Acta Mathematica Scientia

, Volume 39, Issue 3, pp 747–763 | Cite as

Euler Scheme for Fractional Delay Stochastic Differential Equations by Rough Paths Techniques

  • Johanna Garzón
  • Samy TindelEmail author
  • Soledad Torres


In this note, we study a discrete time approximation for the solution of a class of delayed stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H ∈ (1/2,1). In order to prove convergence, we use rough paths techniques. Theoretical bounds are established and numerical simulations are displayed.


Fractional Brownian motion stochastic differential equations rough paths discrete time approximation 

2010 MR Subject Classification

60H15 65C30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bardina X, Bascompte D, Rovira C, Tindel S. An analysis of a stochastic model for bacteriophage systems. Math Biosci, 2013, 241(1): 99–108MathSciNetzbMATHGoogle Scholar
  2. [2]
    Calsina A, Palmada J-M, Ripoll J. Optimal latent period in a bacteriophage population model structured by infection-age. Math Models and Methods in Appl Sc, 2011, 21(4): 1–26MathSciNetzbMATHGoogle Scholar
  3. [3]
    Carletti M. Mean-square stability of a stochastic model for bacteriophage infection with time delays. Math Biosci, 2007, 210(2): 395–414MathSciNetzbMATHGoogle Scholar
  4. [4]
    Cheridito P. Arbitrage in fractional Brownian motion models. Finance Stoch, 2003, 7(4): 533–553MathSciNetzbMATHGoogle Scholar
  5. [5]
    Dasgupta A, Kallianpur G. Arbitrage opportunities for a class of Gladyshev processes. Appl Math Optim, 2000 41 (3): 377–385MathSciNetzbMATHGoogle Scholar
  6. [6]
    Denk G, Meintrup D, Schäffler S. Transient noise simulation: Modeling and simulation of 1/f-noise//Antreich K, et al. Modeling, simulation, and optimization of integrated circuits. Birkhauser. Int Ser Numer Math, 2003, 146: 251–267zbMATHGoogle Scholar
  7. [7]
    Deya A, Neuenkirch A, Tindel, S. A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion. Ann Inst Henri Poincaré Probab Stat, 2012, 48(2): 518–550MathSciNetzbMATHGoogle Scholar
  8. [8]
    Fernique X M. Regularité des trajectoires de fonctions alétoires gaussiennes//Hennequin P L. Ècole d’Èté de Saint-Flour IV, Lecture Notes in Mathematics. Berlin: Springer, 1974, 480: 2–95Google Scholar
  9. [9]
    Ferrante M, Rovira C. Convergence of delay differential equations driven by fractional Brownian motion. Bernoulli, 2006, 12(1): 85–100MathSciNetzbMATHGoogle Scholar
  10. [10]
    Ferrante M, Rovira C. Convergence of delay differential equations driven by fractional Brownian motion. J Evol Equ, 2010, 10(4): 761–783MathSciNetzbMATHGoogle Scholar
  11. [11]
    Friz P, Victoir N. Multidimensional dimensional processes seen as rough paths. Cambridge: Cambridge University Press, 2010zbMATHGoogle Scholar
  12. [12]
    Friz P K, Riedel S. Convergence rates for the full Gaussian rough paths. Ann Inst Henri Poincaré Probab Stat, 2014, 50(1): 154–194MathSciNetzbMATHGoogle Scholar
  13. [13]
    Guasoni P. No arbitrage under transaction costs, with fractional Brownian motion and beyond. Mathematical Finance, 2006, 16(3): 569–582MathSciNetzbMATHGoogle Scholar
  14. [14]
    Gubinelli M. Controlling rough paths. J Funct Anal, 2004, 216(1): 86–140MathSciNetzbMATHGoogle Scholar
  15. [15]
    Hu Y, Oksendal B. Fractional white noise calculus and applications to finance. Infinite dimensional analysis, quantum probability and related topics, 2003, 6(1): 1–32MathSciNetzbMATHGoogle Scholar
  16. [16]
    Hu Y, Oksendal B, Sulem A. Optimal consumption and portfolio in a Black-Scholes market driven by fractional Brownian motion. Infinite dimensional analysis, quantum probability and related topics, 2003, 6(4): 519–536MathSciNetzbMATHGoogle Scholar
  17. [17]
    Kou S, Sunney-Xie X. Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule. Phys Rev Lett, 2004, 93(18)Google Scholar
  18. [18]
    Küchle U, Platen E. Strong Discrete Time Approximation of Stochastic Differential Equations with Time Delay. Math Comput Simulation, 2000, 54(1/3): 189–205MathSciNetGoogle Scholar
  19. [19]
    Leon J, Tindel S. Malliavin calculus for fractional delay equations. J Theoret Probab, 2011, 25(3): 854–889MathSciNetzbMATHGoogle Scholar
  20. [20]
    Liu Y, Tindel S. First-order Euler scheme for SDEs driven by fractional Brownian motions: the rough case. Arxiv, 2017Google Scholar
  21. [21]
    Mishura Y, Shevchenko G. The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion. Stochastics, 2008, 80(5): 489–511MathSciNetzbMATHGoogle Scholar
  22. [22]
    Mohammed S. Stochastic functional differential equations. Research Notes in Mathematics 99. Boston: Pitman Advanced Publishing Program, 1984Google Scholar
  23. [23]
    Neuenkirch A, Nourdin I. Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion. J Theoret Probab, 2007, 20(4): 871–899MathSciNetzbMATHGoogle Scholar
  24. [24]
    Neuenkirch A, Nourdin I, Tindel S. Delay equations driven by rough paths. Electron J Probab, 2008,13(67): 2031–2068Google Scholar
  25. [25]
    Nualart D, Răşcanu A. Differential equations driven by fractional Brownian motion. Collect Mat, 2002, 53(1): 55–81MathSciNetzbMATHGoogle Scholar
  26. [26]
    Smith H. Models of virulent phage growth with application to phage therapy. SIAM J Appl Math, 2008, 68(6): 1717–1737MathSciNetzbMATHGoogle Scholar
  27. [27]
    Szymanski J, Weiss M. Elucidating the origin of anomalous diffusion in crowded fluids. Phys Rev Lett, 2009, 103(3)Google Scholar
  28. [28]
    Tejedor V, Bénichou O, Voituriez R, et al. Quantitative Analysis of Single Particle Trajectories: Mean Maximal Excursion Method. Biophysical J, 2010, 98(7): 1364–1372Google Scholar
  29. [29]
    Willinger W, Taqqu M S, Teverovsky V. Stock market prices and long-range dependence. Finance Stoch, 1999, 3 (1): 1–13zbMATHGoogle Scholar
  30. [30]
    Young L. An inequality of the Höolder type, connected with Stieltjes integration. Acta Math, 1936, 67(1): 251–282MathSciNetzbMATHGoogle Scholar
  31. [31]
    Zähle M. Integration with respect to fractal functions and stochastic calculus I. Prob Theory Relat Fields, 1998, 111 (3): 333–374MathSciNetzbMATHGoogle Scholar

Copyright information

© Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Nacional de ColombiaBogotá D.C.Colombia
  2. 2.Department of MathematicsPurdue UniversityW. LafayetteUSA
  3. 3.Facultad de IngenieríaCIMFAV Universidad de ValparaísoValparaisoChile

Personalised recommendations