The implementation of approximate coupling in two-dimensional SDEs with invertible diffusion terms


We explain and prove some lemmas of the approximate coupling and we give some details of the Matlab implementation of this method. A particular invertible SDEs is used to show the convergence result for this method for general d, which will give an order one error bounds..

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  1. [1]

    Alnafisah Y. First-order numerical schemes for stochastic differential equations using coupling, PhD thesis, University of Edinburgh 2016.

  2. [2]

    Alnafisah Y. The Implementation of Milstein Scheme in Two-Dimensional SDEs Using the Fourier Method, Abstract and Applied Analysis, vol 2018.

  3. [3]

    Alnafisah Y. Two-Level Bound for Stochastic Differential Equations Using the Exact Coupling with an Explicit Coefficients, J Comput Theor Nanosci, 2018, 1954–1964.

  4. [4]

    Alfonsi A, Jourdain B, Kohatsu-Higa A. Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme, Ann Appl Probab, 2014, 24: 1049–1080.

    MathSciNet  Article  Google Scholar 

  5. [5]

    Alfonsi A, Jourdain B, Kohatsu-Higa A. Optimal transport bounds between the time-marginals of multidimensional diffusion and its Euler scheme, arXiv: 1405–7007.

  6. [6]

    Charbonneau B, Svyrydov Y, Tupper P. Weak convergence in the Prokhorov metric of methods for stochastic differential equations, IMA J Numer Anal, 2010, 30: 579–594.

    MathSciNet  Article  Google Scholar 

  7. [7]

    Cruzeiro A B, Malliavin P, Thalmaier A. Geometrization of Monte-Carlo numerical analysis of an elliptic operator: strong approximation, C R Math Acad Sci Paris, 2004, 338: 481–486.

    MathSciNet  Article  Google Scholar 

  8. [8]

    Davie A. Pathwise approximation of stochastic differential equations using coupling, preprint:, 2015.

  9. [9]

    Davie A. KMT theory applied to approximations of SDE, Stochastic Analysis and Applications, 2014, 185–201.

  10. [10]

    Fournier N. Simulation and approximation of Lévy-driven SDEs, ESIAM Probab Stat, 2011, 15: 233–248.

    Article  Google Scholar 

  11. [11]

    Gyongy I, Krylov N. Existence of strong solutions for Itô’s stochastic equations via approximations, Probab Theory Related Fields, 1996, 105: 143–158.

    MathSciNet  Article  Google Scholar 

  12. [12]

    Hormander L. Hypoelliptic second order differential equations, Acta Math, 1967, 119: 147–171.

    MathSciNet  Article  Google Scholar 

  13. [13]

    Kanagawa S. The rate of convergence for the approximate solutions of SDEs, Tokyo J Math, 1989, 12: 33–48.

    MathSciNet  Article  Google Scholar 

  14. [14]

    Kloeden P E, Platen E. Numerical Solution of Stochastic Differential Equations, Springer-Verlag 1995.

  15. [15]

    Kloeden P E, Platen E, Wright I. The approximation of multiple stochastic integrals, J Stoch Anal Appl, 1992, 10: 431–441.

    MathSciNet  Article  Google Scholar 

  16. [16]

    Komlós J, Major P, Tusnády G. An approximation of partial sums of independent RV’s and the sample DF. I, Z Wahr und Wer Gebiete, 1975, 32: 111–131.

    MathSciNet  Article  Google Scholar 

  17. [17]

    Alhojilan Y. Using Weak Bound Simulation for Testing Accuracy of Strong Approximation of Order 2 for Stochastic Differential Equations, J Comput Theor Nanosci, 2018, 15: 3184–3186.

    Article  Google Scholar 

  18. [18]

    Hairer M. Malliavin’s proof of Hörmander’s theorem, Bull Math Sci, 2011, 135: 650–666.

    MathSciNet  Article  Google Scholar 

  19. [19]

    Rachev S T, Ruschendorff L. Mass Transportation Problems, 1998, 1.

  20. [20]

    Rio E. Upper bounds for minimal distances in the central limit theorem, Ann Inst Henri Poincaré Probab Stat, 2009, 45: 802–817.

    MathSciNet  Article  Google Scholar 

  21. [21]

    Rio E. Asymptotic constants for minimal distances in the central limit theorem, Elect, Comm in Probab, 2011, 16: 96–103.

    MathSciNet  Article  Google Scholar 

  22. [22]

    Rydén T, Wiktrosson M. On the simulation of iteraled Itô integrals, Stochastic Processes Appl, 2001, 91: 151–168.

    MathSciNet  Article  Google Scholar 

  23. [23]

    Vaserstein L N. Markov processes over denumerable products of spaces describing large system of automata (Russian), Problemy Peredaci Informacii, 1969, 5: 64–72.

    Google Scholar 

  24. [24]

    Wiktorsson M. Joint characteristic function and simultaneous simulation of iterated Itô integrals for multiple independent Brownian motions, Ann Appl Probab, 2001, 11: 470–487.

    MathSciNet  Article  Google Scholar 

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The author would first like to thank Professor Sandy Davie, Professor Istvan Gyongy and Dr Sotirios Sabanis from Edinburgh University for their useful discussions and for their suggestions and improvements to this version of this paper.

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Correspondence to Yousef Alnafisah.

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The author gratefully acknowledge Qassim University, represented by the deanship of scientific research, on the material support for this research under the number (3871-cos-2018-1-14-S) during the academic year 2018.

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Alnafisah, Y. The implementation of approximate coupling in two-dimensional SDEs with invertible diffusion terms. Appl. Math. J. Chin. Univ. 35, 166–183 (2020).

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  • stochastic differential equation
  • milstein method
  • euler method

MR Subject Classification

  • 60H10