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 Approach to the Series Expansion of Iterated Stratonovich Stochastic Integrals with Respect to Components of a Multidimensional Wiener Process. The Case of Arbitrary Complete Orthonormal Systems in Hilbert Space

Author(s):

Dmitriy Feliksovich Kuznetsov

Peter the Great Saint-Petersburg Polytechnic University

sde_kuznetsov@inbox.ru

Abstract:

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 a multidimensional Wiener process. This approach was proposed by the author in 2022 and is based on generalized multiple Fourier series in complete orthonormal systems of functions in Hilbert space. In the previous parts of this work, expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 6 were obtained. At that, the expansions were constructed using two specific bases in Hilbert space. More precisely, Legendre polynomials and the trigonometric Fourier basis were used. In this paper, expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 4 are obtained on the base of arbitrary complete orthonormal systems of functions in Hilbert space. Sufficient conditions for the expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity are formulated in terms of trace series. The results of the article will be useful for construction of strong numerical methods with orders 1.0, 1.5 and 2.0 (based on the Taylor-Stratonovich expansion) for Ito stochastic differential equations with 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., 1988. 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
  10. Kuznetsov, D. F. New representations of the Taylor-Stratonovich expansion. J. Math. Sci. (N. Y. ). 118, 6 (2003), 5586-5596
  11. Kuznetsov, D. F. Chislennoye integrirovanie stokhasticheskih differencial'nyh uravnenii. 2. [Numerical integration of stochastic differential equations. 2. ]. St. -Petersburg, Polytechn. Univ. Publ., 2006. 764 p
  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
  13. 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. )
  14. 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. 14184v52 [math. PR]. 2024, 1138 p
  15. 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. Differencialnie Uravnenia i Protsesy Upravlenia, 2022, no. 2, 83-186 (In English)
  16. 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. II. Differencialnie Uravnenia i Protsesy Upravlenia, 2022, no. 4, 135-194 (In English)
  17. Allen, E. Approximation of triple stochastic integrals through region subdivision. Commun. in Appl. Anal. (Special Tribute Issue to Professor V. Lakshmikantham), 17 (2013), 355-366
  18. 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
  19. 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
  20. Gaines, J. G., Lyons, T. J. Random generation of stochastic area integrals. SIAM J. Appl. Math. 54 (1994), 1132-1146
  21. 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
  22. Ryden, T., Wiktorsson, M. On the simulation of iterated Ito integrals. Stoch. Proc. and their Appl. , 91:1 (2001), 151-168
  23. 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. )
  24. 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. )
  25. Kloeden, P. E., Platen, E., Wright, I. W. The approximation of multiple stochastic integrals. Stoch. Anal. Appl., 10:4 (1992), 431-441
  26. Rybakov, K. Spectral representations of iterated stochastic integrals and their application for modeling nonlinear stochastic dynamics. Mathematics. 11:19 (2023), 4047
  27. Rybakov, K. A. Using spectral form of mathematical description to represent Stratonovich iterated stochastic integrals. Smart Innovation, Systems and Technologies, vol 217. Singapore, Springer Publ., 2021, pp. 287-304
  28. Rybakov, K. A. Using spectral form of mathematical description to represent iterated Ito stochastic integrals. Smart Innovation, Systems and Technologies, vol 274. Singapore, Springer Publ., 2022, pp. 331-344
  29. 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. Differencial’nie uravnenia i processy upravlenia. 2019, no. 4, 32-52 (In Russ. )
  30. 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
  31. Kastner, F., Robler, 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
  32. Malham, S. J. A., Wiese A. Efficient almost-exact Lé vy area sampling. Stat. Prob. Letters, 88 (2014), 50-55
  33. 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
  34. Platen, E., Bruti-Liberati, N. Numerical solution of stochastic differential equations with jumps in finance. Berlin, Heidelberg, Springer-Verlag Publ., 2010. 868 p
  35. Rybakov, K. A. Orthogonal expansion of multiple Ito stochastic integrals. Differencialnie Uravnenia i Protsesy Upravlenia, 2021, no. 3, 109-140 (In Russ. )
  36. Rybakov, K. A. Orthogonal expansion of multiple Stratonovich stochastic integrals. Differencialnie Uravnenia i Protsesy Upravlenia, 2021, no. 4, 81-115 (In Russ. )
  37. Kuznetsov, D. F. Multiple Ito and Stratonovich stochastic integrals: approximations, properties, formulas. (In English). St. -Petersburg, Polytechn. Univ. Publ., 2013, 382 p
  38. 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
  39. 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
  40. 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
  41. Kuznetsov D. F. Explicit one-step numerical method with the strong convergence order of 2. 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
  42. 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
  43. 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
  44. Kuznetsov, D. F. 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. Differencial’nie uravnenia i processy upravlenia. 2023, no. 4, 67-124 (In English)
  45. Kuznetsov, D. F. Application of the method of approximation of iterated stochastic Ito integrals based on generalized multiple Fourier series to the high-order strong numerical methods for non-commutative semilinear stochastic partial differential equations. Differencial’nie uravnenia i processy upravlenia. 2019, no. 3, 18-62 (In English)
  46. Kuznetsov, D. F. Application of multiple Fourier-Legendre series to strong exponential Milstein and Wagner-Platen methods for non-commutative semilinear stochastic partial differential equations. Differencialnie Uravnenia i Protsesy Upravlenia, 2020, no. 3, 129-162 (In English)
  47. 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. Cham, Springer Publ., 2021, pp. 17-32
  48. Kuznetsov, D. F. Expansion of iterated Stratonovich stochastic integrals, based on generalized multiple Fourier series. Ufa Math. J. , 11:4 (2019), 49-77
  49. Kuznetsov, D. F. The hypotheses on expansions of iterated Stratonovich stochastic integrals of arbitrary multiplicity and their partial proof. (In English). arXiv:1801. 03195v32 [math. PR]. 2024, 271 p
  50. 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. 09516v34 [math. PR]. 2024, 343 p
  51. Kuznetsov, D. F. Expansion of iterated Stratonovich stochastic integrals of fifth and sixth multiplicity based on generalized multiple Fourier series. (In English). arXiv:1802. 00643v28 [math. PR]. 2024, 260 p
  52. 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. Differencialnie Uravnenia i Protsesy Upravlenia. 2023, no. 1, A. 1-A. 947 (In English)
  53. 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
  54. Kuznetsov, D. F. [A method of expansion and approximation of repeated stochastic Stratonovich integrals based on multiple Fourier series on full orthonormal systems]. Differencial’nie uravnenia i processy upravlenia, 1997, no. 1, 18-77 (In Russ. )
  55. Ito, K. Multiple Wiener integral. J. Math. Soc. Japan. 3, 1 (1951), 157-169
  56. Kuo, H. -H. Introduction to Stochastic Integration. N. Y., Springer Publ., 2006. 289 p
  57. Rybakov, K. On traces of linear operators with symmetrized Volterra-type kernels. Symmetry, 15:10, 1821 (2023), 1-18
  58. Gohberg, I. C., Krein, M. G. Vvedenie v teoriju lineinyh nesamosoprjazhennyh operatorov v gilbertovom prostranstve [Introduction the theory of linear nonselfadjoint operators in Hilbert space]. Moscow, Fizmatlit Publ., 1965. 448 p
  59. Gohberg, I., Goldberg, S., Krupnik, N. Traces and determinants of linear operators. Basel, Boston, Berlin, Birkhauser-Verlag Publ., 2000. 258 p
  60. Pugachev, V. S. Lectures on fuctional analysis. Moscow, MAI Publ., 1996, 744 p
  61. Reed, M., Simon, B. Functional analysis. Vol 1. San Diego, Academic Press Publ., 1980, 400 p
  62. Brislawn, C. Kernels of trace class operators. Proc. Amer. Math. Soc., 104, 4 (1988), 1181-1190
  63. Sole, J. LL., Utzet, F. Stratonovich integral and trace. Stoch. : Int. J. Prob. Stoch. Proc., 29, 2 (1990), 203-220
  64. Bardina, X., Jolis, M. Weak convergence to the multiple Stratonovich integral. Stoch. Proc. Appl., 90, 2 (2000), 277-300
  65. Bardina, X., Rovira, C. On the strong convergence of multiple ordinary integrals to multiple Stratonovich integrals. Publicacions Matematiques, 65 (2021), 859-876
  66. 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. Differencial’nie uravnenia i processy upravlenia, 2021, no. 1, 93-422 (In English)

Full text (pdf)