Русская версия
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

About History Editorial Page Addresses Scope Editorial Staff Submission Review For Authors Publication Ethics Issues Русская версия

A New Proof of the Expansion of Iterated Ito Stochastic Integrals with Respect to the Components of a Multidimensional Wiener Process Based on Generalized Multiple Fourier Series and Hermite Polynomials

Author(s):

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

Abstract:

The article is devoted to a new proof of the expansion for iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process. The above expansion is based on Hermite polynomials and generalized multiple Fourier series in arbitrary complete orthonormal systems of functions in a Hilbert space. In 2006, the author obtained a similar expansion, but with a lesser degree of generality, namely, for the case of continuous or piecewise continuous complete orthonornal systems of functions in a Hilbert space. In this article, the author generalizes the expansion of iterated Ito stochastic integrals obtained by him in 2006 to the case of an arbitrary complete orthonormal systems of functions in a Hilbert space using a new approach based on the Ito formula. The obtained expansion of iterated Ito stochastic integrals is useful for constructing of high-order strong numerical methods for systems of Ito stochastic differential equations with multidimensional non-commutative noise.

Keywords

References:

  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. 596 p
  5. Kloeden, P. E., Platen, E., Schurz, H. Numerical solution of SDE through computer experiments. Berlin, Springer-Verlag Publ., 1994. 292 p
  6. 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
  7. Kuznetsov, D. F. Stokhasticheskie differencial’nye uravnenia: teoriya i practika chislennogo resheniya. S programmami dl’ja PC v sisteme MATLAB 7. 0. Izd. 2-e. [Stochastic differential equations: theory and practice of numerical solution. With MATLAB 7. 0 programs. 2nd ed. ]. St. -Petersburg, Polytechnic Univ. Publ., 2007. 770 p
  8. 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
  9. Kuznetsov, D. F. Mean-square approximation of iterated Ito and Stratonovich stochastic integrals based on generalized multiple Fourier series. Application to numerical integration of Ito SDEs and semilinear SPDEs (Third edition). Differencialnie Uravnenia i Protsesy Upravlenia. 2023, no. 1, A. 1-A. 947 (In English)
  10. 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. 14184v46 [math. PR]. 2023, 998 p
  11. Kuznetsov, D. F. Multiple Ito and Stratonovich stochastic integrals: approximations, properties, formulas. St. -Petersburg, Polytechnic Univ. Publ., 2013. 382 p. (In English)
  12. 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. )
  13. 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)
  14. 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
  15. 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. 09746v30 [math. PR]. 2023, 144 p
  16. 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
  17. 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. Automat. Remote Control, 79, 7 (2018), 1240-1254
  18. Kuznetsov, D. F. On Numerical modeling of the multidimentional dynamic systems under random perturbations with the 2. 5 order of strong convergence. Automat. Remote Control, 80, 5 (2019), 867-881
  19. 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
  20. 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
  21. Allen, E. Approximation of triple stochastic integrals through region subdivision. Commun. Appl. Analysis (Special Tribute Issue to Professor V. Lakshmikantham), 17 (2013), 355-366
  22. 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
  23. 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
  24. Gaines, J. G., Lyons, T. J. Random generation of stochastic area integrals. SIAM J. Appl. Math. 54 (1994), 1132-1146
  25. 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
  26. Ryden, T., Wiktorsson, M. On the simulation of iterated Ito integrals. Stoch. Proc. Appl. , 91, 1 (2001), 151-168
  27. 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. )
  28. 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. )
  29. Kloeden, P. E., Platen, E., Wright, I. W. The approximation of multiple stochastic integrals. Stoch. Anal. Appl. 10, 4 (1992), 431-441
  30. Rybakov, K. Application of Walsh series to represent iterated Stratonovich stochastic integrals. IOP Conf. Ser. : Mater. Sci. Eng. 927. 2020, (2020), article id: 012080, 10 p
  31. Rybakov, K. Spectral representations of iterated stochastic integrals and their application for modeling nonlinear stochastic dynamics. Mathematics. 11, 19 (2023), 4047
  32. Rybakov, K. A. Using spectral form of mathematical description to represent Ito iterated stochastic integrals. vol 274. Springer Publ., 2022, pp. 331-344
  33. 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. )
  34. Foster, J., Habermann, K. Brownian bridge expansions for Levy area approximations and particular values of the Riemann zeta function. Combin. Prob. Comp. (2022), 1-28
  35. 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
  36. Malham, S. J. A., Wiese A. Efficient almost-exact Levy area sampling. Stat. Prob. Letters, 88 (2014), 50-55
  37. Stump, D. M., Hill J. M. On an infinite integral arising in the numerical integration of stochastic differential equations. Proc. Royal Soc. of London. Series A. Math., Phys. Eng. Sci., 461, 2054 (2005), 397-413
  38. Platen, E., Bruti-Liberati, N. Numerical solution of stochastic differential equations with jumps in finance. Berlin, Heidelberg, Springer-Verlag Publ., 2010. 868 p
  39. Rybakov, K. A. Orthogonal expansion of multiple Ito stochastic integrals. Differencialnie Uravnenia i Protsesy Upravlenia, 2021, no. 3, 109-140 (In Russ. )
  40. Ito, K. Multiple Wiener integral. J. Math. Soc. Japan. 3, 1 (1951), 157-169
  41. Chung, K. L., Williams, R. J. Introduction to stochastic integration. 2nd Ed. Probability and its Applications. Ed. Liggett T., Newman C., Pitt L. Boston, Basel, Berlin, Birkhauser Publ., 1990. 276 p
  42. Kuo, H. -H. Introduction to Stochastic Integration. Universitext (UTX). N. Y., Springer Publ., 2006. 289 p
  43. Fox, R., Taqqu, M. S. Multiple stochastic integrals with dependent integrators. J. Multivariate Anal. 21 (1987), 105-127
  44. Major, P. The theory of Wiener-Ito integrals in vector valued Gaussian stationary random fields. Part I. Moscow Math. J. 20, 4 (2020), 749-812
  45. Major, P. Multiple Wiener-Ito integrals with applications to limit theorems. Second ed. Cham, Heidelberg, New York, Dordrecht, London, Springer Publ., 2014. 126 p
  46. Major, P. Wiener-Ito integral representation in vector valued Gaussian stationary random fields. arXiv:1901. 04084v1 [math. PR]. 2019, 90 p

Full text (pdf)