Asymptotical Stability of Differential Equations Driven by Hölder Continuous Paths

Abstract

In this manuscript, we establish local exponential stability of the trivial solution of differential equations driven by Hölder continuous paths with Hölder exponent greater than 1/2. This applies in particular to stochastic differential equations driven by fractional Brownian motion with Hurst parameter greater than 1/2. We motivate the study of local stability by giving a particular example of a scalar equation, where global stability of the trivial solution can be obtained.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Amann, H.: Ordinary Differential Equations. An Introduction to Nonlinear Analysis. Walter de Gruyter, Berlin (1990)

    Book  MATH  Google Scholar 

  2. 2.

    Arnold, L.: Stochastic systems: qualitative theory and Lyapunov exponents. In: Fluctuations and Sensitivity in Nonequilibrium Systems. Springer Proc. Phys., 1, pp. 11–18, Springer, Berlin (1984)

  3. 3.

    Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  4. 4.

    Boufoussi, B., Hajji, S.: Functional differential equations driven by a fractional Brownian motion. Comput. Math. Appl. 62, 746–754 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Deya, A., Panloup, F., Tindel, S.: Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise. Preprint (2016)

  6. 6.

    Doss, H.: Liens entre équations différentielles stochastiques et ordinaires. Ann. Inst. Henri Poincaré Nouv. Sér. Sect. B 13, 99–124 (1977)

    MATH  Google Scholar 

  7. 7.

    Dragomir, S.S.: Some Gronwall Type Inequalities and Applications. Nova Science Publishers, New York (2003)

    MATH  Google Scholar 

  8. 8.

    Friz, P., Victoir, N.: Multidimensional Stochastic Processes as Rough Paths. Theory and Applications. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  9. 9.

    Gao, H., Garrido-Atienza, M.J., Schmalfuß, B.: Random attractors for stochastic evolution equations driven by fractional Brownian motion. SIAM J. Math. Anal. 46(4), 2281–2309 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Garrido-Atienza, M.J., Kloeden, P., Neuenkirch, A.: Discretization of stationary solutions of stochastic systems driven by fractional Brownian motion. Appl. Math. Optim. 60(2), 151–172 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Garrido-Atienza, M.J., Maslowski, B., Schmalfuß, B.: Random attractors for stochastic equations driven by a fractional Brownian motion. Int. J. Bifurc. Chaos 20(9), 1–22 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Hairer, M.: Ergodicity of stochastic differential equations driven by fractional Brownian motion. Ann. Probab. 33(2), 703–758 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Hairer, M., Ohashi, A.: Ergodic theory for SDEs with extrinsic memory. Ann. Probab. 35(5), 1950–1977 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Hairer, M., Pillai, N.S.: Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion. Ann. Inst. Henri Poincaré 47(2), 601–628 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Hairer, M., Pillai, N.S.: Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths. Ann. Probab. 41(4), 2544–2598 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Khasminskii, R.Z.: On the stability of nonlinear stochastic systems. J. Appl. Math. Mech. 30, 1082–1089 (1967)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  18. 18.

    Lejay, A.: An introduction to rough paths. In: Séminaire de Probabilités XXXVII, Volume 1832 of Lecture Notes in Mathematics, pp. 1–59. Springer, Berlin (2003)

  19. 19.

    Lyons, T., Qian, Z.: System Control and Rough Paths. Oxford University Press, London (2002)

    Book  MATH  Google Scholar 

  20. 20.

    Mao, X.: Stability of Stochastic Differential Equations with Respect to Semimartingales. Longman Scientific & Technical, Harlow (1991)

    MATH  Google Scholar 

  21. 21.

    Mao, X.: Exponential Stability of Stochastic Differential Equations. Marcel Dekker, New York (1994)

    MATH  Google Scholar 

  22. 22.

    Maslowski, B., Schmalfuß, B.: Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion. Stoch. Anal. Appl. 22, 1577–1607 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  24. 24.

    Nualart, D., Răşcanu, A.: Differential equations driven by fractional Brownian motion. Collect. Math. 53(1), 55–81 (2002)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Tan, L.: Exponential stability of fractional stochastic differential equations with distributed delay. Adv. Differ. Equ. 2014, 321 (2014)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Young, L.C.G.: An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67, 251–282 (1936)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Zähle, M.: Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Relat. Fields 111(3), 333–374 (1998)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to María J. Garrido-Atienza.

Additional information

This work was partially supported by MTM2015-63723-P, MINECO/FEDER funding (M. J. Garrido-Atienza and B. Schmalfuß).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Garrido-Atienza, M.J., Neuenkirch, A. & Schmalfuß, B. Asymptotical Stability of Differential Equations Driven by Hölder Continuous Paths. J Dyn Diff Equat 30, 359–377 (2018). https://doi.org/10.1007/s10884-017-9574-6

Download citation

Keywords

  • Differential equations
  • Hölder continuous driving signal
  • Fractional Brownian motion
  • Exponential stability

Mathematics Subject Classification

  • Primary: 37L15
  • Secondary: 34A34
  • 34F05