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Young Differential Delay Equations Driven by Hölder Continuous Paths

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Modern Mathematics and Mechanics

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

In this paper we prove the existence and uniqueness of the solution of Young differential delay equations under weaker conditions than it is known in the literature. We also prove the continuity and differentiability of the solution with respect to the initial function and give an estimate for the growth of the solution. The proofs use techniques of stopping times, Shauder-Tychonoff fixed point theorem and a Gronwall-type lemma.

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References

  1. Boufoussi, B., Hajji, S.: Functional differential equations driven by a fractional Brownian motion. Comput. Math. Appl. 62(1), 746–754 (2011)

    Article  MathSciNet  Google Scholar 

  2. Boufoussi, B., Hajji, S.: Neutral stochastic functional differential equations driven by a fractional Brownian motion in Hilbert space. Stat. Probab. Lett. 82, 1549–1558 (2012)

    Article  MathSciNet  Google Scholar 

  3. Boufoussi, B., Hajji, S., Lakhel, E.H.: Functional differential equations in Hilbert spaces driven by a fractional Brownian motion. Afr. Math. 23(2), 173–194 (2012)

    Article  MathSciNet  Google Scholar 

  4. Chen, Y., Gao, H., Garrido-Arienza, M.J., Schmalfuß, B.: Pathwise solutions of stochastic partial differential equations driven by Hölder-continuous integrators with exponent larger than \(\frac {1}{2}\) and random dynamical systems. Discrete Contin. Dyn. Syst. 34(1), 79–98 (2013)

    Google Scholar 

  5. Cong, N.D., Duc, L.H., Hong, P.T.: Nonautonomous Young differential equations revisited. J. Dyn. Diff. Equat. 262, 1–23 (2017). https://doi.org/10.1007/s10884-017-9634-y

    Article  Google Scholar 

  6. Duc, L.H., Schmalfuß, B., Siegmund, S.: A note on the generation of random dynamical systems from fractional stochastic delay differential equations. Stoch. Dyn. 15(3), 1550018 (2015). https://doi.org/10.1142/S0219493715500185

    Article  MathSciNet  Google Scholar 

  7. Duc, L.H., Garrido-Atienza, M.J., Neuenkirch, A., Schmalfuß, B.: Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in \((\frac {1}{2},1)\). J. Differ. Equ. 264(2), 1119–1145 (2018)

    Google Scholar 

  8. Friz, P., Victoir, N.: Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge Studies in Advanced Mathematics, vol. 120. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  9. Garrido-Atienza, M.J., Lu, K., Schmalfuß, B.: Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete Contin. Dyn. Syst. B 14(2), 473–493 (2010)

    Article  MathSciNet  Google Scholar 

  10. Lejay, A.: Controlled differential equations as Young integrals: a simple approach. J. Differ. Equ. 249, 1777–1798 (2010)

    Article  MathSciNet  Google Scholar 

  11. Lyons, T.: Differential equations driven by rough signals, I, An extension of an inequality of L.C. Young. Math. Res. Lett. 1, 45–464 (1994)

    Google Scholar 

  12. Lyons, T.: Differential equations driven by rough signals. Rev. Mat. Iberoam. 14(2), 215–310 (1998)

    Article  MathSciNet  Google Scholar 

  13. Lyons, T., Qian, Zh.: System Control and Rough Paths. Oxford Mathematical Monographs. Clarendon Press, Oxford (2002)

    Book  Google Scholar 

  14. Lyons, T., Caruana, M., Lévy, T.: Differential Equations Driven by Rough Paths. Lecture Notes in Mathematics, vol. 1908. Springer, Berlin (2007)

    Google Scholar 

  15. Mandelbrot, B., van Ness, J.: Fractional Brownian motion, fractional noises and applications. SIAM Rev. 4(10), 422–437 (1968)

    Article  MathSciNet  Google Scholar 

  16. Maslowski, B., Nualart, D.: Evolution equations driven by a fractional Brownian motion. J. Funct. Anal. 202(1), 277–305 (2003)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  18. Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Application. Gordon and Breach Science Publishers, Yvendon (1993)

    MATH  Google Scholar 

  19. Shevchenko, G.: Mixed stochastic delay differential equations. Theory Probab. Math. Stat. 89, 181–195 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Zähle, M.: Integration with respect to fractal functions and stochastic calculus. II. Math. Nachr. 225, 145–183 (2001)

    Article  MathSciNet  Google Scholar 

  23. Zeidler, E.: Nonlinear Functional Analysis and Its Applications I. Springer, Berlin (1986)

    Book  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous referees for their careful reading and insightful remarks which led to improvement of our manuscript. This research is partly funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) and by Thang Long University.

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Authors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

    Luu Hoang Duc

  2. Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

    Luu Hoang Duc

  3. Thang Long University, Hanoi, Vietnam

    Phan Thanh Hong

Authors
  1. Luu Hoang Duc
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  2. Phan Thanh Hong
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Corresponding author

Correspondence to Luu Hoang Duc .

Editor information

Editors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    Victor A. Sadovnichiy

  2. National Technical University of Ukraine, “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine

    Michael Z. Zgurovsky

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Duc, L.H., Hong, P.T. (2019). Young Differential Delay Equations Driven by Hölder Continuous Paths. In: Sadovnichiy, V., Zgurovsky, M. (eds) Modern Mathematics and Mechanics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-96755-4_17

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  • DOI: https://doi.org/10.1007/978-3-319-96755-4_17

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-96754-7

  • Online ISBN: 978-3-319-96755-4

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