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

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 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-Legendre series as well as multiple trigonometric Fourier series. The theorem on the mean-square convergent expansion for the iterated Stratonovich stochastic integrals of arbitrary multiplicity is formulated and proved under the condition of convergence of trace series. This condition has been verified for integrals of multiplicities 1 to 5 and complete orthonormal systems of Legendre polynomials and trigonometric functions in Hilbert space. The Hu-Meyer formula and multiple Wiener stochastic integral were used in the proof of the mentioned theorem. The rate of mean-square convergence of the obtained expansions is found. The results of the article can be applied to the numerical integration of 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. 14184v28 [math. PR]. 2022, 869 p. DOI: http://doi.org/10.48550/arXiv.2003.14184
  14. Allen, E. Approximation of triple stochastic integrals through region subdivision. Commun. Appl. Analysis (Special Tribute Issue to Professor V. Lakshmikantham), 17 (2013), 355-366
  15. 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
  16. 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
  17. Gaines, J. G., Lyons, T. J. Random generation of stochastic area integrals. SIAM J. Appl. Math. 54 (1994), 1132-1146
  18. 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
  19. Ryden, T., Wiktorsson, M. On the simulation of iterated Ito integrals. Stoch. Proc. and their Appl. , 91, 1 (2001), 151-168
  20. 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. )
  21. 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. )
  22. Kloeden, P. E., Platen, E., Wright, I. W. The approximation of multiple stochastic integrals. Stoch. Anal. Appl. 10, 4 (1992), 431-441
  23. 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: https://diffjournal.spbu.ru/EN/numbers/2019.4/article.1.1.html
  24. 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
  25. 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
  26. 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
  27. Foster, J., Habermann, K. Brownian bridge expansions for Lé vy area approximations and particular values of the Riemann zeta function. arXiv:2102. 10095v1 [math. PR]. 2021, 26 p. DOI: http://doi.org/10.48550/arXiv.2102.10095
  28. 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
  29. Malham, S. J. A., Wiese, A. Efficient almost-exact Lé vy area sampling. Stat. Prob. Letters. 88 (2014), 50-55
  30. 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
  31. Platen, E., Bruti-Liberati, N. Numerical solution of stochastic differential equations with jumps in finance. Berlin, Heidelberg, Springer-Verlag Publ., 2010. 868 p
  32. 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
  33. 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
  34. 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
  35. 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
  36. 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
  37. 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
  38. 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
  39. 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
  40. 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
  41. 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
  42. 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
  43. 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
  44. 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. 09746v22 [math. PR]. 2022, 111 p. DOI: http://doi.org/10.48550/arXiv.1712.09746
  45. 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. 01079v13 [math. PR]. 2022, 67 p. DOI: http://doi.org/10.48550/arXiv.1801.01079
  46. 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. 00231v18 [math. PR]. 2022, 105 p. DOI: http://doi.org/10.48550/arXiv.1801.00231
  47. Kuznetsov, D. F. The hypotheses on expansions of iterated Stratonovich stochastic integrals of arbitrary multiplicity and their partial proof. (In English). arXiv:1801. 03195v13 [math. PR]. 2022, 104 p. DOI: http://doi.org/10.48550/arXiv.1801.03195
  48. Kuznetsov, D. F. Expansions of iterated Stratonovich stochastic integrals based on generalized multiple Fourier series: multiplicities 1 to 5 and beyond. (In English). arXiv:1712. 09516v16 [math. PR]. 2022, 174 p. DOI: http://doi.org/10.48550/arXiv.1712.09516
  49. Kuznetsov, D. F. Expansion of iterated Stratonovich stochastic integrals of fifth multiplicity based on generalized multiple Fourier series. (In English). arXiv:1802. 00643v9 [math. PR]. 2022, 94 p. DOI: http://doi.org/10.48550/arXiv.1802.00643
  50. 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
  51. 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
  52. 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: https://doi.org/10.1007/978-3-030-83266-7_2
  53. Rybakov, K. A. Orthogonal expansion of multiple Ito stochastic integrals. Differencialnie Uravnenia i Protsesy Upravlenia, 2021, no. 3, 109-140 (In Russ. ) Available at: https://diffjournal.spbu.ru/EN/numbers/2021.3/article.1.8.html
  54. 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
  55. Ito, K. Multiple Wiener integral. J. Math. Soc. Japan. 3, 1 (1951), 157-169
  56. Budhiraja, A. Multiple stochastic integrals and Hilbert space valued traces with applications to asymptotic statistics and non-linear filtering. Ph. D., The University of North Caroline at Chapel Hill, 1994. 132 p
  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: https://diffjournal.spbu.ru/EN/numbers/2021.4/article.1.5.html
  58. Johnson, G. W., Kallianpur, G. Homogeneous chaos, p-forms, scaling and the Feynman integral. Trans. Amer. Math. Soc. 340 (1993), 503-548
  59. Budhiraja, A., Kallianpur, G. Two results on multiple Stratonovich integrals. Statistica Sinica. 7 (1997), 907-922
  60. Wong, E., Zakai, M. On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Stat. 36, 5 (1965), 1560-1564
  61. Wong, E., Zakai, M. On the relation between ordinary and stochastic differential equations. Int. J. Eng. Sci. 3 (1965), 213-229
  62. Ikeda, N., Watanabe, S. Stochastic Differential Equations and Diffusion Processes. 2nd Ed. Amsterdam, Oxford, New-York, North-Holland Publishing Company, 1989. 555 p
  63. Liptser, R. Sh., Shirjaev, A. N. Statistika sluchainyh processov: nelineinaya filtracia i smezhnye voprosy [Statistics of Stochastic Processes: Nonlinear Filtering and Related Problems]. Moscow, Nauka Publ., 1974. 696 p
  64. Luo, W. Wiener chaos expansion and numerical solutions of stochastic partial differential equations. Ph. D., California Institute of Technology, 2006. 225 p

Полный текст (pdf)