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On Numerical Modeling of the Multidimensional Dynamic Systems under Random Perturbations with the 1.5 and 2.0 Orders of Strong Convergence

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Abstract

The paper was devoted to developing numerical methods with the orders 1.5 and 2.0 of strong convergence for the multidimensional dynamic systems under random perturbations obeying stochastic differential Ito equations. Under the assumption of a special mean-square convergence criterion, attention was paid to the methods of numerical modeling of the iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 4 that are required to realize the aforementioned numerical methods.

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References

  1. Kloeden, P.E. and Platen, E., Numerical Solution of Stochastic Differential Equations, Berlin: Springer, 1992.

    Book  MATH  Google Scholar 

  2. Arato, M., Linear Stochastic Systems with Constant Coefficients. A Statistical Approach, Berlin: Springer, 1982.

    Book  MATH  Google Scholar 

  3. Shiryaev, A.N., Osnovy stokhasticheskoi finansovoi matematiki (Fundamentals of Stochastic Financial Mathematics), Moscow: Fazis, 1998, vols. 1,2.

  4. Liptser, R.Sh. and Shiryaev, A.N., Statistika sluchainykh protsessov, Moscow: Nauka, 1974. Translated into English under the title Statistics of Random Processes, Berlin: Springer, 1978.

    Google Scholar 

  5. Milstein, G.N., Chislennoe integrirovanie stokhasticheskikh differentsial’nykh uravnenii (Numerical Integration of Stochastic Differential Equations), Sverdlovsk: Ural’sk. Univ., 1988.

    Google Scholar 

  6. Milstein, G.N. and Tretyakov, M.V., Stochastic Numerics for Mathematical Physics, Berlin: Springer, 2004.

    Book  MATH  Google Scholar 

  7. Kloeden, P.E., Platen, E., and Schurz, H., Numerical Solution of SDE Through Computer Experiments, Berlin: Springer, 1994.

    Book  MATH  Google Scholar 

  8. Platen, E. and Wagner, W., On a Taylor Formula for a Class of Ito Processes, Probab. Math. Statist., 1982, no. 3, pp. 37–51.

    MathSciNet  MATH  Google Scholar 

  9. Kloeden, P.E. and Platen, E., The Stratonovich and Ito–Taylor Expansions, Math. Nachr., 1991, vol. 151, pp. 33–50.

    Article  MathSciNet  MATH  Google Scholar 

  10. Kul’chitskii, O.Yu. and Kuznetsov, D.F., Unified Taylor–Ito Expansion, Zap. Nauchn. Seminarov POMI RAN, Veroyatn. Statist., 2, 1997, vol. 244, pp. 186–204.

    Google Scholar 

  11. Kuznetsov, D.F., New Representations of the Taylor–Stratonovich Expansion, Zap. Nauchn. Seminarov POMI RAN, Veroyatn. Statist., 4, 2001, vol. 278, pp. 141–158.

    MATH  Google Scholar 

  12. Gikhman, I.I. and Skorokhod, A.V., Stokhasticheskie differentsial’nye uravneniya i ikh prilozheniya (Stochastic Differential Equations and Their Applications), Kiev: Naukova Dumka, 1982.

    MATH  Google Scholar 

  13. Kuznetsov, D.F., Chislennoe integrirovanie stokhasticheskikh differentsial’nykh uravnenii. 2 (Numerical Integration of Stochastic Differential Equations. 2), St. Petersburg: Polytekhn. Univ., 2006.

    Google Scholar 

  14. Kuznetsov, D.F., New Representations of the Explicit One-step Numerical Methods for Stochastic Differential Equations with Transition Component, Zh. Vychisl. Mat. Mat. Fiz., 2001, vol. 41, no. 6, pp. 922–937.

    MathSciNet  Google Scholar 

  15. Kloeden, P.E., Platen, E., and Wright, I.W., The Approximation of Multiple Stochastic Integrals, Stoch. Anal. Appl., 1992, vol. 10, no. 4, pp. 431–441.

    Article  MathSciNet  MATH  Google Scholar 

  16. Allen, E., Approximation of Triple Stochastic Integrals Through Region Subdivision, Commun. Appl. Anal. (Special Tribute Issue to Prof. V. Lakshmikantham), 2013, vol. 17, pp. 355–366.

    MathSciNet  MATH  Google Scholar 

  17. Kuznetsov, D.F., Stokhasticheskie differentsial’nye uravneniya: teoriya i praktika chislennogo resheniya (Stochastic Differential Equations: Theory and Practice of Numerical Solution), St. Petersburg: Polytekhn. Univ., 2007.

  18. Kuznetsov, D.F., Strong Approximation of Multiple Ito and Stratonovich Stochastic Integrals: Multiple Fourier Series Approach, St. Petersburg: Polytekhn. Univ., 2011, 2nd ed.

    MATH  Google Scholar 

  19. Kuznetsov, D.F., Multiple Ito and Stratonovich Stochastic Integrals: Fourier–Legendre and Trigonometric Expansions, Approximations, Formulas, Electr. J. Differ. Equat. Control Process, 2017, no. 1, pp. A.1–A.385.

    Google Scholar 

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Authors and Affiliations

  1. Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

    D. F. Kuznetsov

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  1. D. F. Kuznetsov
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Correspondence to D. F. Kuznetsov.

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Original Russian Text © D.F. Kuznetsov, 2018, published in Avtomatika i Telemekhanika, 2018, No. 7, pp. 80–98.

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Kuznetsov, D.F. On Numerical Modeling of the Multidimensional Dynamic Systems under Random Perturbations with the 1.5 and 2.0 Orders of Strong Convergence. Autom Remote Control 79, 1240–1254 (2018). https://doi.org/10.1134/S0005117918070056

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  • DOI: https://doi.org/10.1134/S0005117918070056

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