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

Mean-square Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Integration of Ito SDEs and Semilinear SPDEs

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 monograph is devoted to the mean-square approximation of iterated Ito and Stratonovich stochastic integrals in the context of numerical integration of Ito stochastic differential equations and non-commutative semilinear stochastic partial differential equations with nonlinear multiplicative trace class noise. The presented book opens up a new direction in researching of iterated stochastic integrals with respect to components of the multidimensional Wiener process. For the first time we successfully use the generalized multiple Fourier series (multiple Fourier-Legendre series as well as multiple trigonometric Fourier series) converging in the sense of norm in Hilbert space for the expansion and mean-square approximation of iterated Ito stochastic integrals of arbitrary multiplicity k (Chapter 1). The convergence with probability 1 as well as the convergence in the sense of n-th moment for the mentioned expansion have been proved (n=2, 3,...). For these two types of convergence, the convergence rate was found. Moreover, the expansion for iterated Ito stochastic integrals is adapted for iterated Stratonovich stochastic integrals of multiplicities 1 to 5 (Chapter 2) as well as for some other types of iterated stochastic integrals (Chapter 1). Two theorems on expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k based on generalized iterated Fourier series with pointwise convergence are formulated and proved (Chapter 2). The results of Chapters 1 and 2 can be considered from the point of view of the Wong-Zakai approximation for the case of a multidimensional Wiener process and the Wiener process approximation based on its series expansion using Legendre polynomials and trigonometric functions. The integration order replacement technique for the class of iterated Ito stochastic integrals has been introduced (Chapter 3). Exact expressions are obtained for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k (Chapter 1) and iterated Stratonovich stochastic integrals of multiplicities 1 to 4 (Chapter 5). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the unified Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4) using the system of Legendre polynomials and the system of trigonometric functions. The methods formulated in this book have been compared with some existing methods (Chapter 6). The results of Chapter 1 were applied (Chapter 7) to the approximation of iterated stochastic integrals with respect to the finite-dimensional approximation of the Q-Wiener process (for integrals of multiplicity k) and with respect to the infinite-dimensional Q-Wiener process (for integrals of multiplicities 1 to 3).

Ключевые слова

Ссылки:

  1. 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
  2. Kuznetsov, D. F. Stokhasticheskie differencial’nye uravnenia: teoriya i practika chislennogo resheniya. S programmoi dl’ja PC v sisteme MATLAB 7. 0. [Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MATLAB 7. 0 program]. St. -Petersburg, Polytechnic Univ. Publ., 2007. 778 p. DOI: http://doi.org/10.18720/SPBPU/2/s17-228
  3. 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. xxxii+770 p. DOI: http://doi.org/10.18720/SPBPU/2/s17-229
  4. 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. xxxiv+768 p. DOI: http://doi.org/10.18720/SPBPU/2/s17-230
  5. Kuznetsov, D. F. Stokhasticheskie differencial’nye uravnenia: teoriya i practika chislennogo resheniya. . S programmami dl’ja PC v sisteme MATLAB 7. 0. Izd. 4-e. [Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MATLAB 7. 0 programs. 4th Ed. ]. St. -Petersburg, Polytechnic Univ. Publ., 2010. xxx+786 p. DOI: http://doi.org/10.18720/SPBPU/2/s17-231
  6. Kuznetsov, D. F. [Multiple Stochastic Ito and Stratonovich Integrals and Multiple Fourier Series]. Differencialnie Uravnenia i Protsesy Upravlenia, 2010, no. 3, A. 1-A. 257 (In Russ. ) Available at: https://diffjournal.spbu.ru/EN/numbers/2010.3/article.2.1.html
  7. Kuznetsov, D. F. Strong approximation of multiple Ito and Stratonovich stochastic integrals: multiple Fourier series approach. St. -Petersburg, Polytechnic Univ. Publ., 2011. 250 p. (In English). DOI: http://doi.org/10.18720/SPBPU/2/s17-232
  8. Kuznetsov, D. F. Strong approximation of multiple Ito and Stratonovich stochastic integrals: multiple Fourier series approach. 2nd Ed. St. -Petersburg, Polytechnic Univ. Publ., 2011. 284 p. (In English). DOI: http://doi.org/10.18720/SPBPU/2/s17-233
  9. Kuznetsov, D. F. Approximation of Multiple Ito and Stratonovich Stochastic Integrals. Multiple Fourier Series Approach. Saarbrü cken, Lambert Academic Publ., 2012. 409 p. (In English). Available at: http://www.sde-kuznetsov.spb.ru/12a.pdf
  10. 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
  11. Kuznetsov, D. F. Multiple Ito and Stratonovich stochastic integrals: Fourier-Legendre and trigonometric expansions, approximations, formulas. Differencialnie Uravnenia i Protsesy Upravlenia, 2017, no. 1, A. 1-A. 385 (In English) Available at: http://diffjournal.spbu.ru/EN/numbers/2017.1/article.2.1.html
  12. Kuznetsov, D. F. [Stochastic differential equations: theory and practice of numerical solution. With programs on MATLAB. 5-th Ed. ]. Differencialnie Uravnenia i Protsesy Upravlenia, 2017, no. 2, A. 1-A. 1000 (In Russ. ) Available at: https://diffjournal.spbu.ru/EN/numbers/2017.2/article.2.1.html
  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. ) Available at: http://diffjournal.spbu.ru/EN/numbers/2018.4/article.2.1.html
  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. Differencialnie Uravnenia i Protsesy Upravlenia. 2020, no. 4, A. 1-A. 606 (In English) Available at: http://diffjournal.spbu.ru/EN/numbers/2020.4/article.1.8.html
  15. 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. 14184 [math. PR]. 2020, 788 p. Available at: https://arxiv.org/abs/2003.14184
  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. DOI: http://doi.org/10.1134/S0965542518070096
  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. DOI: http://doi.org/10.1134/S0005117918070056
  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. DOI: http://doi.org/10.1134/S0005117919050060
  19. 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
  20. 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
  21. Kuznetsov, D. F. Explicit one-step numerical method with the strong convergence order of
  22. 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
  23. 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. Differencialnie Uravnenia i Protsesy Upravlenia, 2019, no. 3, 18-62 (In English) Available at: http://diffjournal.spbu.ru/EN/numbers/2019.3/article.1.2.html
  24. Kuznetsov, D. F. Application of multiple Fourier-Legendre series to strong exponentialMilstein and Wagner-Platen methods for non-commutative semilinear stochastic partial differential equations. Differencialnie Uravnenia i Protsesy Upravlenia, 2020, no. 3, 129-162 (In English) Available at: https://diffjournal.spbu.ru/RU/numbers/2020.3/article.1.6.html
  25. Kuznetsov, D. F. Strong approximation of iterated Ito and Stratonovich stochastic integrals. Abstracts of talks given at the 4th International Conference on Stochastic Methods. Theory Prob. Appl. , 65, 1 (2020), 141-142 (In English). DOI: http://doi.org/10.1137/S0040585X97T989878
  26. 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
  27. Kuznetsov, D. F. [Expansion of multiple Stratonovich stochastic integrals of second multiplicity, based on double Fourier-Legendre series summarized by Prinsheim method]. Differencialnie Uravnenia i Protsesy Upravlenia, 2018, no. 1, 1-34 (In Russ. ) Available at: http://diffjournal.spbu.ru/EN/numbers/2018.1/article.1.1.html
  28. 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. 09746 [math. PR]. 2017, 97 p. Available at: https://arxiv.org/abs/1712.09746
  29. Kuznetsov, D. F. Development and application of the Fourier method to the mean-square approximation of iterated Ito and Stratonovich stochastic integrals. (In English). arXiv:1712. 08991 [math. PR]. 2017, 46 p. Available at: https://arxiv.org/abs/1712.08991
  30. 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. 01079 [math. PR]. 2018, 57 p. Available at: https://arxiv.org/abs/1801.01079
  31. 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. 00231 [math. PR]. 2017, 100 p. Available at: https://arxiv.org/abs/1801.00231
  32. Kuznetsov, D. F. Expansions of Iterated Stratonovich stochastic integrals of multiplicities 1 to 4 based on generalized multiple Fourier series. (In English). arXiv:1712. 09516 [math. PR]. 2017, 109 p. Available at: https://arxiv.org/abs/1712.09516
  33. Kuznetsov, D. F. Expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity based on generalized iterated Fourier series converging pointwise. (In English). arXiv:1801. 00784 [math. PR]. 2018, 74 p. Available at: https://arxiv.org/abs/1801.00784
  34. Kuznetsov, D. F. Expansion of iterated Stratonovich stochastic integrals of multiplicity 3 based on generalized multiple Fourier series converging in the mean: general case of series summation. (In English). arXiv:1801. 01564 [math. PR]. 2018, 59 p. Available at: https://arxiv.org/abs/1801.01564
  35. Kuznetsov, D. F. Expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 4. Combined approach based on generalized multiple and iterated Fourier series. (In English). arXiv:1801. 05654 [math. PR]. 2018, 41 p. Available at: https://arxiv.org/abs/1801.05654
  36. Kuznetsov, D. F. Expansion of iterated Stratonovich stochastic integrals of multiplicity 2. Combined approach based on generalized multiple and iterated Fourier series. (In English). arXiv:1801. 07248 [math. PR]. 2018, 18 p. Available at: https://arxiv.org/abs/1801.07248
  37. Kuznetsov, D. F. Expansion of iterated Stratonovich stochastic integrals of fifth multiplicity based on generalized multiple Fourier series. (In English). arXiv:1802. 00643 [math. PR]. 2018, 38 p. Available at: https://arxiv.org/abs/1802.00643
  38. Kuznetsov, D. F. The hypotheses on expansions of iterated Stratonovich stochastic integrals of arbitrary multiplicity and their partial proof. (In English). arXiv:1801. 03195 [math. PR]. 2018, 35 p. Available at: https://arxiv.org/abs/1801.03195
  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. (In English). arXiv:1901. 02345 [math. GM]. 2019, 34 p. Available at: https://arxiv.org/abs/1901.02345
  40. Kuznetsov, D. F. Expansion of iterated stochastic integrals with respect to martingale Poisson measures and with respect to martingales based on generalized multiple Fourier series. (In English). arXiv:1801. 06501 [math. PR]. 2018, 37 p. Available at: https://arxiv.org/abs/1801.06501
  41. Kuznetsov, D. F. To numerical modeling with strong orders 1. 0, 1. 5, and 2. 0 of convergence for multidimensional dynamical systems with random disturbances. (In English). arXiv:1802. 00888 [math. PR]. 2018, 22 p. Available at: https://arxiv.org/abs/1802.00888
  42. Kuznetsov, D. F, Numerical simulation of 2. 5-set of iterated Stratonovich stochastic integrals of multiplicities 1 to 5 from the Taylor-Stratonovich expansion. (In English). arXiv:1806. 10705[math. PR]. 2018, 24 p. Available at: https://arxiv.org/abs/1806.10705
  43. Kuznetsov, D. F, Numerical simulation of 2. 5-set of iterated Ito stochastic integrals of multiplicities 1 to 5 from the Taylor-Ito expansion. (In English). arXiv:1805. 12527 [math. PR]. 2018, 23 p. Available at: https://arxiv.org/abs/1805.12527
  44. Kuznetsov, D. F. Explicit one-step strong numerical methods of orders 2. 0 and 2. 5 for Ito stochastic differential equations based on the unified Taylor-Ito and Taylor-Stratonovich expansions. (In English). arXiv:1802. 04844 [math. PR]. 2018, 31 p. Available at: https://arxiv.org/abs/1802.04844
  45. Kuznetsov, D. F. Strong numerical methods of orders 2. 0, 2. 5, and 3. 0 for Ito stochastic differential equations based on the unified stochastic Taylor expansions and multiple Fourier-Legendre series. (In English). arXiv:1807. 02190 [math. PR]. 2018, 38 p. Available at: https://arxiv.org/abs/1807.02190
  46. Kuznetsov, D. F. Expansion of iterated Stratonovich stochastic integrals of multiplicity 2 based on double Fourier-Legendre series summarized by Pringsheim method. (In English). arXiv:1801. 01962 [math. PR]. 2018, 30 p. Available at: https://arxiv.org/abs/1801.01962
  47. 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. 03724 [math. GM]. 2019, 41 p. Available at: https://arxiv.org/abs/1905.03724
  48. 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. 02612 [math. PR]. 2019, 32 p. Available at: https://arxiv.org/abs/1912.02612
  49. Kuznetsov, D. F. Expansions of iterated Stratonovich stochastic integrals from the Taylor-Stratonovich expansion based on multiple trigonometric Fourier series. Comparison with the Milstein expansion. (In English). arXiv:1801. 08862 [math. PR]. 2018, 30 p. Available at: https://arxiv.org/abs/1801.08862
  50. Kuznetsov, D. F. New simple method of expansion of iterated Ito stochastic integrals of multiplicity 2 based on expansion of the Brownian motion using Legendre polynomials and trigonometric functions. (In English). arXiv:1807. 00409 [math. PR], 2019, 20 p. Available at: https://arxiv.org/abs/1807.00409
  51. Kuznetsov, D. F. Four new forms of the Taylor-Ito and Taylor-Stratonovich expansions and its application to the high-order strong numerical methods for Ito stochastic differential equations. (In English). arXiv:2001. 10192 [math. PR], 2020, 80 p. Available at https://arxiv.org/abs/2001.10192
  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, M. D., Kuznetsov, D. F. Implementation of strong numerical methods of orders 0. 5, 1. 0, 1. 5, 2. 0, 2. 5, and 3. 0 for Ito SDEs with non-commutative noise based on the unified Taylor-Ito and Taylor-Stratonovich expansions and multiple Fourier-Legendre series. (In English). arXiv:2009. 14011 [math. PR], 2020, 336 p. Available at: https://arxiv.org/abs/2009.14011
  54. Kuznetsov, M. D., Kuznetsov, D. F. Optimization of the mean-square approximation procedures for iterated Ito stochastic integrals of multiplicities 1 to 5 from the unified Taylor-Ito expansion based on multiple Fourier-Legendre series. (In English). arXiv:2010. 13564 [math. PR], 2020, 59 p. Available at: https://arxiv.org/abs/2010.13564
  55. Kuznetsov, D. F., Kuznetsov, M. D. Optimization of the mean-square approximation procedures for iterated Ito stochastic integrals based on multiple Fourier-Legendre series. J. Phys. : Conf. Series. 1925, 1 (2021), article id: 012010, 12 p. DOI: http://doi.org/10.1088/1742-6596/1925/1/012010
  56. Kuznetsov, D. F. Application of multiple Fourier-Legendre series to the implementation of strong exponential Milstein and Wagner-Platen methods for non-commutative semilinear SPDEs. Proc. of the XIII Int. Conf. on Appl. Math. and Mech. in the Aerospace Ind. (AMMAI-2020). Moscow, MAI Publ., 2020, 451-453. Available at: http://www.sde-kuznetsov.spb.ru/20e.pdf
  57. 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
  58. Kuznetsov, D. F. The proof of convergence with probability 1 in the method of expansion of iterated Ito stochastic integrals based on generalized multiple Fourier series. (In English). arXiv:2006. 16040 [math. PR], 2020, 29 pp. Available at: https://arxiv.org/abs/2006.16040
  59. Kuznetsov, D. F., Kuznetsov, M. D. A software package for implementation of strong numerical methods of convergence orders 0. 5, 1. 0, 1. 5, 2. 0, 2. 5, and 3. 0 for Ito SDEs with non-commutative multi-dimensional noise. Abstracts of the 19th Int. Conf. Aviation and Cosmonautics (AviaSpace-2020). Moscow, Pero Publ., 2020, 569-570. Available at: http://www.sde-kuznetsov.spb.ru/20h.pdf
  60. Kuznetsov, D. F. Application of multiple Fourier-Legendre series to the implementation of strong exponential Milstein and Wagner-Platen methods for non-commutative semilinear SPDEs with nonlinear multiplicative trace class noise. Proc. of the 5th Int. Conf. on Stochastic Methods (ICSM-5). Moscow, RUDN Publ., 2020, 88-92. Available at: http://www.sde-kuznetsov.spb.ru/20i.pdf
  61. Kuznetsov, D. F. Application of the Fourier method for the numerical solution of stochastic differential equations. Abstracts of the 2nd Int. Conf. on Mathematical Modeling in Applied Sciences (ICMMAS 19). Belgorod State Univ. & Alpha-Publishing Publ., 2019, 236-237. Available at: http://www.sde-kuznetsov.spb.ru/19c.pdf
  62. Kuznetsov, D. F. Strong approximation of multiple Ito and Stratonovich stochastic integrals. Abstracts of the Int. Conf. on Mathematical Modeling in Applied Sciences(ICMMAS 17). St. -Petersburg, Polytechnic Univ. Publ., 2017, 141-142. Available at: http://www.sde-kuznetsov.spb.ru/17.pdf
  63. 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
  64. Clark J. M. C., Cameron R. J. The maximum rate of convergence of discrete approximations for stochastic differential equations. Stochastic Differential Systems Filtering and Control. Lecture Notes in Control and Information Sciences, vol 25. Ed. Grigelionis B. Berlin, Heidelberg, Springer Publ., 1980. 162-171
  65. Wong, E., Zakai, M. On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Stat.. 36, 5 (1965), 1560-1564
  66. Wong, E., Zakai, M. On the relation between ordinary and stochastic differential equations. Int. J. Eng. Sci. 3 (1965), 213-229
  67. Ikeda, N., Watanabe, S. Stochastic Differential Equations and Diffusion Processes. 2nd Ed. Amsterdam, Oxford, New-York, North-Holland Publishing Company, 1989. 555 p
  68. 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
  69. Kuznetsov, D. F. [Some problems of the theory of numerical solution of Ito stochastic differential equations]. Differencialnie Uravnenia i Protsesy Upravlenia, 1998, no. 1, 66-367 (In Russ. ) Available at: http://diffjournal.spbu.ru/EN/numbers/1998.1/article.1.3.html
  70. 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
  71. Kuznetsov, D. F. New representations of explicit one-step numerical methods for jump diffusion stochastic differential equations. Comp. Math. Math. Phys. , 41, 6 (2001), 874-888 Available at: http://www.sde-kuznetsov.spb.ru/01b.pdf
  72. Kulchitsky, O. Yu., Kuznetsov, D. F. Approximation of multiple Ito stochastic integrals. VINITI, 1678-V94 (1994), 42 p. (In Russ. )
  73. Kuznetsov, D. F. Metody chislennogo modelirovanija reshenii system stokhasticheskih differencial'nyh uravnenii Ito v zadachah mekhaniki. Kand. Diss. [Methods of numerical simulation of stochastic differential Ito equations solutions in problems of mechanics. Ph. D. ], St. -Petersburg, 1996. 260 p
  74. Milstein G. N. Chislennoye integrirovaniye stokhasticheskih differencial’nyh uravnenii [Numerical integration of stochastic differential equations], Sverdlovsk, Ural. Univ. Publ
  75. 225 p
  76. Kloeden, P. E., Platen, E., Wright, I. W. The approximation of multiple stochastic integrals. Stoch. Anal. Appl. 10, 4 (1992), 431-441
  77. Kloeden, P. E., Platen, E. Numerical solution of stochastic differential equations. Berlin, Springer-Verlag Publ., 1992. 632 p
  78. Kloeden, P. E., Platen, E., Schurz, H. Numerical solution of SDE through computer experiments. Berlin, Springer-Verlag Publ., 1994. 292 p
  79. 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. )
  80. 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. )
  81. 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
  82. Ryden, T., Wiktorsson, M. On the simulation of iterated Ito integrals. Stoch. Proc. and their Appl. , 91, 1 (2001), 151-168
  83. Gaines, J. G., Lyons, T. J. Random generation of stochastic area integrals. SIAM J. Appl. Math. 54 (1994), 1132-1146
  84. Gilsing, H., Shardlow, T. SDELab: A package for solving stochastic differential equations in MATLAB. J. Comp. Appl. Math. , 205 (2007), 1002-1018
  85. Milstein, G. N., Tretyakov, M. V. Stochastic numerics for mathematical physics. Berlin, Springer-Verlag Publ., 2004. xx+596 p
  86. Platen, E., Bruti-Liberati, N. Numerical solution of stochastic differential equations with jumps in finance. Berlin, Heidelberg, Springer-Verlag Publ., 2010. 868 p
  87. Allen, E. Approximation of triple stochastic integrals through region subdivision. Commun. Appl. Analysis (Special Tribute Issue to Professor V. Lakshmikantham), 17 (2013), 355-366
  88. 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
  89. 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
  90. Zahri, M. Multidimensional Milstein scheme for solving a stochastic model for prebiotic evolution. J. of Taibah Univ. for Science. 8, 2 (2014), 186-198
  91. Chernykh, N. V., Pakshin, P. V. Numerical solution algorithms for stochastic differential systems with switching diffusion. Autom. Remote Control. 74, 12 (2013), 2037-2063. DOI: https://doi.org/10.1134/S0005117913120072
  92. 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
  93. Gihman, I. I., Skorohod, A. V. Stokhasticheskie differencial’nye uravnenia [Stochastic differential equations]. Kiev, Naukova Dumka Publ., 1968. 354 p
  94. 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
  95. Skorohod, A. V. Sluchainye protsesy s nezavisimymi proraschenyami [Stochastic processes with independent increments]. Moscow, Nauka Publ., 1964. 280 p
  96. Koroluk, V. S., Portenko, N. I., Skorohod, A. V., Turbin, A. F. Spravochnik po teorii veroyatnostei i matematicheskoi statistike [Handbook on the probability theory and mathematical statistics]. Moscow, Nauka Publ., 1985. 640 p
  97. Shiryaev, A. N. Probability. Springer-Verlag, New York, 1996. 624 p
  98. Kloeden, P. E., Neuenkirch, A. The pathwise convergence of approximation schemes for stochastic differential equations. LMS J. Comp. Math. 10 (2007), 235-253
  99. Ito, K. Multiple Wiener integral. J. Math. Soc. Japan. 3, 1 (1951), 157-169
  100. 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
  101. 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
  102. 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
  103. Hobson, E. W. The theory of spherical and ellipsoidal harmonics. Cambridge, Cambridge Univ. Press Publ., 1931. 502 p
  104. Starchenko, T. K. [About conditions of convergence of double Fourier-Legendre series]. Trudy inst. matematiki NAN Belarusi “Analiticheskije metody analiza i differencial'nyh uravnenii” [Proc. of the Mathematical inst. of NAS of Belarus “Analitical methods of analysis and differential equations”]. Minsk, 2005, no. 5, pp. 124-126. (in Russ. )
  105. Suetin, P. K. Klassicheskije ortogonal'nye mnogochleny. Izd 3-e. [Classical orthogonal polynomials. 3rd Ed. ]. Moskow, Fizmatlit, 2005. 480 p
  106. Il’in, V. A., Poznyak, E. G. Osnovy matematicheskogo analiza. Chast’ 2 [Foundations of mathematical analysis. Part II]. Moscow, Nauka Publ., 1973. 448 p
  107. Kuznetsov, D. F. Theorems about integration order replacement in multiple stochastic integrals. VINITI, 3607-V97 (1997), 31 p. (In Russ. )
  108. Kuznetsov, D. F. Integration order replacement technique for iterated Ito stochastic integrals and iterated stochastic integrals with respect to martingales. (In English). arXiv:1801. 04634 [math. PR]. 2018, 27 p. Available at: https://arxiv.org/abs/1801.04634
  109. Sjö lin, P. Convergence almost everywhere of certain singular integrals and multiple Fourier series. Ark. Mat. 9, 1-2 (1971), 65-90
  110. 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
  111. Luo, W. Wiener chaos expansion and numerical solutions of stochastic partial differential equations. Ph. D., California Institute of Technology, 2006. 225 p
  112. Wong, E., Zakai, M. Riemann-Stieltjes approximations of stochastic integrals. Z. Warsch. Verw. Gebiete. 12 (1969), 87-97
  113. Ikeda, N., Nakao, S., Yamato, Y. A class of approximations of Brownian motion. Publ. RIMS Kyoto Univ. 13, 1 (1977), 285-300
  114. Koneczny, F. On the Wong-Zakai approximation of stochastic differential equations. J. Mult. Anal. 13 (1983), 605-611
  115. Karatzas, I., Shreve, S. E. Brownian Motion and Stochastic Calculus. 2nd Ed. N. Y., Berlin, Heidelberg, London, Paris, Tokyo, Hong Kong, Barcelona, Springer-Verlag Publ., 1991. 470 p
  116. Mackevicius, V., Zibaitis, B. Gaussian approximations of Brownian motion in a stochastic integral. Lith. Math. J. 33 (1993), 393-406. DOI: http://doi.org/10.1007/BF00995993
  117. Twardowska, K. Wong-Zakai approximations for stochastic differential equations. Acta Appl. Math. 43 (1996), 317-359. DOI: http://doi.org/10.1007/BF00047670
  118. Gyö ngy, I., Michaletzky, G. On Wong-Zakai approximations with δ -martingales. Royal Soc. London Proc. Ser. 460, 2041 (2004), 309-324
  119. Gyö ngy, I., Shmatkov, A. Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations. Appl. Math. Optim. 54 (2006), 315-341
  120. 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. vii+132 p
  121. Budhiraja, A., Kallianpur, G. Two results on multiple Stratonovich integrals. Statistica Sinica. 7 (1997), 907-922
  122. Johnson, G. W., Kallianpur, G. Homogeneous chaos, p-forms, scaling and the Feynman integral. Trans. Amer. Math. Soc. 340 (1993), 503-548
  123. 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
  124. Kuznetsov, D. F. Replacement the order of integration in iterated stochastic integrals with respect to martingale. Preprint. St. -Petersburg, St. -Petersburg St. Techn. Univ., SPbGTU Publ., 1999, 11 p. (In Russ. ) DOI: http://doi.org/10.13140/RG.2.2.19936.58889
  125. Arato, M. Linear stochastic systems with constant coefficients. A statistical approach. Berlin, Heidelberg, N. Y., Springer-Verlag Publ., 1982. 289 p
  126. Shiryaev, A. N. Essentials of stochastic finance: facts, models, theory. World Scientific Publ. Co United States, 1999. 852 p
  127. Chugai, K. N., Kosachev, I. M., Rybakov, K. A. Approximate filtering methods in continuous-time stochastic systems. Smart Innovation, Systems and Technologies, vol. 173, Eds. Jain L. C., Favorskaya M. N., Nikitin I. S., Reviznikov D. L. Springer Publ., 2020, pp. 351-371. DOI: https://doi.org/10.1007/978-981-15-2600-8_24
  128. 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
  129. Nasyrov, F. S. Lokal'nye vremena, simmetrichnyje integraly i stochasticheskiy analiz [Local times, symmetric integrals and stochastic analysis]. Moscow, Fizmatlit Publ
  130. 212 p
  131. Kagirova, G. R., Nasyrov, F. S. On an optimal filtration problem for one-dimensional diffusion processes. Siberian Adv. Math. 28, 3 (2018), 155-165
  132. Asadullin, E. M., Nasyrov, F. S. About filtering problem of diffusion processes. Ufa Math. J. 3, 2 (2011), 3-9
  133. Kloeden, P. E., Platen, E. The Stratonovich and Ito-Taylor expansions. Math. Nachr. 151 (1991), 33-50
  134. Platen, E., Wagner, W. On a Taylor formula for a class of Ito processes. Probab. Math. Statist. 3 (1982), 37-51
  135. 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
  136. Kulchitskiy, O. Yu., Kuznetsov, D. F. Unified Taylor-Ito expansion. Differencialnie Uravnenia i Protsesy Upravlenia, 1997, no. 1, 1-17 (In Russ. ) Available at: http://diffjournal.spbu.ru/EN/numbers/1997.1/article.1.1.html
  137. 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
  138. Kulchitsky, O. Yu., Kuznetsov, D. F. Expansion of Ito processes into Taylor-Ito series at the neighborhood of the fixed time moment. VINITI, 2637-V93 (1993), 26 p. (In Russ. )
  139. Kuznetsov, D. F. Two new representations of the Taylor-Stratonovich expansion. Preprint. St. -Petersburg, St. -Petersburg St. Techn. Univ., SPbGTU Publ., 1999, 13 p. DOI: http://doi.org/10.13140/RG.2.2.18258.86729
  140. Wiener, N. Un problѐ me de probabilité s dé nombrables. Bull. Soc. Math. France. 52 (1924), 569-578
  141. Lé vy, P. Wiener's random function and other Laplacian random functions. Proc. 2nd Berkeley Symp. Math. Stat. Prob. 1951, 171-187
  142. Ito, K., McKean, H. Diffusion processes and their sample paths. Berlin, Heidelberg, New York, Springer-Verlag Publ., 1965. 395 p
  143. Neckel, T., Parra Hinojosa, A., Rupp, F. Path-wise algorithms for random and stochastic ODEs with applications to ground-motion-induced excitations of multi-storey buildings. Report number TUM-I1758. Munchen, TU Munchen, 2017, 33 p
  144. Kuznetsov, D. F, Combined method of strong approximation of multiple stochastic integrals. J. Automat. Inform. Sciences. 34, 8 (2002), 6 p. DOI: http://doi.org/10.1615/JAutomatInfScien.v34.i8.40
  145. Kuznetsov, D. F Weak numerical method of order 4. 0 for stochastic differential Ito equations. Vestnik Molodykh Uchenykh. Serya: Prikl. Mat. i Mekh. 4 (2000), 47-52. (In Russ. ) Available at: http://www.sde-kuznetsov.spb.ru/02b.pdf
  146. Gyö gy, I. Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. I. Potential Anal. 9, 1 (1998), 1-25
  147. Gyö ngy, I. Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. II. Potential Anal. 11, 1 (1999), 1-37
  148. Gyö ngy, I. and Krylov, N. An accelerated splitting-up method for parabolic equations. SIAM J. Math. Anal.. 37, 4 (2005), 1070-1097
  149. Hausenblas, E. Numerical analysis of semilinear stochastic evolution equations in Banach spaces. J. Comp. Appl. Math. 147, 2 (2002), 485-516
  150. Hausenblas, E. Approximation for semilinear stochastic evolution equations. Potential Anal. 18, 2 (2003), 141-186
  151. Jentzen, A. Pathwise numerical approximations of SPDEs with additive noise under non global Lipschitz coefficients. Potential Anal. 31, 4 (2009), 375-404
  152. Jentzen, A. Taylor expansions of solutions of stochastic partial differential equations. arXiv:0904. 2232 [math. NA]. 2009, 32 p. Available at: https://arxiv.org/abs/0904.2232
  153. Jentzen, A. and Kloeden, P. E. Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise. Proc. R. Soc. Lond. Ser. Math. Phys. Eng. Sci. 465, Issue 2102 (2009), 649-667
  154. Jentzen, A. and Kloeden, P. E. Taylor expansions of solutions of stochastic partial differential equations with additive noise. Ann. Prob. 38, 2 (2010), 532-569
  155. Jentzen, A., Kloeden, P. E. Taylor Approximations for Stochastic Partial Differential Equations. Philadelphia, SIAM Publ., 2011. 224 p
  156. Jentzen, A. and Rö ckner, M. A Milstein scheme for SPDEs. Foundations Comp. Math. 15, 2 (2015), 313-362
  157. Becker, S., Jentzen, A. and Kloeden, P. E. An exponential Wagner-Platen type scheme for SPDEs. SIAM J. Numer. Anal. 54, 4 (2016), 2389-2426
  158. Zhang, Z., Karniadakis, G. Numerical Methods for Stochastic Partial Differential Equations with White Noise. Springer Publ., 2017. 398 p
  159. Jentzen, A. and Rö ckner, M. Regularity analysis of stochastic partial differential equations with nonlinear multiplicative trace class noise. J. Differ. Eq. 252, 1 (2012), 114-136
  160. Lord, G. J. and Tambue, A. A modified semi-implict Euler-Maruyama scheme for finite element discretization of SPDEs. arXiv:1004. 1998 [math. NA]. 2010, 23 p. Available at: https://arxiv.org/abs/1004.1998
  161. Mishura, Y. S. and Shevchenko, G. M. Approximation schemes for stochastic differential equations in a Hilbert space. Theor. Prob. Appl. 51, 3 (2007), 442-458
  162. Mü ller-Gronbach, T., Ritter, K., Wagner, T. Optimal pointwise approximation of a linear stochastic heat equation with additive space-time white noise. Monte Carlo and quasi-Monte Carlo methods 2006. Berlin, Springer Publ., 2007, 577-589
  163. Pré vô t, C., Rö ckner, M. A concise course on stochastic partial differential equations, vol. 1905 of Lecture Notes in Mathematics. Berlin, Springer Publ., 2007. 148 p
  164. Shardlow T. Numerical methods for stochastic parabolic PDEs. Numer. Funct. Anal. Optim. 20, 1-2 (1999), 121-145
  165. Da Prato, G., Jentzen, A. and Rö ckner, M. A mild Ito formula for SPDEs. arXiv:1009. 3526[math. PR]. 2012, 39 p. Available at: https://arxiv.org/abs/1009.3526
  166. Da Prato, G. and Zabczyk, J. Stochastic equations in infinite dimensions. 2nd Ed. Cambridge, Cambridge University Press Publ., 2014. 493 p
  167. Kruse, R. Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise. IMA J. Numer. Anal. 34, 1 (2014), 217-251
  168. Kruse, R. Consistency and stability of a Milstein-Galerkin finite element scheme for semilinear SPDE. Stoch. PDE: Anal. Comp. 2, 4 (2014), 471-516
  169. Leonhard, C. Derivative-free numerical schemes for stochastic partial differential equations. Ph. D., Institute of Mathematics of the University of Lü beck, 2017. 131 p
  170. Leonhard, C., Rö ß ler, A. Iterated stochastic integrals in infinite dimensions: approximation and error estimates. Stoch. PDE: Anal. Comp. 7, 2 (2019), 209-239
  171. Bao, J., Reisinger, C., Renz, P., Stockinger, W. First order convergence of Milstein schemes for McKean equations and interacting particle systems. arXiv:2004. 03325v1 [math. PR], 2020, 27 pp. Available at: https://arxiv.org/abs/2004.03325
  172. Son, L. N., Tuan, A. H., Dung, T. N., Yin, G. Milstein-type procedures for numerical solutions of stochastic differential equations with Markovian switching. SIAM J. Numer. Anal. 55, 2 (2017), 953-979
  173. Sun, Y., Yang, J., Zhao, W. Ito-Taylor schemes for solving mean-field stochastic differential equations. Numer. Math. Theor. Meth. Appl. 10, 4 (2017), 798-828
  174. Rybakov, K. A. Spectral method of analysis and optimal estimation in linear stochastic systems. Int. J. Model. Simul. Sci. Comput. 11, 3 (2020), 2050022. DOI: https://doi.org/10.1142/S1793962320500221
  175. Rybakov, K. A. Modeling linear nonstationary stochastic systems by spectral method. Differencialnie Uravnenia i Protsesy Upravlenia, 2020, no. 3, 98-128 (In Russ. ) Available at: http://diffjournal.spbu.ru/EN/numbers/2020.3/article.1.5.html
  176. Averina, T. A., Rybakov, K. A. Using maximum cross section method for filtering jump-diffusion random processes. Russ. J. Numer. Anal. Math. Modelling. 35, 2 (2020), 55-67. DOI: http://doi.org/10.1515/rnam-2020-0005
  177. Rybakov, K. A. Modeling and analysis of output processes of linear continuous stochastic systems based on orthogonal expansions of random functions. J. Computer and Systems Sci. Int. , 59, 3 (2020), 322-337. DOI: http://doi.org/10.1134/S1064230720030156
  178. Kuznetsov, D. F. The three-step strong numerical methods of the orders of accuracy 1. 0 and 1. 5 for Ito stochastic differential equations. J. Autom. Inform. Sci. 34, 12 (2002), 14 p. DOI: http://doi.org/10.1615/JAutomatInfScien.v34.i12.30
  179. Kuznetsov, D. F. Finite-difference strong numerical methods of order 1. 5 and 2. 0 for stochastic differential Ito equations with nonadditive multidimensional noise. J. Autom. Inform. Sci. 33, 5-8 (2001), 13 p. DOI: http://doi.org/10.1615/JAutomatInfScien.v33.i5-8.180
  180. Kuznetsov, D. F. Expansion of the Stratonovich multiple stochastic integrals based on the Fourier multiple series. J. Math. Sci. (N. Y. ). 109, 6 (2002), 2148-2165. DOI: http://doi.org/10.1023/A:1014581416903
  181. Kuznetsov, D. F. Nekotorie voprosy teorii chislennogo resheniia stokhasticheskih differencial'nyh uravnenii. [Problems of the numerical analysis of Ito stochastic differential equations]. St. -Petersburg, SPbGTU Publ., 1998. 204 p
  182. Rybakov, K. A. Using spectral form of mathematical description to represent iterated Stratonovich stochastic integrals. Applied mathematics and computational mechanics for smart applications. Smart Innovation, Systems and Technologies, vol 217. Ed. Jain, L. C., Favorskaya, M. N., Nikitin, I. S., Reviznikov, D. L. Singapore, Springer Publ., 2021, pp. 287-304. DOI: http://doi.org/10.1007/978-981-33-4826-4_20
  183. Yang, G., Burrage, K., Komori, Y., Burrage, P., Ding, X. A class of new Magnus-type methods for semi-linear non-commutative Ito stochastic differential equations. Numer. Algor. (2021). DOI: http://doi.org/10.1007/s11075-021-01089-7
  184. Cao, W., Zhang, Z., Karniadakis, G. Numerical methods for stochastic delay differential equations via the Wong-Zakai approximation. SIAM J. Sci. Comput. 37, 1 (2015), A295-A318. DOI: http://doi.org/10.1137/130942024
  185. Schurz, H. Basic concepts of numerical analysis of stochastic differential equations explained by balanced implicit theta methods. Stochastic differential equations and processes. Springer Proceedings in Mathematics, vol. 7. Ed. Zili, M., Filatova, D. Berlin, Heidelberg, Springer Publ., 2012, pp. 1-139. DOI: http://doi.org/10.1007/978-3-642-22368-6_1
  186. Mishura, Y. S., Shvai, A. V. An estimate of the rate of convergence of an approximating scheme applied to a stochastic differential equation with an additional parameter. Theor. Prob. Math. Stat. 82 (2011), 75-85. DOI: http://doi.org/10.1090/S0094-9000-2011-00828-9
  187. Bardina, X., Rovira, C. On the strong convergence of multiple ordinary integrals to multiple Stratonovich integrals. Publ. Mat. 65 (2021), 859-876. DOI: http://doi.org/10.5565/PUBLMAT6522114
  188. Gaines, J. G. A basis for iterated stochastic integrals. Math. and Computers in Simul. 38 (1995), 7-11
  189. Foster, J., Habermann, K. Brownian bridge expansions for Lé vy area approximations and particular values of the Riemann zeta function. arXiv:2102. 10095 [math. PR]. 2021, 26 p. Available at: https://arxiv.org/abs/2102.10095
  190. Fihtengolc, G. M. Kurs differencial’nogo i integral’nogo schislenija. T. 2 [Differential and integral calculus course. Vol II]. Moscow, Fizmatlit, 1970, 800 p

Полный текст (pdf)