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

Features of the Expansion of Multiple Stochastic Stratonovich Integrals Using Walsh and Haar Functions

Author(s):

Konstantin Rybakov

Moscow Aviation Institute (National Research University)

rkoffice@mail.ru

Abstract:

The problem of the root-mean-square convergence for approximations of multiple stochastic Stratonovich integrals based on the generalized multiple Fourier series method using Walsh and Haar functions is considered. It is shown that when they are chosen to expand multiple stochastic integrals, the proof of the root-mean-square convergence of a subsequence of series partial sums, which is formed in a way that is quite natural for these functions, does not require the explicit fulfillment of any additional conditions, except for the condition of the existence of the multiple stochastic Stratonovich integral.

Keywords

References:

  1. Kloeden, P. E., Platen, E. Numerical Solution of Stochastic Differential Equations. Springer-Verlag, 1995
  2. 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. Differencialnie Uravnenia i Protsesy Upravlenia, 2023, no. 1, pp. A. 1-A. 947
  3. 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, pp. 83-186
  4. 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, pp. 135-194
  5. Rybakov, K. A. Orthogonal expansion of multiple Ito stochastic integrals. Differencialnie Uravnenia i Protsesy Upravlenia, 2021, no. 3, pp. 109-140. (In Russian)
  6. Rybakov, K. A. Orthogonal expansion of multiple Stratonovich stochastic integrals. Differencialnie Uravnenia i Protsesy Upravlenia, 2021, no. 4, pp. 81-115. (In Russian)
  7. Delgado, R. Multiple Ogawa, Stratonovich and Skorohod anticipating integrals. Stochastic Analysis and Applications, 1998, vol. 16, no. 5, pp. 859-872
  8. Farre, M., Jolis, M., Utzet, F. Multiple Stratonovich integral and Hu-Meyer formula for Levy processes. The Annals of Probability, 2010, vol. 38, no. 6, pp. 2136-2169
  9. Kolmogorov, A. N., Fomin, S. V. Elementy teorii funktsiy i funktsional'nogo analiza [Elements of the theory of functions and functional analysis]. Moscow, Nauka Publ., 1976
  10. Gikhman, I. I., Skorokhod, A. V. Introduction to the Theory of Random Processes. Dover Publ., 1997
  11. Lapin, S. V., Egupov, N. D. Teoriya matrichnykh operatorov i ee prilozhenie k zadacham avtomaticheskogo upravleniya [Theory of matrix operators and its application to problems of automatic control]. Moscow, BMSTU Publ., 1997
  12. Rybin, V. V. Modelirovaniye nestatsionarnykh sistem upravleniya tselogo i drobnogo poryadka proyektsionno-setochnym spektral'nym metodom [Modeling of nonstationary integer-order and fractional-order control systems by grid-projection spectral method]. Moscow, MAI Publ., 2013
  13. Golubov, B., Efimov, A., Skvortsov, V. Walsh series and transforms: Theory and applications. Springer
  14. Rybakov, K. A. On the orthogonal expansion of the iterated Stratonovich stochastic integrals. Herald of Dagestan State University. Series 1. Natural Sciences, 2022, vol. 35, no. 3, pp. 17-23. (In Russian)

Full text (pdf)