The exponential stability of trivial solution and numerical solution for neutral stochastic functional differential equations (NSFDEs) with jumps is considered. The stability includes thealmost sure exponential stability and the mean-square exponential stability.New conditions for jumps are proposed by means of the Borel measurable function toensure stability. It is shown that if the drift coefficientsatisfies the linear growth condition,the Euler-Maruyama method can reproduce the corresponding exponentialstability of the trivial solution.A numerical example is constructed to illustrate our theory.
This work was supported by National Natural Science Foundation of China (Grant Nos. 61573156, 61503142) and Key Youth Research Fund of Guangdong University of Technology (Grant No. 17ZK0010).
[1] Kolmanovsky V B, Nosov V R. Stability of neutral-type functional differential equations. Nonlinear Anal-Theor Methods Appl, 1982, 6: 873-910 CrossRef Google Scholar
[2] Mao X R. Exponential stability in mean square of neutral stochastic differential functional equations. Syst Control Lett, 1995, 26: 245-251 CrossRef Google Scholar
[3] Hu R. Asymptotic properties of several types of neutral stochastic functional differential equations. Dissertation for Ph.D. Degree. Wuhan: Huazhong University of Science & Technology, 2009. Google Scholar
[4] Wu F K, Hu S G, Huang C M. Robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay. Syst Control Lett, 2010, 59: 195-202 CrossRef Google Scholar
[5] Jankovi\'{c} S, Jovanovi\'{c} M. The $p$th moment exponential stability of neutral stochastic functional differential equations. Filomat, 2006, 20: 59-72 CrossRef Google Scholar
[6] Mao X R. Razumikhin-type theorems on exponential stability of neutral stochastic differential equations. SIAM J Math Anal, 1997, 28: 389-401 CrossRef Google Scholar
[7] Luo Q, Mao X R, Shen Y. New criteria on exponential stability of neutral stochastic differential delay equations. Syst Control Lett, 2006, 55: 826-834 CrossRef Google Scholar
[8] Mao X R. Asymptotic properties of neutral stochastic differential delay equations. Stochastics Stochastic Rep, 2000, 68: 273-295 CrossRef Google Scholar
[9] Jiang F, Shen Y, Wu F K. A note on order of convergence of numerical method for neutral stochastic functional differential equations. Commun Nonlinear Sci Numer Simul, 2012, 17: 1194-1200 CrossRef ADS Google Scholar
[10] Yu Z H. The improved stability analysis of the backward Euler method for neutral stochastic delay differential equations. Int J Comput Math, 2013, 90: 1489-1494 CrossRef Google Scholar
[11] Zong X F, Wu F K, Huang C M. Exponential mean square stability of the theta approximations for neutral stochastic differential delay equations. J Comput Appl Math, 2015, 286: 172-185 CrossRef Google Scholar
[12] Wu F K, Mao X R. Numerical solutions of neutral stochastic functional differential equations. SIAM J Numer Anal, 2008, 46: 1821-1841 CrossRef Google Scholar
[13] Wu F K, Mao X R, Szpruch L. Almost sure exponential stability of numerical solutions for stochastic delay differential equations. Numer Math, 2010, 115: 681-697 CrossRef Google Scholar
[14] Wu F K, Mao X R, Kloeden P E. Almost sure exponential stability of the Euler-Maruyama approximations for stochastic functional differential equations. Random Oper Stoch Equat, 2011, 19: 165--186. Google Scholar
[15] Zhou S B, Xie S F, Fang Z. Almost sure exponential stability of the backward Euler-Maruyama discretization for highly nonlinear stochastic functional differential equation. Appl Math Comput, 2014, 236: 150--160. Google Scholar
[16] Zhou S B. Exponential stability of numerical solution to neutral stochastic functional differential equation. Appl Math Comput, 2015, 266: 441--461. Google Scholar
[17] Li Q Y, Gan S. Almost sure exponential stability of numerical solutions for stochastic delay differential equations with jumps. J Appl Math Comput, 2011, 37: 541-557 CrossRef Google Scholar
[18] Yu Z H. Almost sure and mean square exponential stability of numerical solutions for neutral stochastic functional differential equations. Int J Comput Math, 2015, 92: 132-150 CrossRef Google Scholar
[19] Tan J G, Wang H L, Guo Y F. Existence and uniqueness of solutions to neutral stochastic functional differential equations with Poisson jumps. Abstr Appl Anal, 2012, 2012: 371239. Google Scholar
[20] Liu D Z, Yang G Y, Zhang W. The stability of neutral stochastic delay differential equations with Poisson jumps by fixed points. J Comput Appl Math, 2011, 235: 3115-3120 CrossRef Google Scholar
[21] Tan J G, Wang H L, Guo Y F, et al. Numerical solutions to neutral stochastic delay differential equations with Poisson jumps under local Lipschitz condition. Math Probl Eng, 2014, 2014: 976183. Google Scholar
[22] Mo H Y, Zhao X Y, Deng F Q. Exponential mean-square stability of the θ-method for neutral stochastic delay differential equations with jumps. Int J Syst Sci, 2017, 48: 462-470 CrossRef Google Scholar
[23] Zhu Q X. Asymptotic stability in the $p$th moment for stochastic differential equations with Lévy noise. J Math Anal Appl, 2014, 416: 126-142 CrossRef Google Scholar
[24] Mao W, You S R, Mao X R. On the asymptotic stability and numerical analysis of solutions to nonlinear stochastic differential equations with jumps. J Comput Appl Math, 2016, 301: 1-15 CrossRef Google Scholar
[25] Mao X R. Stochastic Differential Equations and Applications. Chichester: Horwood, 1997. Google Scholar
[26] Higham D J, Mao X R, Yuan C G. Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations. SIAM J Numer Anal, 2007, 45: 592-609 CrossRef Google Scholar
[27] Tan J G, Wang H L. Mean-square stability of the Euler-Maruyama method for stochastic differential delay equations with jumps. Int J Comput Math, 2011, 88: 421--429. Google Scholar
Figure 1
(Color online) Mean square stability of EM numerical solution $X_{k}$ with $\Delta=0.004$.
Figure 2
(Color online) A sample path of EM numerical solution $X_{k}$ with $\Delta=0.004$.
Figure 3
(Color online) Mean square stability of EM numerical solution $X_{k}$ with $\Delta=0.01$.
Figure 4
(Color online) Mean square stability of EM numerical solution $X_{k}$ with $\Delta=0.001$.
Figure 5
(Color online) Mean square stability of EM numerical solution $X_{k}$ with $\Delta=0.3$.
Figure 6
(Color online) A sample path of EM numerical solution $X_{k}$ with $\Delta=0.3$.
Copyright 2019 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有
京ICP备18024590号-1