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On Numerical Modeling of the Multidimentional Dynamic Systems under Random Perturbations with the 2.5 Order of Strong Convergence

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Abstract

Numerical modeling methods with a strong convergence of order 2.5 are developed for the multidimensional dynamic systems under random perturbations described by Itô stochastic differential equations. Special attention is paid to the numerical modeling methods of the multiple Itô stochastic integrals of multiplicities 1–5 in terms of the mean-square convergence criterion, which are required to implement the former methods.

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References

  1. 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, 2018, vol. 79, no. 7, pp. 1240–1254.

    Article  MathSciNet  MATH  Google Scholar 

  2. Liptser, R.Sh. and Shiryaev, A.N., Statistika sluchainykh protsessov: nelineinaya fil’tratsiya i smezhnye voprosy (Statistics of Random Processes: Nonlinear Filtering and Related Problems), Moscow: Nauka, 1974.

    Google Scholar 

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

    Book  MATH  Google Scholar 

  4. Kloeden, P.E., Platen, E., and Schurz, H., Numerical Solution of SDE through Computer Experiments, Berlin: Springer, 2003.

    MATH  Google Scholar 

  5. Shiryaev, A.N., Osnovy stokhasticheskoi finansovoi matematiki. T. 1, 2 (Foundations of Stochastic Mathematical Finance, vols. 1,2), Moscow: Fazis, 1998.

    Google Scholar 

  6. Platen, E. and Bruti-Liberati, N., Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Berlin-Heidelberg: Springer, 2010.

    Book  MATH  Google Scholar 

  7. Stratonovich, R.L., Izbrannye voprosy teorii fluktuatsii v radiotekhnike (Selected Problems of Fluctuations Theory in Radio Engineering), Moscow: Sovetskoe Radio, 1961.

    Google Scholar 

  8. Arato, M., Linear Stochastic Systems with Constant Coefficients. A Statistical Approach, Berlin-Heidelberg-New York: Springer, 1982.

    Book  MATH  Google Scholar 

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

    Google Scholar 

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

    Book  MATH  Google Scholar 

  11. Han, X. and Kloeden, P.E., Random Ordinary Differential Equations and Their Numerical Solution, Singapore: Springer, 2017.

    Book  MATH  Google Scholar 

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

  13. 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 

  14. Kul’chitskii, O.Yu. and Kuznetsov, D.F., Unified Taylor-Itô Expansion, Zap. Nauchn. Semin. Peterburg. Otd. Mat. Inst. Ross. Akad. Nauk, Veroyatn. Stat. 2, 1997, vol. 244, pp. 186–204.

    MATH  Google Scholar 

  15. Kuznetsov, D.F., Chislennoe integrirovanie stokhasticheskikh differentsial’nykh uravnenii. 2 (Numerical Integration of Stochastic Differential Equations. 2), St. Petersburg: Politekh. Univ., 2006. DOI: https://doi.org/10.18720/SPBPU/2/s17-227

    Google Scholar 

  16. Shiryaev, A.N., Veroyatnost’ (Probability), Moscow: Nauka, 1989.

    MATH  Google Scholar 

  17. Kuznetsov, D.F., Strong Approximation of Multiple Ito and Stratonovich Stochastic Integrals: Multiple Fourier Series Approach, St. Petersburg: Polytechnical University, 2011, 2nd ed. DOI: https://doi.org/10.18720/SPBPU/2/s17-233

    MATH  Google Scholar 

  18. Kuznetsov, D.F., Multiple Ito and Stratonovich Stochastic Integrals: Fourier-Legendre and Trigonometric Expansions, Approximations, Formulas, Electr. J. Differ. Equat. Control Proc., 2017, no. 1, pp. A.1–A.385. DOI: https://doi.org/10.18720/SPBPU/2/z17-3

  19. Kuznetsov, D.F., Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on MATLAB, Differ. Uravn. Prots. Uprav., 2017, no. 2, pp. A.1–A.1000. DOI: https://doi.org/10.18720/SPBPU/2/z17-4

  20. Allen, E., Approximation of Triple Stochastic Integrals through Region Subdivision, Commun. Appl. Anal., 2013, vol. 17, pp. 355–366.

    MathSciNet  MATH  Google Scholar 

  21. 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 

  22. Kuznetsov, D.F., Expansion of Multiple Stratonovich Stochastic Integrals of Fifth Multiplicity, Based on Generalized Multiple Fourier Series, arXiv:1802.00643, 2018.

    Google Scholar 

  23. Suetin, P.K., Klassicheskie ortogonal’nye mnogochleny (Classical Orthogonal Polynomials), Moscow: Fizmatlit, 2005.

    MATH  Google Scholar 

  24. 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 

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  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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Russian Text © The Author(s), 2019, published in Avtomatika i Telemekhanika, 2019, No. 5, pp. 99–117.

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Kuznetsov, D.F. On Numerical Modeling of the Multidimentional Dynamic Systems under Random Perturbations with the 2.5 Order of Strong Convergence. Autom Remote Control 80, 867–881 (2019). https://doi.org/10.1134/S0005117919050060

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  • DOI: https://doi.org/10.1134/S0005117919050060

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