengming_huang@hotmail.com)
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Extract

Based on the martingale theory and large deviation techniques, we investigate the pth moment exponential stability criterion of the exact and numerical solutions to hybrid stochastic differential equations (SDEs) under the local Lipschitz condition. This new stability criterion shows that Markovian switching can serve as a stochastic stabilizing factor by its logarithmic moment-generating function. We also investigate the pth moment exponential stability of Euler–Maruyama (EM), backward EM (BEM) and split-step backward EM (SSBEM) approximations for hybrid SDEs and show that, under the additional linear growth condition, the EM method can share the mean-square exponential stability of the exact solution for sufficiently small step size. However, the BEM method can work without the linear growth condition. We further investigate the SSBEM method under a coupled condition.

Keywords

hybrid SDEsmoment exponential stabilityMarkov chainEuler–Maruyama approximationbackward Euler–Maruyama approximationsplit-step backward Euler–Maruyama approximation

MSC classification

Secondary: 60H10: Stochastic ordinary differential equations 60H35: Computational methods for stochastic equations 34K34: Hybrid systems 93D20: Asymptotic stability
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 6 , December 2016 , pp. 1303 - 1328
Copyright
Copyright © Royal Society of Edinburgh 2016 

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