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

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.

Keywords

Iterated stochastic Ito integral, Fourier series, numerical method, mean-square convergence

Received

29.09.2018

Publication date

29.09.2018

Number of characters

460

Cite Download pdf To download PDF you should sign in

GOST	Kuznetsov D. On Numerical Modeling of the Multidimensional Dynamic Systems under Random Perturbations with the 1.5 and 2.0 Orders of Strong Convergence // Avtomatika i Telemekhanika – 2018. – Issue 7 C. 80-98 [Electronic resource]. URL: http://ras.jes.su/ait/s000523100000268-8-1-en (circulation date: 04.07.2024). DOI: 10.31857/S000523100000268-8
MLA	Kuznetsov, D. "On Numerical Modeling of the Multidimensional Dynamic Systems under Random Perturbations with the 1.5 and 2.0 Orders of Strong Convergence." Avtomatika i Telemekhanika 7 (2018):80-98. DOI: 10.31857/S000523100000268-8
APA	Kuznetsov D. (2018). On Numerical Modeling of the Multidimensional Dynamic Systems under Random Perturbations with the 1.5 and 2.0 Orders of Strong Convergence. Avtomatika i Telemekhanika (7), pp.80-98 DOI: 10.31857/S000523100000268-8

Размещенный ниже текст является ознакомительной версией и может не соответствовать печатной

Readers community rating: votes 0

1. Kloeden P.E., Platen E. Numerical solution of stochastic differential equations. Berlin: Springer, 1992.

2. Arato M. Linear stochastic systems with constant coefficients. A statistical approach. Berlin–Heidelberg–N.Y.: Springer, 1982.

3. Shiryaev A.N. Osnovy stokhasticheskoj finansovoj matematiki. T. 1, 2. M.: Fazis, 1998.

4. Liptser R.Sh., Shiryaev A.N. Statistika sluchajnykh protsessov: Nelinejnaya fil'tratsiya i smezhnye voprosy. M.: Nauka, 1974.

5. Mil'shtejn G.N. Chislennoe integrirovanie stokhasticheskikh differentsial'nykh uravnenij. Sverdlovsk: Izd-vo Ural'sk. un-ta, 1988.

6. Milstein G.N., Tretyakov M.V. Stochastic numerics for mathematical physics. Berlin: Springer, 2004.

7. Kloeden P.E., Platen E., Schurz H. Numerical solution of SDE through computer experiments. Berlin: Springer, 1994.

8. Platen E., Wagner W. On a Taylor formula for a class of Ito processes // Probab. Math. Statist. 1982. No. 3. P. 37–51.

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

10. Kul'chitskij O.Yu., Kuznetsov D.F. Unifitsirovannoe razlozhenie Tejlora– Ito // Zap. nauchn. seminarov POMI RAN. Veroyatnost' i statistika. 2. SPb., 1997. T. 244. S. 186–204.

11. Kuznetsov D.F. Novye predstavleniya razlozheniya Tejlora-Stratonovicha // Zap. nauchn. seminarov POMI RAN. Veroyatnost' i statistika. 4. SPb., 2001. T. 278. S. 141–158.

12. Gikhman I.I., Skorokhod A.V. Stokhasticheskie differentsial'nye uravneniya i ikh prilozheniya. Kiev: Nauk. dumka, 1982.

13. Kuznetsov D.F. Chislennoe integrirovanie stokhasticheskikh differentsial'nykh uravnenij. 2. SPb.: Izd-vo Politekhn. un-ta, 2006.

14. Kuznetsov D.F. Novye predstavleniya yavnykh odnoshagovykh chislennykh metodov dlya stokhasticheskikh differentsial'nykh uravnenij so skachkoobraznoj komponentoj // Zhurn. vychisl. matem. i mat. fiz. 2001. T. 41. № 6. S. 922–937.

15. Kloeden P.E., Platen E., Wright I.W. The Approximation of Multiple Stochastic Integrals // Stoch. Anal. Appl. 1992. V. 10. No. 4. P. 431–441.

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

17. Kuznetsov D.F. Stokhasticheskie differentsial'nye uravneniya: teoriya i praktika chislennogo resheniya. SPb.: Izd-vo Politekhn. un-ta, 2007.

18. Kuznetsov D.F. Strong approximation of multiple Ito and Stratonovich stochastic integrals: multiple Fourier series approach. 2nd Ed. St. Petersburg: Polytechn. Univ. Publ. House, 2011.

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. P. A.1–A.385.