English version
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Дифференциальные Уравнения
и
Процессы Управления

О журнале История Журнала Издатель Адреса Тематика Редакторский состав Рецензирование Для авторов Издательская этика Коллекция номеров English version

A New Approach to the Series Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity with Respect to Components of the Multidimensional Wiener Process. Ii

Автор(ы):

Dmitriy Feliksovich Kuznetsov

Peter the Great Saint-Petersburg Polytechnic University
195251, Saint-Petersburg, Polytechnicheskaya ul., 29
Department of Higher Mathematics
Dr. Sc., Professor

sde_kuznetsov@inbox.ru

Аннотация:

The article is devoted to the development of a new approach to the series expansion of iterated Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. This approach is based on multiple Fourier series in complete orthonormal systems of Legendre polynomials and trigonometric functions in Hilbert space. The theorem on the mean-square convergent expansion for the iterated Stratonovich stochastic integrals of multiplicity 6 is formulated and proved. In the first part of the paper, expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 5 were obtained. These results allow us to construct efficient approximation procedures for iterated Stratonovich stochastic integrals that are necessary for the implementation of strong numerical methods with orders 1.0, 1.5, 2.0, 2.5, and 3.0 for Ito stochastic differential equations with non-commutative noise (in the framework of the approach based on the Taylor-Stratonovich expansion).

Ключевые слова

Ссылки:

  1. Gihman, I. I., Skorohod, A. V. Stokhasticheskie differencial’nye uravnenia i ih prilozhenia [Stochastic differential equations and its applications]. Kiev, Naukova Dumka Publ., 1982. 612 p
  2. Kloeden, P. E., Platen, E. Numerical solution of stochastic differential equations. Berlin, Springer-Verlag Publ., 1992. 632 p
  3. Milstein G. N. Chislennoye integrirovaniye stokhasticheskih differencial’nyh uravnenii [Numerical integration of stochastic differential equations]. Sverdlovsk, Ural. Univ. Publ. 225 p
  4. Milstein, G. N., Tretyakov, M. V. Stochastic numerics for mathematical physics. Berlin, Springer-Verlag Publ., 2004. 616 p
  5. Kloeden, P. E., Platen, E., Schurz, H. Numerical solution of SDE through computer experiments. Berlin, Springer-Verlag Publ., 1994. 292 p
  6. Platen, E., Wagner, W. On a Taylor formula for a class of Ito processes. Probab. Math. Statist. 3 (1982), 37-51
  7. Kloeden, P. E., Platen, E. The Stratonovich and Ito-Taylor expansions. Math. Nachr. 151 (1991), 33-50
  8. Averina, T. A. Statisticheskoje modelirovanie reshenii stokhasticheskih differencial'nyh uravnenii i system so sluchainoi strukturoi [Statistical modeling of solutions of stochastic differential equations and systems with random structure]. Novosibirsk, Siberian Branch of the Russian Academy of Sciences Publ., 2019. 350 p
  9. Kulchitskiy, O. Yu., Kuznetsov, D. F. The unified Taylor-Ito expansion. J. Math. Sci. (N. Y. ). 99, 2 (2000), 1130-1140. DOI: http://doi.org/10.1007/BF02673635
  10. Kuznetsov, D. F. New representations of the Taylor-Stratonovich expansion. J. Math. Sci. (N. Y. ). 118, 6 (2003), 5586-5596. DOI: http://doi.org/10.1023/A:1026138522239
  11. Kuznetsov, D. F. Chislennoye integrirovanie stokhasticheskih differencial'nyh uravnenii. 2 [Numerical integration of stochastic differential equations. 2]. St. -Petersburg, Polytechnic Univ. Publ., 2006. 764 p. DOI: http://doi.org/10.18720/SPBPU/2/s17-227
  12. Kuznetsov, D. F. Stokhasticheskie differencial’nye uravnenia: teoriya i practika chislennogo resheniya. S programmami dl’ja PC v sisteme MATLAB 7. 0. Izd. 3-e [Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MATLAB 7. 0 programs. 3rd Ed. ]. St. -Petersburg, Polytechnic Univ. Publ., 2009. 768 p. DOI: http://doi.org/10.18720/SPBPU/2/s17-230
  13. Kuznetsov, D. F. Strong approximation of iterated Ito and Stratonovich stochastic integrals based on generalized multiple Fourier series. Application to numerical solution of Ito SDEs and semilinear SPDEs. (In English). arXiv:2003. 14184v34 [math. PR]. 2022, 928 p. DOI: http://doi.org/10.48550/arXiv.2003.14184
  14. Kuznetsov, D. F. A new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity with respect to components of the multidimensional Wiener process. [In English]. Differencialnie Uravnenia i Protsesy Upravlenia, 2022, no. 2, 83-186 (In English) Available at: http://diffjournal.spbu.ru/EN/numbers/2022.2/article.1.6.html
  15. Allen, E. Approximation of triple stochastic integrals through region subdivision. Commun. Appl. Analysis (Special Tribute Issue to Professor V. Lakshmikantham), 17 (2013), 355-366
  16. Li C. W., Liu X. Q. Approximation of multiple stochastic integrals and its application to stochastic differential equations. Nonlinear Anal. Theor. Meth. Appl. 30, 2 (1997), 697-708
  17. Tang, X., Xiao, A. Asymptotically optimal approximation of some stochastic integrals and its applications to the strong second-order methods. Adv. Comp. Math. 45 (2019), 813-846
  18. Gaines, J. G., Lyons, T. J. Random generation of stochastic area integrals. SIAM J. Appl. Math. 54 (1994), 1132-1146
  19. Wiktorsson, M. Joint characteristic function and simultaneous simulation of iterated Ito integrals for multiple independent Brownian motions. Ann. Appl. Prob. 11, 2 (2001), 470-487
  20. Ryden, T., Wiktorsson, M. On the simulation of iterated Ito integrals. Stoch. Proc. and their Appl. , 91, 1 (2001), 151-168
  21. Averina, T. A., Prigarin, S. M. Calculation of stochastic integrals of Wiener processes. Preprint 1048. Novosibirsk, Inst. of Comp. Math. Math. Geophys. of Siberian Branch of the Russian Academy of Sciences., 1995, 15 pp. (In Russ. )
  22. Prigarin, S. M., Belov, S. M. One application of series expansions of Wiener process. Preprint 1107. Novosibirsk, Inst. of Comp. Math. Math. Geophys. of Siberian Branch of the Russian Academy of Sciences, 1998, 16 p. (In Russ. )
  23. Kloeden, P. E., Platen, E., Wright, I. W. The approximation of multiple stochastic integrals. Stoch. Anal. Appl. 10, 4 (1992), 431-441
  24. Rybakov, K. A. Applying spectral form of mathematical description for representation of iterated stochastic integrals. Differencialnie Uravnenia i Protsesy Upravlenia, 2019, no. 4, 1-31 (In Russ. ) Available at: http://diffjournal.spbu.ru/EN/numbers/2019.4/article.1.1.html
  25. Rybakov, K. A. Using spectral form of mathematical description to represent Stratonovich iterated stochastic integrals. Smart Innovation, Systems and Technologies, vol 217. Springer Publ., 2021, pp. 287-304. DOI: http://doi.org/10.1007/978-981-33-4826-4_20
  26. Rybakov, K. A. Using spectral form of mathematical description to represent Ito iterated stochastic integrals. Smart Innovation, Systems and Technologies, vol 274. Springer Publ., 2022, pp. 331-344. DOI: http://doi.org/10.1007/978-981-16-8926-0_22
  27. Kuznetsov, D. F. Approximation of iterated Ito stochastic integrals of the second multiplicity based on the Wiener process expansion using Legendre polynomials and trigonometric functions (In Russ). Differencialnie Uravnenia i Protsesy Upravlenia, 2019, no. 4, 32-52 (In Russ. ) Available at: http://diffjournal.spbu.ru/EN/numbers/2019.4/article.1.2.html
  28. Foster, J., Habermann, K. Brownian bridge expansions for Lé vy area approximations and particular values of the Riemann zeta function. Combin. Prob. Comp. (2022), 1-28. DOI: http://doi.org/10.1017/S096354832200030X
  29. Kastner, F., Rö ß ler, A. An analysis of approximation algorithms for iterated stochastic integrals and a Julia and MATLAB simulation toolbox. arXiv:2201. 08424v1 [math. NA]. 2022, 43 p. DOI: http://doi.org/10.48550/arXiv.2201.08424
  30. Malham, S. J. A., Wiese, A. Efficient almost-exact Lé vy area sampling. Stat. Prob. Letters. 88 (2014), 50-55
  31. Stump, D. M., Hill J. M. On an infinite integral arising in the numerical integration of stochastic differential equations. Proc. Royal Soc. Series A. Math. Phys. Eng. Sci. 461, 2054 (2005), 397-413
  32. Platen, E., Bruti-Liberati, N. Numerical solution of stochastic differential equations with jumps in finance. Berlin, Heidelberg, Springer-Verlag Publ., 2010. 868 p
  33. Kuznetsov, D. F. Multiple Ito and Stratonovich stochastic integrals: approximations, properties, formulas. St. -Petersburg, Polytechnic Univ. Publ., 2013. 382 p. (In English). DOI: http://doi.org/10.18720/SPBPU/2/s17-234
  34. Kuznetsov, D. F. [Stochastic differential equations: theory and practice of numerical solution. With MATLAB programs. 6-th Ed. ]. Differencialnie Uravnenia i Protsesy Upravlenia, 2018, no. 4, A. 1-A. 1073 (In Russ. ) Available at: http://diffjournal.spbu.ru/EN/numbers/2018.4/article.2.1.html
  35. Kuznetsov, D. F. Development and application of the Fourier method for the numerical solution of Ito stochastic differential equations. Comp. Math. Math. Phys. , 58, 7 (2018), 1058-1070. DOI: http://doi.org/10.1134/S0965542518070096
  36. 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, 7 (2018), 1240-1254. DOI: http://doi.org/10.1134/S0005117918070056
  37. 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, 5 (2019), 867-881. DOI: http://doi.org/10.1134/S0005117919050060
  38. Kuznetsov, D. F. Explicit one-step numerical method with the strong convergence order of 5 for Ito stochastic differential equations with a multi-dimensional nonadditive noise based on the Taylor-Stratonovich expansion. Comp. Math. Math. Phys. 60, 3 (2020), 379-389. DOI: http://doi.org/10.1134/S0965542520030100
  39. Kuznetsov, D. F. Comparative analysis of the efficiency of application of Legendre polynomials and trigonometric functions to the numerical integration of Ito stochastic differential equations. Comp. Math. Math. Phys. , 59, 8 (2019), 1236-1250. DOI: http://doi.org/10.1134/S0965542519080116
  40. Kuznetsov, D. F. Application of the method of approximation of iterated Ito stochastic integrals based on generalized multiple Fourier series to the high-order strong numerical methods for non-commutative semilinear stochastic partial differential equations. (In English). arXiv:1905. 03724v15 [math. GM]. 2022, 41 p. DOI: http://doi.org/10.48550/arXiv.1905.03724
  41. Kuznetsov, D. F. Application of multiple Fourier-Legendre series to implementation of strong exponential Milstein and Wagner-Platen methods for non-commutative semilinear stochastic partial differential equations. (In English). arXiv:1912. 02612v6 [math. PR]. 2022, 32 p. DOI: http://doi.org/10.48550/arXiv.1912.02612
  42. Kuznetsov, D. F. [A method of expansion and approximation of repeated stochastic Stratonovich integrals based on multiple Fourier series on full orthonormal systems]. Differencialnie Uravnenia i Protsesy Upravlenia, 1997, no. 1, 18-77 (In Russ. ) Available at: http://diffjournal.spbu.ru/EN/numbers/1997.1/article.1.2.html
  43. Kuznetsov, D. F. Mean square approximation of solutions of stochastic differential equations using Legendre’s polynomials. J. Automat. Inform. Sciences. 32, 12 (2000), 69-86. DOI: http://doi.org/10.1615/JAutomatInfScien.v32.i12.80
  44. Kuznetsov, D. F. Expansion of iterated Stratonovich stochastic integrals based on generalized multiple Fourier series. Ufa Math. J. , 11, 4 (2019), 49-77. DOI: http://doi.org/10.13108/2019-11-4-49
  45. Kuznetsov, D. F. Expansion of iterated Ito stochastic integrals of arbitrary multiplicity based on generalized multiple Fourier series converging in the mean. (In English). arXiv:1712. 09746v24 [math. PR]. 2022, 111 p. DOI: http://doi.org/10.48550/arXiv.1712.09746
  46. Kuznetsov, D. F. Exact calculation of the mean-square error in the method of approximation of iterated Ito stochastic integrals, based on generalized multiple Fourier series. (In English). arXiv:1801. 01079v14 [math. PR]. 2022, 68 p. DOI: http://doi.org/10.48550/arXiv.1801.01079
  47. Kuznetsov, D. F. Mean-square approximation of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the Taylor-Ito and Taylor-Stratonovich expansions using Legendre polynomials. (In English). arXiv:1801. 00231v19 [math. PR]. 2022, 106 p. DOI: http://doi.org/10.48550/arXiv.1801.00231
  48. Kuznetsov, D. F. The hypotheses on expansions of iterated Stratonovich stochastic integrals of arbitrary multiplicity and their partial proof. (In English). arXiv:1801. 03195v18 [math. PR]. 2022, 138 p. DOI: http://doi.org/10.48550/arXiv.1801.03195
  49. Kuznetsov, D. F. Expansions of iterated Stratonovich stochastic integrals based on generalized multiple Fourier series: multiplicities 1 to 6 and beyond. (In English). arXiv:1712. 09516v21 [math. PR]. 2022, 204 p. DOI: http://doi.org/10.48550/arXiv.1712.09516
  50. Kuznetsov, D. F. Expansion of iterated Stratonovich stochastic integrals of fifth and sixth multiplicity based on generalized multiple Fourier series. (In English). arXiv:1802. 00643v14 [math. PR]. 2022, 129 p. DOI: http://doi.org/10.48550/arXiv.1802.00643
  51. Kuznetsov, D. F. The proof of convergence with probability 1 in the method of expansionof iterated Ito stochastic integrals based on generalized multiple Fourier series. Differencialnie Uravnenia i Protsesy Upravlenia, 2020, no. 2, 89-117 (In English) Available at: http://diffjournal.spbu.ru/RU/numbers/2020.2/article.1.6.html
  52. Kuznetsov, M. D., Kuznetsov, D. F. SDE-MATH: A software package for the implementation of strong high-order numerical methods for Ito SDEs with multidimensional non-commutative noise based on multiple Fourier-Legendre series. Differencialnie Uravnenia i Protsesy Upravlenia, 2021, no. 1, 93-422 (In English) Available at: http://diffjournal.spbu.ru/EN/numbers/2021.1/article.1.5.html
  53. Kuznetsov, D. F., Kuznetsov, M. D. Mean-square approximation of iterated stochastic integrals from strong exponential Milstein and Wagner-Platen methods for non-commutative semilinear SPDEs based on multiple Fourier-Legendre series. Recent developments in stochastic methods and applications. ICSM-5 2020. Springer Proceedings in Mathematics & Statistics, vol 371, Ed. Shiryaev, A. N., Samouylov, K. E., Kozyrev, D. V. Cham, Springer Publ., 2021, pp. 17-32. DOI: http://doi.org/10.1007/978-3-030-83266-7_2
  54. Rybakov, K. A. Orthogonal expansion of multiple Ito stochastic integrals. Differencialnie Uravnenia i Protsesy Upravlenia, 2021, no. 3, 109-140 (In Russ. ) Available at: http://diffjournal.spbu.ru/EN/numbers/2021.3/article.1.8.html
  55. Stratonovich, R. L. Uslovnye markpvskie processy i ih primenenie k teorii optimal’nogo upravlenia [Conditional Markov’s processes and its application to the theory of optimal control]. Moscow, Moscow St. Univ. Publ, 1966. 320 p
  56. Johnson, G. W., Kallianpur, G. Homogeneous chaos, p-forms, scaling and the Feynman integral. Trans. Amer. Math. Soc. 340 (1993), 503-548
  57. Rybakov, K. A. Orthogonal expansion of multiple Stratonovich stochastic integrals. Differencialnie Uravnenia i Protsesy Upravlenia, 2021, no. 4, 81-115 (In Russ. ) Available at: http://diffjournal.spbu.ru/EN/numbers/2021.4/article.1.5.html

Полный текст (pdf)