173
Views
0
CrossRef citations to date
0
Altmetric
Original Articles

Almost sure exponential stability of semi-Euler numerical scheme for nonlinear stochastic functional differential equation

&
Pages 1166-1182 | Received 23 Dec 2019, Accepted 06 Aug 2020, Published online: 27 Aug 2020

References

  • X. Mao, Stochastic Differential Equations and Their Applications, 2nd ed., Horwood Pub, Chichester, 2007.
  • F. Wu and X. Mao, Numerical solutions of neutral stochastic functional differential equations, SIAM J. Numer. Anal. 46(4) (2008), pp. 1821–1841. doi: 10.1137/070697021
  • X. Mao, Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions, Appl. Math. Comput. 217 (2011), pp. 5512–5524.
  • F. Wu, X. Mao, and P. Kloeden, Almost sure exponential stability of the Euler–Maruyama approximations for stochastic functional differential equations, Random Oper. Stoch. Equ. 19(2) (2011), pp. 165–186. doi: 10.1515/ROSE.2011.010
  • E. Buckwar, Introduction to the numerical analysis of stochastic delay differential equations, J. Comput. Appl. Math. 125 (2000), pp. 297–307. doi: 10.1016/S0377-0427(00)00475-1
  • C. Baker and E. Buckwar, Exponential stability in p-th mean of solutions and of convergent Euler-type solutions of stochastic delay differential equations, J. Comput. Appl. Math. 184 (2005), pp. 404–427. doi: 10.1016/j.cam.2005.01.018
  • F. Wu, X. Mao, and P. Kloeden, Almost sure exponential stability of the Euler–Maruyama approximate solutions to stochastic functional differential equations, Random Oper. Stoch. Equ. 19 (2011), pp. 165–186. doi: 10.1515/ROSE.2011.010
  • F. Wu, X. Mao, and L. Szpruch, Almost sure exponential stability of numerical solutions for stochastic delay differential equations, Numer. Math. 115(4) (2010), pp. 681–697. doi: 10.1007/s00211-010-0294-7
  • C. Huang, Mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations, J. Comput. Appl. Math. 259 (2014), pp. 77–86. doi: 10.1016/j.cam.2013.03.038
  • L. Liu and Q. Zhu, Almost sure exponential stability of numerical solutions to stochastic delay Hopfield neural networks, Appl. Math. Comput. 266 (2015), pp. 698–712.
  • X. Zong, F. Wu, and C. Huang, Exponential mean square stability of the theta approximations for neutral stochastic differential delay equations, J. Comput. Appl. Math. 286 (2015), pp. 172–185. doi: 10.1016/j.cam.2015.03.016
  • L. Liu and Q. Zhu, Mean square stability of two classes of theta method for neutral stochastic differential delay equations, J. Comput. Appl. Math. 305 (2016), pp. 55–67. doi: 10.1016/j.cam.2016.03.021
  • H. Mo, X. Zhao, and F. Deng, Exponential mean-square stability of the θ-method for neutral stochastic delay differential equations with jumps, Int. J. Syst. Sci. 48 (2016), pp. 462–470. doi: 10.1080/00207721.2016.1186245
  • M. Obradovi and Miloevi, Stability of a class of neutral stochastic differential equations with unbounded delay and Markovian switching and the Euler-Maruyama method, J. Comput. Appl. Math.309 (2017), pp. 244–266. doi: 10.1016/j.cam.2016.06.038
  • Y. Lu, M. Song, and M. Liu, Convergence and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments, J. Comput. Appl. Math. 317 (2017), pp. 55–71. doi: 10.1016/j.cam.2016.11.033
  • L. Liu, F. Deng, and T. Hou, Almost sure exponential stability of implicit numerical solution for stochastic functional differential equation with extended polynomial growth condition, Appl. Math. Comput. 330 (2018), pp. 201–212.
  • H. Schurz, The invariance of asymptotic laws of linear stochastic systems under discretization, Z. Angew. Math. Mech. 79 (1999), pp. 375–382. doi: 10.1002/(SICI)1521-4001(199906)79:6<375::AID-ZAMM375>3.0.CO;2-7
  • M. Milo evi, The Euler–Maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments, J. Comput. Appl. Math. 298 (2016), pp. 1–12. doi: 10.1016/j.cam.2015.11.019
  • D. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal. 38 (2000), pp. 753–769. doi: 10.1137/S003614299834736X
  • D. Higham, X. Mao, and A. Stuart, Exponential mean-square stability of numerical solutions to stochastic differential equations, LMS J. Comput. Math. 6 (2003), pp. 297–313. doi: 10.1112/S1461157000000462
  • D. Higham, X. Mao, and C. Yuan, Almost sure and moment exponential stability in numerical simulation of stochastic differential equation, SIAM J. Numer. Anal. 45 (2007), pp. 592–607. doi: 10.1137/060658138
  • Q. Guo, X. Mao, and R. Yue, Almost sure exponential stability of stochastic differential delay equations, SIAM J. Control Optim. 54 (2016), pp. 1919–1933. doi: 10.1137/15M1019465
  • M. Miloevi, Highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler-Maruyama method, Math. Comp. Model. 54 (2011), pp. 2235–2251. doi: 10.1016/j.mcm.2011.05.033
  • M. Miloevi, Almost sure exponential stability of solutions to highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler–Maruyama approximation, Math. Comput. Model. 57 (2013), pp. 887–899. doi: 10.1016/j.mcm.2012.09.016
  • W. Fei, L. Hu, X. Mao, and M. Shen, Delay dependent stability of highly nonlinear hybrid stochastic systems, Automatica, 82 (2017), pp. 165–170. doi: 10.1016/j.automatica.2017.04.050
  • M. Shen, W. Fei, X. Mao, and Y. Liang, Stability of highly nonlinear neutral stochastic differential delay equations, Syst. Control Lett. 115 (2018), pp. 1–8. doi: 10.1016/j.sysconle.2018.02.013
  • L. Feng and S. Li, The pth moment asymptotic stability and exponential stability of stochastic functional differential equations with polynomial growth condition, Adv. Differ. Equ. 0 (2014), pp. 302. doi: 10.1186/1687-1847-2014-302
  • S. Zhou, S. Xie, and Z. Fang, Almost sure exponential stability of the backward Euler–Maruyama discretization for highly nonlinear stochastic functional differential equation, J. Comput. Appl. Math.236 (2014), pp. 150–160. doi: 10.1016/j.amc.2014.03.010

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.