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

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

Автор(ы):

Dmitriy Feliksovich Kuznetsov

Peter the Great Saint-Petersburg Polytechnic University
Department of Higher Mathematics
Dr. Sc., Professor

sde_kuznetsov@inbox.ru

Аннотация:

The book is devoted to the strong approximation of iterated 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 monograph open a new direction in researching of iterated Ito and Stratonovich 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,...). 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 integration order replacement technique for the class of iterated Ito stochastic integrals has been introduced (Chapter 3). We derived the exact and approximate expressions for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k (Chapter 1). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and 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 of arbitrary multiplicity k with respect to the infinite-dimensional Q-Wiener process.

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. 4-th 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. (In English). arXiv:2003. 14184 [math. PR]. 2020, 607 p. Available at: https://arxiv.org/abs/2003.14184
  15. 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
  16. 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
  17. 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
  18. 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
  19. 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
  20. 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
  21. 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
  22. 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
  23. 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
  24. 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
  25. 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
  26. 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, 70 p. Available at: https://arxiv.org/abs/1712.09746
  27. 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, 45 p. Available at: https://arxiv.org/abs/1712.08991
  28. 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, 49 p. Available at: https://arxiv.org/abs/1801.01079
  29. Kuznetsov, D. F. Mean-square approximation of iterated Ito and Stratonovich stochastic integrals from the Taylor-Ito and Taylor-Stratonovich expansions using Legendre polynomials. (In English). arXiv:1801. 00231 [math. PR]. 2017, 77 p. Available at: https://arxiv.org/abs/1801.00231
  30. 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, 50 p. Available at: https://arxiv.org/abs/1712.09516
  31. 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, 60 p. Available at: https://arxiv.org/abs/1801.00784
  32. 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, 54 p. Available at: https://arxiv.org/abs/1801.01564
  33. Kuznetsov, D. F. Expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 4. Combined approach based on generalized multiple and repeated Fourier series. (In English). arXiv:1801. 05654 [math. PR]. 2018, 40 p. Available at: https://arxiv.org/abs/1801.05654
  34. 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
  35. 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, 37 p. Available at: https://arxiv.org/abs/1802.00643
  36. 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, 29 p. Available at: https://arxiv.org/abs/1801.03195
  37. 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
  38. 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
  39. 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
  40. 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
  41. Kuznetsov, D. F, Numerical simulation of 2. 5-set of iterated Ito stochastic integrals of multiplicities 1 to 5 from the Taylor-Stratonovich expansion. (In English). arXiv:1805. 12527 [math. PR]. 2018, 22 p. Available at: https://arxiv.org/abs/1805.12527
  42. 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, 30 p. Available at: https://arxiv.org/abs/1802.04844
  43. 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, 37 p. Available at: https://arxiv.org/abs/1807.02190
  44. 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
  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. (In English). arXiv:1905. 03724 [math. GM]. 2019, 40 p. Available at: https://arxiv.org/abs/1905.03724
  46. 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
  47. 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
  48. Kuznetsov, D. F. New simple method for obtainment an expansion of double stochastic Ito integrals based on the expansion of 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
  49. 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
  50. Kuznetsov, M. D., Kuznetsov, D. F. SDE-MATH: A software package for the implementation of strong high-order numerical methods with the convergence orders 0. 5, 1. 0, 1. 5, 2. 0, 2. 5, and 3. 0 for Ito SDEs with multidimensional non-commutative noise based on multiple Fourier-Legendre series and unified Taylor-Ito and Taylor-Stratonovich expansions. Differencialnie Uravnenia i Protsesy Upravlenia, 2021, no. 1 (to appear)
  51. 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, 188 p. Available at: https://arxiv.org/abs/2009.14011
  52. 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, 50 p. Available at: https://arxiv.org/abs/2010.13564
  53. 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
  54. 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
  55. Wong, E., Zakai, M. On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Stat.. 36, 5 (1965), 1560-1564
  56. Wong, E., Zakai, M. On the relation between ordinary and stochastic differential equations. Int. J. Eng. Sci. 3 (1965), 213-229
  57. Ikeda, N., Watanabe, S. Stochastic Differential Equations and Diffusion Processes. 2nd Ed. Amsterdam, Oxford, New-York, North-Holland Publishing Company, 1989. 555 p
  58. 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
  59. 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
  60. Kuznetsov, D. F. Mean square approximation of solutions of stochastic differential equations using Legendre’s polynomials. J. Automat. Inform. Sciences (Begell House), 32, 12 (2000), 69-86. DOI: http://doi.org/10.1615/JAutomatInfScien.v32.i12.80
  61. 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
  62. Kulchitsky, O. Yu., Kuznetsov, D. F. Approximation of multiple Ito stochastic integrals. VINITI, 1678-V94 (1994), 42 p. (In Russ. )
  63. 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
  64. Milstein G. N. Chislennoye integrirovaniye stokhasticheskih differencial’nyh uravnenii [Numerical integration of stochastic differential equations], Sverdlovsk, Ural. Univ. Publ. 225 p
  65. Kloeden, P. E., Platen, E., Wright, I. W. The approximation of multiple stochastic integrals. Stoch. Anal. Appl. 10, 4 (1992), 431-441
  66. Kloeden, P. E., Platen, E. Numerical solution of stochastic differential equations. Berlin, Springer-Verlag Publ., 1992. 632 p
  67. Kloeden, P. E., Platen, E., Schurz, H. Numerical solution of SDE through computer experiments. Berlin, Springer-Verlag Publ., 1994. 292 p
  68. 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. )
  69. 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. )
  70. 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
  71. Ryden, T., Wiktorsson, M. On the simulation of iterated Ito integrals. Stoch. Proc. and their Appl. , 91, 1 (2001), 151-168
  72. Gaines, J. G., Lyons, T. J. Random generation of stochastic area integrals. SIAM J. Appl. Math. 54 (1994), 1132-1146
  73. Gilsing, H., Shardlow, T. SDELab: A package for solving stochastic differential equations in MATLAB. J. Comp. Appl. Math. , 205 (2007), 1002-1018
  74. Milstein, G. N., Tretyakov, M. V. Stochastic numerics for mathematical physics. Berlin, Springer-Verlag Publ., 2004. xx+596 p
  75. Platen, E., Bruti-Liberati, N. Numerical solution of stochastic differential equations with jumps in finance. Berlin, Heidelberg, Springer-Verlag Publ., 2010. 868 p
  76. Allen, E. Approximation of triple stochastic integrals through region subdivision. Commun. Appl. Analysis (Special Tribute Issue to Professor V. Lakshmikantham), 17 (2013), 355-366
  77. 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
  78. 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
  79. Zahri, M. Multidimensional Milstein scheme for solving a stochastic model for prebiotic evolution. J. of Taibah Univ. for Science. 8, 2 (2014), 186-198
  80. 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
  81. 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
  82. Gihman, I. I., Skorohod, A. V. Stokhasticheskie differencial’nye uravnenia [Stochastic differential equations]. Kiev, Naukova Dumka Publ., 1968. 354 p
  83. 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
  84. Skorohod, A. V. Sluchainye protsesy s nezavisimymi proraschenyami [Stochastic processes with independent increments]. Moscow, Nauka Publ., 1964. 280 p
  85. 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
  86. Hobson, E. W. The theory of spherical and ellipsoidal harmonics. Cambridge, Cambridge Univ. Press Publ., 1931. 502 p
  87. Suetin, P. K. Klassicheskije ortogonal'nye mnogochleny. Izd 3-e. [Classical orthogonal polynomials. 3rd Ed. ]. Moskow, Fizmatlit, 2005. 480 p
  88. 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. )
  89. Il’in, V. A., Poznyak, E. G. Osnovy matematicheskogo analiza. Chast’ 2 [Foundations of mathematical analysis. Part II]. Moscow, Nauka Publ., 1973. 448 p
  90. Kuznetsov, D. F. Theorems about integration order replacement in multiple stochastic integrals. VINITI, 3607-V97 (1997), 31 p. (In Russ. )
  91. 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
  92. Sjö lin, P. Convergence almost everywhere of certain singular integrals and multiple Fourier series. Ark. Mat. 9, 1-2 (1971), 65-90
  93. 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
  94. Arato, M. Linear stochastic systems with constant coefficients. A statistical approach. Berlin, Heidelberg, N. Y., Springer-Verlag Publ., 1982. 289 p
  95. Shiryaev, A. N. Essentials of Stochastic Finance: Facts, Models, Theory. World Scientific Publishing Co United States, 1999. 852 p
  96. 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
  97. Chung, K. L., Williams, R. J. Introduction to stochastic integration. Progress in Probability and Stochastics. Vol. 4, Ed. Huber P., Rosenblatt M. Boston, Basel, Stuttgart, Birkhauser Publ., 1983. 152 p
  98. 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
  99. Shiryaev A. N. Probability. Springer-Verlag, New York, 1996. 624 p
  100. Fihtengolc G. M. Kurs differencial’nogo i integral’nogo schislenija. T. 2 [Differential and integral calculus course. Vol II]. Moscow, Fizmatlit, 1970, 800 p
  101. 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
  102. 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
  103. 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
  104. 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
  105. Nasyrov, F. S. Lokal'nye vremena, simmetrichnyje integraly i stochasticheskiy analiz [Local times, symmetric integrals and stochastic analysis]. Moscow, Fizmatlit Publ., 2011. 212 p
  106. Kagirova, G. R., Nasyrov, F. S. On an optimal filtration problem for one-dimensional diffusion processes. Siberian Adv. Math. 28, 3 (2018), 155-165
  107. Asadullin, E. M., Nasyrov, F. S. About filtering problem of diffusion processes. Ufa Math. J. 3, 2 (2011), 3-9
  108. 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
  109. Rybakov, K. A. Modeling linear nonstationary stochastic systems by spectral method. Differencialnie Uravnenia i Protsesy Upravlenia, 2020, no. 3, 98-128
  110. Kloeden, P. E., Platen, E. The Stratonovich and Ito-Taylor expansions. Math. Nachr. 151 (1991), 33-50
  111. 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
  112. 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
  113. 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. )
  114. Platen, E., Wagner, W. On a Taylor formula for a class of Ito processes. Probab. Math. Statist. 3 (1982), 37-51
  115. 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
  116. Wiener, N. Un problѐ me de probabilité s dé nombrables. Bull. Soc. Math. France. 52 (1924), 569-578
  117. Lé vy, P. Wiener's random function and other Laplacian random functions. Proc. 2nd Berkeley Symp. Math. Stat. Prob. 1951, 171-187
  118. Ito, K., McKean, H. Diffusion processes and their sample paths. Berlin, Heidelberg, New York, Springer-Verlag Publ., 1965. 395 p
  119. Luo, W. Wiener chaos expansion and numerical solutions of stochastic partial differential equations. Ph. D., California Institute of Technology, 2006. 225 p
  120. 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, no. 4, 2019, 32-52. Available at: http://diffjournal.spbu.ru/EN/numbers/2019.4/article.1.2.html
  121. 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
  122. Kuznetsov, D. F, Combined method of strong approximation of multiple stochastic integrals. J. Automat. Inform. Sciences (Begell House), 34, 8 (2002), 6 p. DOI: http://doi.org/10.1615/JAutomatInfScien.v34.i8.40
  123. 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
  124. 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
  125. 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
  126. Gyö ngy, I. and Krylov, N. An accelerated splitting-up method for parabolic equations. SIAM J. Math. Anal.. 37, 4 (2005), 1070-1097
  127. Hausenblas, E. Numerical analysis of semilinear stochastic evolution equations in Banach spaces. J. Comp. Appl. Math. 147, 2 (2002), 485-516
  128. Hausenblas, E. Approximation for semilinear stochastic evolution equations. Potential Anal. 18, 2 (2003), 141-186
  129. Jentzen, A. Pathwise numerical approximations of SPDEs with additive noise under non global Lipschitz coefficients. Potential Anal. 31, 4 (2009), 375-404
  130. 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
  131. 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
  132. 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
  133. Jentzen, A., Kloeden, P. E. Taylor Approximations for Stochastic Partial Differential Equations. Philadelphia, SIAM Publ., 2011. 224 p
  134. Jentzen, A. and Rö ckner, M. A Milstein scheme for SPDEs. Foundations Comp. Math. 15, 2 (2015), 313-362
  135. 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
  136. Zhang, Z., Karniadakis, G. Numerical Methods for Stochastic Partial Differential Equations with White Noise. Springer Publ., 2017. 398 p
  137. 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
  138. 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
  139. 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
  140. 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
  141. 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
  142. Shardlow T. Numerical methods for stochastic parabolic PDEs. Numer. Funct. Anal. Optim. 20, 1-2 (1999), 121-145
  143. 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
  144. Da Prato, G. and Zabczyk, J. Stochastic equations in infinite dimensions. 2nd Ed. Cambridge, Cambridge University Press Publ., 2014. 493 p
  145. 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
  146. Kruse, R. Consistency and stability of a Milstein-Galerkin finite element scheme for semilinear SPDE. Stoch. PDE: Anal. Comp. 2, 4 (2014), 471-516
  147. 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
  148. 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
  149. 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
  150. 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
  151. 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

Полный текст (pdf)