Advertisement

Science China Mathematics

, Volume 62, Issue 3, pp 597–616 | Cite as

Backward Euler-Maruyama method applied to nonlinear hybrid stochastic differential equations with time-variable delay

  • Chengjian Zhang
  • Ying XieEmail author
Articles
  • 62 Downloads

Abstract

In this paper, we consider strong convergence and almost sure exponential stability of the backward Euler-Maruyama method for nonlinear hybrid stochastic differential equations with time-variable delay. Under the local Lipschitz condition and polynomial growth condition, it is proved that the backward Euler-Maruyama method is strongly convergent. Additionally, the moment estimates and almost sure exponential stability for the analytical solution are proved. Also, under the appropriate condition, we show that the numerical solutions for the backward Euler-Maruyama methods are almost surely exponentially stable. A numerical experiment is given to illustrate the computational effectiveness and the theoretical results of the method.

Keywords

nonlinear hybrid stochastic differential equations time-variable delay backward Euler-Maruyama method strong convergence almost surely exponential stability 

MSC(2010)

65C20 60H35 65L20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11571128).

References

  1. 1.
    Anderson W J. Continuous-Time Markov Chains. Berlin: Springer-Verlag, 1991CrossRefzbMATHGoogle Scholar
  2. 2.
    Bahar A, Mao X. Stochastic delay Lotka-Volterra model. J Math Anal Appl, 2004, 292: 364–380MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bao J, Hou Z. An analytic approximation of solutions of stochastic differential delay equations with Markovian switch-ing. Math Comput Modelling, 2009, 50: 1379–1384MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Basak G K, Bisi A, Ghosh M K. Stability of a random diffusion with linear drift. J Math Anal Appl, 1996, 202: 604–622MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hu L, Mao X, Zhang L. Robust stability and boundedness of nonlinear hybrid stochastic differential delay equations. IEEE Trans Automat Control, 2013, 58: 2319–2332MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Li H, Xiao L, Ye J. Strong predictor-corrector Euler-Maruyama methods for stochastic differential equations with Markovian switching. J Comput Appl Math, 2013, 237: 5–17MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Li R, Hou Y. Convergence and stability of numerical solutions to SDDEs with Markovian switching. Appl Math Comput, 2006, 175: 1080–1091MathSciNetzbMATHGoogle Scholar
  8. 8.
    Li R, Meng H, Qin C. Exponential stability of numerical solutions to SDDEs with Markovian switching. Appl Math Comput, 2006, 174: 1302–1313MathSciNetzbMATHGoogle Scholar
  9. 9.
    Mao X. Stability of stochastic differential equations with Markovian switching. Stochastic Process Appl, 1999, 79: 45–67MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mao X. Stochastic Differential Equations and Applications. Cambridge: Woodhead Publishing, 2007zbMATHGoogle Scholar
  11. 11.
    Mao X. Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type condi-tions. Appl Math Comput, 2011, 217: 5512–5524MathSciNetzbMATHGoogle Scholar
  12. 12.
    Mao X, Shen Y, Gray A. Almost sure exponential stability of backward Euler-Maruyama discretizations for hybrid stochastic differential equations. J Comput Appl Math, 2011, 235: 1213–1226MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mao X, Szpruch L. Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coeffcients. J Comput Appl Math, 2013, 238: 14–28MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mao X, Szpruch L. Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coeffcients. Stochastics, 2013, 85: 144–171MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mao X, Yuan C. Stochastic Differential Equations with Markovian Switching. London: Imperial College Press, 2006CrossRefzbMATHGoogle Scholar
  16. 16.
    Milošević M. Highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler-Maruyama method. Math Comput Modelling, 2011, 54: 2235–2251MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Milošević M. Almost sure exponential stability of solutions to highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler-Maruyama approximation. Math Comput Modelling, 2013, 57: 887–899MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Milošević M. Implicit numerical methods for highly nonlinear neutral stochastic differential equations with time-dependent delay. Appl Math Comput, 2014, 244: 741–760MathSciNetzbMATHGoogle Scholar
  19. 19.
    Niu Y, Burrage K, Zhang C. Multi-scale approach for simulating time-delay biochemical reaction systems. IET Syst Biol, 2015, 9: 31–38CrossRefGoogle Scholar
  20. 20.
    Niu Y, Zhang C, Burrage K. Strong predictor-corrector approximation for stochastic delay differential equations. J Comput Math, 2015, 33: 587–605MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pang S, Deng F, Mao X. Almost sure and moment exponential stability of Euler-Maruyama discretizations for hybrid stochastic differential equations. J Comput Appl Math, 2008, 213: 127–141MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rodkina A, Schurz H. Almost sure asymptotic stability of drift-implicit θ-methods for bilinear ordinary stochastic differential equations in R 1. J Comput Appl Math, 2005, 180: 13–31MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Szpruch L, Mao X, Higham D J, et al. Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model. BIT, 2011, 51: 405–425MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Xie Y, Zhang C. Asymptotical boundedness and moment exponential stability for stochastic neutral differential equa-tions with time-variable delay and markovian switching. Appl Math Lett, 2017, 70: 46–51MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Yuan C, Glover W. Approximate solutions of stochastic differential delay equations with Markovian switching. J Comput Appl Math, 2006, 194: 207–226MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhou S. Strong convergence and stability of backward Euler-Maruyama scheme for highly nonlinear hybrid stochastic differential delay equation. Calcolo, 2015, 52: 445–473MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zhou S, Hu C. Numerical approximation of stochastic differential delay equation with coeffcients of polynomial growth. Calcolo, 2017, 54: 1–22MathSciNetCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhanChina
  2. 2.Hubei Key Laboratory of Engineering Modeling and Scientific ComputingHuazhong University of Science and TechnologyWuhanChina

Personalised recommendations