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Issues in the Software Implementation of Stochastic Numerical Runge–Kutta

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Distributed Computer and Communication Networks (DCCN 2018)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 919))

Abstract

This paper discusses the application of stochastic Runge-Kutta-like numerical methods with weak and strong convergences for systems of stochastic differential equations in Itô form. At the beginning a brief overview of available publications about stochastic numerical methods and information from the theory of stochastic differential equations are given. Then the difficulties that arise when trying to implement stochastic numerical methods and motivate to use source code generation are described. We discuss some implementation details, such as program languages (Python, Julia) and libraries (Jinja2, Numpy). Also the link to the repository with source code is provided in the article.

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References

  1. Jinja2 official site. http://jinja.pocoo.org

  2. Amiri, S., Hosseini, S.M.: Stochastic Runge-Kutta rosenbrock type methods for SDE systems. Appl. Numer. Math. 115, 1–15 (2017). https://doi.org/10.1016/j.apnum.2016.11.010

    Article  MathSciNet  MATH  Google Scholar 

  3. Bachelier, L.: Théorie de la spéculation. Ann. Sci. l’École Norm. Supér. 3(17), 21–86 (1900)

    Article  Google Scholar 

  4. Bell, J.R.: Algorithm 334: normal random deviates. Commun. ACM 11(7), 498 (1968). https://doi.org/10.1016/j.apnum.2016.11.010

    Article  Google Scholar 

  5. Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.: Julia: a fresh approach to numerical computing. SIAM Rev. 59, 65–98 (2017)

    Article  MathSciNet  Google Scholar 

  6. Box, G.E.P., Muller, M.E.: A note on the generation of random normal deviates. Ann. Math. Stat. 29(2), 610–611 (1958). https://doi.org/10.1214/aoms/1177706645

    Article  MATH  Google Scholar 

  7. Burrage, K., Burrage, P.M.: High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations. Appl. Numer. Math. 22, 81–101 (1996)

    Article  MathSciNet  Google Scholar 

  8. Burrage, K., Burrage, P.M.: General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems. Appl. Numer. Math. 28, 161–177 (1998)

    Article  MathSciNet  Google Scholar 

  9. Burrage, K., Burrage, P.M.: Order conditions of stochastic Runge-Kutta methods by B-series. SIAM J. Numer. Anal. 38, 1626–1646 (2000)

    Article  MathSciNet  Google Scholar 

  10. Burrage, K., Burrage, P.M., Belward, J.A.: A bound on the maximum strong order of stochastic Runge-Kutta methods for stochastic ordinary differential equations. BIT 37, 771–780 (1997)

    Article  MathSciNet  Google Scholar 

  11. Burrage, K., Burrage, P.M.: Low rank Runge-kutta methods, symplecticity and stochastic hamiltonian problems with additive noise. J. Computat. Appl. Math. 236(16), 3920–3930 (2012). https://doi.org/10.1016/j.cam.2012.03.007

    Article  MathSciNet  MATH  Google Scholar 

  12. Burrage, P.M.: Runge-Kutta methods for stochastic differential equations. Ph.D. thesis, University of Qeensland, Australia (1999)

    Google Scholar 

  13. Butcher, J.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, New Zealand (2003)

    Book  Google Scholar 

  14. Debrabant, K., Rößler, A.: Continuous weak approximation for stochastic differential equations. J. Comput. Appl. Math. 214, 259–273 (2008)

    Article  MathSciNet  Google Scholar 

  15. Debrabant, K., Rößler, A.: Classification of stochastic Runge-Kutta methods for the weak approximation of stochastic differential equations, March 2013. arXiv:1303.4510v1

  16. Gevorkyan, M.N., Velieva, T.R., Korolkova, A.V., Kulyabov, D.S., Sevastyanov, L.A.: Stochastic Runge–Kutta software package for stochastic differential equations. In: Zamojski, W., Mazurkiewicz, J., Sugier, J., Walkowiak, T., Kacprzyk, J. (eds.) Dependability Engineering and Complex Systems. AISC, vol. 470, pp. 169–179. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-39639-2_15

    Chapter  Google Scholar 

  17. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I, 2nd edn. Springer, Berlin (2008). https://doi.org/10.1007/978-3-662-12607-3

    Book  MATH  Google Scholar 

  18. Jones, E., Oliphant, T., Peterson, P., et al.: SciPy: open source scientific tools for Python (2001). http://www.scipy.org/. Accessed 08 10 2017

  19. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations, 2nd edn. Springer, Heidelberg (1995). https://doi.org/10.1007/978-3-662-12616-5

    Book  MATH  Google Scholar 

  20. Komori, Y., Mitsuri, T.: Stable ROW-type weak scheme for stochastic differential equations. RIMS Kokyuroku (932), 29–45 (1995)

    Google Scholar 

  21. Ma, Q., Ding, X.: Stochastic symplectic partitioned Runge-Kutta methods for stochastic hamiltonian systems with multiplicative noise. Appl. Math. Comput. 252, 520–534 (2015). https://doi.org/10.1016/j.amc.2014.12.045

    Article  MathSciNet  MATH  Google Scholar 

  22. Mackevičius, V.: Second-order weak approximations for stratonovich stochastic differential equations. Lith. Math. J. 34(2), 183–200 (1994). https://doi.org/10.1007/BF02333416

    Article  MathSciNet  MATH  Google Scholar 

  23. Maruyama, G.: Continuous Markov processes and stochastic equations. Rend. Circ. Mat. 4, 48–90 (1955)

    Article  MathSciNet  Google Scholar 

  24. Matsumoto, M., Nishimura, T.: Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Trans. Model. Comput. Simul. 8(1), 3–30 (1998). https://doi.org/10.1145/272991.272995

    Article  MATH  Google Scholar 

  25. Milstein, G.N.: Approximate integration of stochastic differential equations. Theory Probab. Appl. 19, 557–562 (1974)

    MathSciNet  Google Scholar 

  26. Milstein, G.N.: A method of second-order accuracy integration of stochastic differential equations. Theory Probab. Appl. 23, 396–401 (1979)

    Article  MathSciNet  Google Scholar 

  27. Milstein, G.N.: Weak approximation of solutions of systems of stochastic differential equations. Theory Probab. Appl. 30, 750–766 (1986)

    Article  Google Scholar 

  28. Øksendal, B.: Stochastic Differential Equations. An Introduction with Applications, 6th edn. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-642-14394-6

    Book  MATH  Google Scholar 

  29. Platen, E.: Beiträge zur zeitdiskreten Approximation von Itoprozessen. Ph.D. thesis, Akad. der Wiss., Berlin (1984)

    Google Scholar 

  30. Rossum, G.: Python reference manual. Technical report, Amsterdam, The Netherlands (1995). https://docs.python.org/3/

  31. Rößler, A.: Strong and weak approximation methods for stochastic differential equations—some recent developments. In: Devroye, L., Karasözen, B., Kohler, M., Korn, R. (eds.) Recent Developments in Applied Probability and Statistics, pp. 127–153. Physica-Verlag HD, Heidelberg (2010). https://doi.org/10.1007/978-3-7908-2598-5_6

    Chapter  Google Scholar 

  32. Rößler, A.: Runge-Kutta methods for the numerical solution of stochastic differential equations. Ph.D. thesis, Technischen Universität Darmstadt, Darmstadt, februar 2003

    Google Scholar 

  33. Rümelin, W.: Numerical treatment of stochastic differential equations. SIAM J. Numer. Anal. 19(3), 604–613 (1982)

    Article  MathSciNet  Google Scholar 

  34. Soheili, A.R., Namjoo, M.: Strong approximation of stochastic differential equations with Runge-Kutta methods. World J. Model. Simul. 4(2), 83–93 (2008)

    Google Scholar 

  35. Tocino, A., Ardanuy, R.: Runge-Kutta methods for numerical solution of stochastic differential equations. J. Comput. Appl. Math. 138, 219–241 (2002)

    Article  MathSciNet  Google Scholar 

  36. Wiktorsson, M.: Joint characteristic function and simultaneous simulation of iterated Itô integrals for multiple independent Brownian motions. Ann. Appl. Probab. 11(2), 470–487 (2001)

    Article  MathSciNet  Google Scholar 

  37. Zhou, W., Zhang, J., Hong, J., Song, S.: Stochastic symplectic Runge-Kutta methods for the strong approximation of hamiltonian systems with additive noise. J. Comput. Appl. Math. 325, 134–148 (2017). https://doi.org/10.1016/j.cam.2017.04.050

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The work is partially supported by Russian Foundation for Basic Research (RFBR) grants No 16-07-00556. Also the publication was prepared with the support of the “RUDN University Program 5-100”.

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Authors and Affiliations

  1. Department of Applied Probability and Informatics, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya str., Moscow, 117198, Russia

    Migran N. Gevorkyan, Anastasiya V. Demidova, Anna V. Korolkova & Dmitry S. Kulyabov

  2. Laboratory of Information Technologies, Joint Institute for Nuclear Research, 6 Joliot-Curie, Dubna, Moscow region, 141980, Russia

    Dmitry S. Kulyabov

Authors
  1. Migran N. Gevorkyan
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  2. Anastasiya V. Demidova
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  3. Anna V. Korolkova
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  4. Dmitry S. Kulyabov
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Corresponding author

Correspondence to Dmitry S. Kulyabov .

Editor information

Editors and Affiliations

  1. V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    Vladimir M. Vishnevskiy

  2. V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    Dmitry V. Kozyrev

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Gevorkyan, M.N., Demidova, A.V., Korolkova, A.V., Kulyabov, D.S. (2018). Issues in the Software Implementation of Stochastic Numerical Runge–Kutta. In: Vishnevskiy, V., Kozyrev, D. (eds) Distributed Computer and Communication Networks. DCCN 2018. Communications in Computer and Information Science, vol 919. Springer, Cham. https://doi.org/10.1007/978-3-319-99447-5_46

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  • DOI: https://doi.org/10.1007/978-3-319-99447-5_46

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  • Online ISBN: 978-3-319-99447-5

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