Skip to main content
SpringerLink
Log in

Existence and Stability of Solutions to Highly Nonlinear Stochastic Differential Delay Equations Driven by G-Brownian Motion

  • Published:
Applied Mathematics-A Journal of Chinese Universities Aims and scope Submit manuscript

Abstract

Under linear expectation (or classical probability), the stability for stochastic differential delay equations (SDDEs), where their coefficients are either linear or nonlinear but bounded by linear functions, has been investigated intensively. Recently, the stability of highly nonlinear hybrid stochastic differential equations is studied by some researchers. In this paper, by using Peng’s G-expectation theory, we first prove the existence and uniqueness of solutions to SDDEs driven by G-Brownian motion (G-SDDEs) under local Lipschitz and linear growth conditions. Then the second kind of stability and the dependence of the solutions to G-SDDEs are studied. Finally, we explore the stability and boundedness of highly nonlinear G-SDDEs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Z Chen. Strong laws of large numbers for sub-linear expectations, Sci China Math, 2016, 59(5): 945–954.

    Article  MathSciNet  MATH  Google Scholar 

  2. Z Chen, P Wu, B Li. A strong law of large numbers for non-additive probabilities, International Journal of Approximate Reasoning, 2013, 54: 365–377.

    Article  MathSciNet  MATH  Google Scholar 

  3. X Bai, Y Lin. On the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with integral-Lipschitz coefficients, Acta Math Appl Sin Engl Ser, 2014, 30: 589–610.

    Article  MathSciNet  MATH  Google Scholar 

  4. S Deng, C Fei, W Fei, X Mao. Stability equivalence between the stochastic differential delay equations driven by G-Brownian motion and the Euler-Maruyama method, Applied Mathematics Letters, 2019, 96: 138–146.

    Article  MathSciNet  Google Scholar 

  5. L Denis, M Hu, S Peng. Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths, arXiv:0802.1240, 2008.

    Google Scholar 

  6. C Fei, W Fei. Consistency of least squares estimation to the parameter for stochastic differential equations under distribution uncertainty, Acta Math. Sci., Chinese Ser. (2019) accepted (see arXiv:1904.12701v1).

    Google Scholar 

  7. C Fei, W Fei, X Mao, M Shen, L Yan. Stability analysis of highly nonlinear hybrid multiple-delay stochastic differential equations, Journal of Applied Analysis and Computations, 2019, 9(3): 1–18.

    MATH  Google Scholar 

  8. C Fei, M Shen, W Fei, X Mao, L Yan. Stability of highly nonlinear hybrid stochastic integro-differential delay equations, Nonlinear Anal Hybrid Syst, 2019, 31: 180–199.

    Article  MathSciNet  MATH  Google Scholar 

  9. W Fei, C Fei. Optimal stochastic control and optimal consumption and portfolio with G-Brownian motion, arXiv:1309.0209v1, 2013.

    Google Scholar 

  10. W Fei, C Fei. On exponential stability for stochastic differential equations disturbed by G-Brownian motion, arXiv:1311.7311v1, 2013.

    Google Scholar 

  11. W Fei, L Hu, X Mao, M Shen. Delay dependent stability of highly nonlinear hybrid stochastic systems, Automatica, 2017, 28: 165–170.

    Article  MathSciNet  MATH  Google Scholar 

  12. W Fei, L Hu, X Mao, M Shen. Structured robust stability and boundedness of nonlinear hybrid delay dystems, SIAM Contr Opt, 2018, 56: 2662–2689.

    Article  MATH  Google Scholar 

  13. W Fei, L Hu, X Mao, M Shen. Generalised criteria on delay dependent stability of highly nonlinear hybrid stochastic systems, International Journal of Robust and Nonlinear Control, 2019, 25, 1201–1215.

    Article  MATH  Google Scholar 

  14. F Gao. Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion, Stoch Proc Appl, 2009, 119(10): 3356–3382.

    Article  MathSciNet  MATH  Google Scholar 

  15. L Hu, X Mao, Y Shen. Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Systems & Control Letters, 2013, 62: 178–187.

    Article  MathSciNet  MATH  Google Scholar 

  16. X Li, X Lin, Y Lin. Lyapunov-type conditions and stochastic differential equations driven by G-Brownian motion, J Math Anal Appl, 2016, 439: 235–255.

    Article  MathSciNet  MATH  Google Scholar 

  17. X Li, S Peng. Stopping times and related Itôs calculus with G-Brownian motion, Stoch Proc Appl, 2009, 121: 1492–1508.

    Article  MATH  Google Scholar 

  18. Y Li, L Yan. Stability of delayed Hopfield neural networks under a sublinear expectation framework, Journal of the Franklin Institute, 2018, 355: 4268–4281.

    Article  MathSciNet  MATH  Google Scholar 

  19. Q Lin. Some properties of stochastic differential equations driven by G-Brownian motion, Acta Math Sin (Engl Ser), 2013, 29: 923–942.

    Article  MathSciNet  MATH  Google Scholar 

  20. Y Lin. Stochastic differential eqations driven by G-Brownian motion with reflecting boundary, Electron J Probab, 2013, 18(9): 1–23.

    MathSciNet  Google Scholar 

  21. P Luo, F Wang. Stochastic differential equations driven by G-Brownian motion and ordinary differential equations, Stoch Proc Appl, 2014, 124: 3869–3885.

    Article  MathSciNet  MATH  Google Scholar 

  22. L Ma, Z Wang, R Liu, F Alsaadi. A note on guaranteed cost control for nonlinear stochastic systems with input saturation and mixed time-delays, International Journal of Robust and Nonlinear Control, 2017, 27 (18): 4443–4456.

    Google Scholar 

  23. X Mao. Stochastic Differential Equations and Their Applications, 2nd Edition, Chichester, Horwood Pub, 2007.

    Google Scholar 

  24. X Mao, C Yuan. Stochastic Differential Equations with Markovian Swtching, Imperial College Press, London, 2006.

    Book  Google Scholar 

  25. S Peng. Nonlinear expectations and stochastic calculus under uncertainty, arX-iv:1002.4546v1, 2010.

    Google Scholar 

  26. Y Ren, X Jia, R Sakthivel. The p-th moment stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion, Applicable Analysis, 2017, 96(6): 988–1003.

    Article  MathSciNet  MATH  Google Scholar 

  27. Y Ren, W Yin, R Sakthivel. Stabilization of stochastic differential equations driven by G-Brownian motion with feedback control based on discrete-time state observation, Automatica, 2018, 95: 146–151.

    Article  MathSciNet  MATH  Google Scholar 

  28. B Shen, Z Wang, H Tan. Guaranteed cost control for uncertain nonlinear systems with mixed time-delays: The discrete-time case, European Journal of Control, 2018, 40: 62–67.

    Article  MathSciNet  MATH  Google Scholar 

  29. M Shen, C Fei, W Fei, X Mao. The boundedness and stability of highly nonlinear hybrid neutral stochastic systems with multiple delays, Sci China Inf Sci, DOI: 10.1007/s11432-018-9755-7.

  30. M Shen, W Fei, X Mao, S Deng. Exponential stability of highly nonlinear neutral pantograph stochastic differential equations, Asian J Control, DOI: 10.1002/asjc.1903.

  31. M Shen, W Fei, X Mao, Y Liang. Stability of highly nonlinear neutral stochastic differential delay equations, Systems & Control Letters, 2018, 115: 1–8.

    Article  MathSciNet  MATH  Google Scholar 

  32. L Zhang. Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications, Sci China Math, 2016, 59(4): 751–768.

    Article  MathSciNet  MATH  Google Scholar 

  33. D Zhang, Z Chen. Exponential stability for stochastic differential equation driven by G-Brownian motion, Applied Mathematics Letters, 2012, 25: 1906–1910.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Glorious Sun School of Business and Management, Donghua University, Shanghai, 200051, China

    Chen Fei & Li-tan Yan

  2. School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, Anhui, 241000, China

    Wei-yin Fei

Authors
  1. Chen Fei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wei-yin Fei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Li-tan Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei-yin Fei.

Additional information

Supported by the National Natural Science Foundation of China (71571001).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fei, C., Fei, Wy. & Yan, Lt. Existence and Stability of Solutions to Highly Nonlinear Stochastic Differential Delay Equations Driven by G-Brownian Motion. Appl. Math. J. Chin. Univ. 34, 184–204 (2019). https://doi.org/10.1007/s11766-019-3619-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11766-019-3619-x

MR Subject Classification

Keywords

Search

Navigation