Skip to main content
SpringerLink
Log in

Stochastic Runge–Kutta Software Package for Stochastic Differential Equations

  • Conference paper
  • First Online:
Dependability Engineering and Complex Systems (DepCoS-RELCOMEX 2016)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 470))

Included in the following conference series:

Abstract

As a result of the application of a technique of multistep processes stochastic models construction the range of models, implemented as a self-consistent differential equations, was obtained. These are partial differential equations (master equation, the Fokker–Planck equation) and stochastic differential equations (Langevin equation). However, analytical methods do not always allow to research these equations adequately. It is proposed to use the combined analytical and numerical approach studying these equations. For this purpose the numerical part is realized within the framework of symbolic computation. It is recommended to apply stochastic Runge–Kutta methods for numerical study of stochastic differential equations in the form of the Langevin. Under this approach, a program complex on the basis of analytical calculations metasystem Sage is developed. For model verification logarithmic walks and Black–Scholes two-dimensional model are used. To illustrate the stochastic “predator–prey” type model is used. The utility of the combined numerical-analytical approach is demonstrated.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations, 2nd edn. Springer, Berlin (1995)

    MATH  Google Scholar 

  2. Roßler, A.: Runge-Kutta Methods for the Numerical Solution of Stochastic Differential Equations, Ph.D. thesis, Technischen Universit at Darmstadt, Darmstadt (February 2003)

    Google Scholar 

  3. 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  MATH  Google Scholar 

  4. 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  MATH  Google Scholar 

  5. 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  MATH  Google Scholar 

  6. 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  MATH  Google Scholar 

  7. Roßler, A.: Strong and Weak Approximation Methods for Stochastic Differential Equations—Some Recent Developments, http://www.math.uni-hamburg.de/research/papers/prst/prst2010-02.pdf (2010)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  9. Stein, W., et al.: Sage Mathematics Software (Version 6.5), The Sage Development Team, http://www.sagemath.org (2015)

  10. Eferina, E.G., Korolkova, A.V., Gevorkyan, M.N., Kulyabov, D.S., Sevastyanov, L.A.: One-step stochastic processes simulation software package, bulletin of peoples’ friendship University of Russia. Math. Inf. Sci. Phys. (3), 46–59 (2014). arXiv:1503.07342

  11. Velieva, T.R., Korolkova, A.V., Kulyabov, D.S.: Designing installations for verification of the model of active queue management discipline RED in the GNS3. In: 6th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), IEEE, 2014, pp. 570–577. arXiv:1504.02324, doi:10.1109/ICUMT.2014.7002164

  12. Floyd, S., Jacobson, V.: Random early detection gateways for congestion avoidance. IEEE/ACM Trans. Netw. 1, 397–413 (1993)

    Article  Google Scholar 

  13. Hnatič, M., Eferina, E.G., Korolkova, A.V., Kulyabov, D.S., Sevastyanov, L.A.: Operator approach to the master equation for the one-step process. EPJ Web Conf 108, 02027 (2016). doi:10.1051/epjconf/201610802027

    Article  Google Scholar 

  14. Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525–546 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Malham, S.J.A., Wiese, A.: An introduction to SDE simulation (2010). arXiv:1004.0646

  16. Platen, E., Bruti-Liberati, N.: Numerical Solution of Stochastic Differential Equations with Jumps in Finance. Springer, Heidelberg (2010)

    Book  MATH  Google Scholar 

  17. Demidova, A.V., Korolkova, A.V., Kulyabov, D.S., Sevastyanov, L.A.: The method of constructing models of peer to peer protocols. In: 6th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT). IEEE, pp. 557–562 (2014). doi:10.1109/ICUMT.2014.7002162

  18. Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I, 2nd edn. Springer, Berlin (2008)

    MATH  Google Scholar 

  19. Oksendal, B.: Stochastic differential equations. An introduction with applications, 6th Edition. Springer, Berlin (2003)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Debrabant, K., Rossler, A.: Classification of Stochastic Runge–Kutta Methods for the Weak Approximation of Stochastic Differential Equations (2013). arXiv:1303.4510

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

    Article  MathSciNet  MATH  Google Scholar 

  24. 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 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  27. 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 

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

    Article  Google Scholar 

Download references

Acknowledgments

The work is partially supported by RFBR grants No’s 14-01-00628, and 15-07-08795, and 16-07-00556.

Calculations were carried out on computer cluster “Felix” (RUDN) and Heterogeneous computer cluster “HybriLIT” (Multifunctional center storage, processing and analysis of data JINR).

Author information

Authors and Affiliations

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

    M. N. Gevorkyan, T. R. Velieva, A. V. Korolkova, D. S. Kulyabov & L. A. Sevastyanov

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

    D. S. Kulyabov

  3. Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna, Moscow Region, 141980, Russia

    L. A. Sevastyanov

Authors
  1. M. N. Gevorkyan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. R. Velieva
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. V. Korolkova
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. D. S. Kulyabov
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. L. A. Sevastyanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. N. Gevorkyan .

Editor information

Editors and Affiliations

  1. Department of Computer Engineering, Wrocław University of Technology, Wrocław, Poland

    Wojciech Zamojski

  2. Department of Computer Engineering, Wrocław University of Technology, Wrocław, Poland

    Jacek Mazurkiewicz

  3. Department of Computer Engineering, Wrocław University of Technology, Wroclaw, Poland

    Jarosław Sugier

  4. Department of Computer Engineering, Wrocław University of Technology, Wrocław, Poland

    Tomasz Walkowiak

  5. Polish Academy of Sciences, Systems Research Institute, Warszawa, Poland

    Janusz Kacprzyk

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Gevorkyan, M.N., Velieva, T.R., Korolkova, A.V., Kulyabov, D.S., Sevastyanov, L.A. (2016). 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. DepCoS-RELCOMEX 2016. Advances in Intelligent Systems and Computing, vol 470. Springer, Cham. https://doi.org/10.1007/978-3-319-39639-2_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-39639-2_15

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-39638-5

  • Online ISBN: 978-3-319-39639-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation