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The article is devoted to the mean-square approximation of iterated Ito and Stratonovich stochastic integrals in the context of the numerical integration of Ito stochastic differential equations.
Kuznetsov D.F., Kuznetsov M.D. Optimization of the mean-square approximation procedures for iterated Ito stochastic integrals based on multiple Fourier–Legendre series. [In English]. Journal of Physics: Conference Series, Vol. 1925 (2021), article id: 012010, 12 pp. DOI: http://doi.org/10.1088/1742-6596/1925/1/012010
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Kuznetsov D.F. A method of expansion and approximation of repeated stochastic Stratonovich integrals based on multiple Fourier series on full orthonormal systems. [In Russian]. Electronic Journal ”Differential Equations and Control Processes”. ISSN 1817-2172 (online), 1 (1997), 18-77. Available at: http://diffjournal.spbu.ru/EN/numbers/1997.1/article.1.2.html
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Kuznetsov D.F. Problems of the numerical analysis of Ito stochastic differential equations. [In Russian]. Electronic Journal ”Differential Equations and Control Processes”. ISSN 1817-2172 (online), 1 (1998), 66-367. Available at: http://diffjournal.spbu.ru/EN/numbers/1998.1/article.1.3.html
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Kuznetsov D.F. Mean square approximation of solutions of stochastic differential equations using Legendres polynomials. [In English]. Journal of Automation and Information Sciences (Begell House), 2000, 32 (Issue 12), 69-86. DOI: http://doi.org/10.1615/JAutomatInfScien.v32.i12.80
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Kuznetsov D.F. New representations of explicit one-step numerical methods for jump-diffusion stochastic differential equations. [In English]. Computational Mathematics and Mathematical Physics, 41, 6 (2001), 874-888. Available at: http://www.sde-kuznetsov.spb.ru/01b.pdf
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Wiktorsson M. Joint characteristic function and simultaneous simulation of iterated Ito integrals for multiple independent Brownian motions. The Annals of Applied Probability, 11, 2 (2001), 470-487
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Kuznetsov D.F. New representations of the Taylor-Stratonovich expansion. Journal of Mathematical Sciences (N. Y.), 118, 6 (2003), 5586-5596. DOI: http://doi.org/10.1023/A:1026138522239
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Milstein G.N., Tretyakov M.V. Stochastic Numerics for Mathematical Physics. Springer, Berlin, 2004, 616 pp
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Kuznetsov D.F. Numerical Integration of Stochastic Differential Equations. 2. [In Russian]. Polytechnical University Publishing House, Saint-Petersburg, 2006, 764 pp. DOI: http://doi.org/10.18720/SPBPU/2/s17-227
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Kuo, H.-H. Introduction to Stochastic Integration. Universitext (UTX), Springer. N. Y., 2006, 289 pp
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Luo W. Wiener chaos expansion and numerical solutions of stochastic partial differential equations. PhD thesis, California Inst. of Technology, 2006, 225 pp
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Kuznetsov D.F. Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab Programs, 1st Edition. [In Russian]. Polytechnical University Publishing House, Saint-Petersburg, 2007, 778 pp. DOI: http://doi.org/10.18720/SPBPU/2/s17-228
2007
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Kuznetsov D.F. Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab Programs, 2nd Edition. [In Russian]. Polytechnical University Publishing House, Saint-Petersburg, 2007, XXXII+770 pp. DOI: http://doi.org/10.18720/SPBPU/2/s17-229
2007
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Kuznetsov D.F. Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab Programs, 3rd Edition. [In Russian]. Polytechnical University Publishing House, Saint-Petersburg, 2009, XXXIV+768 pp. DOI: http://doi.org/10.18720/SPBPU/2/s17-230
2009
Cited alongside, same era.
Kuznetsov D.F. Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab Programs. 4th Edition. [In Russian]. Polytechnical University Publishing House, Saint-Petersburg, 2010, XXX+786 pp. DOI: http://doi.org/10.18720/SPBPU/2/s17-231
2010
Cited alongside, same era.
Kuznetsov D.F. Multiple Stochastic Ito and Stratonovich Integrals and Multiple Fourier Series. [In Russian]. Electronic Journal ”Differential Equations and Control Processes”. ISSN 1817-2172 (online), 3 (2010), A.1-A.257. DOI: http://doi.org/10.18720/SPBPU/2/z17-7
2010
Cited alongside, same era.
Platen E., Bruti-Liberati N. Numerical Solution of Stochastic Differential Equations with Jumps in Finance. Springer, Berlin-Heidelberg, 2010, 868 pp
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. Electronic Journal ”Differential Equations and Control Processes”. ISSN 1817-2172 (online), 3 (2020), 129-162. Available at: http://diffjournal.spbu.ru/EN/numbers/2020.3/article.1.6.html
2020
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2020
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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. [In English]. Proceedings of the XIII International Conference on Applied Mathematics and Mechanics in the Aerospace Industry AMMAI-2020 (Crimea, Alushta, 6-13 September, 2020), MAI, Moscow, 2020, pp. 451-453. Available at: http://www.sde-kuznetsov.spb.ru/20e.pdf
2020
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2010
Cited alongside, same era.
Kuznetsov D.F. Strong Approximation of Multiple Ito and Stratonovich Stochastic Integrals: Multiple Fourier Series Approach. 1st Edition. [In English]. Polytechnical University Publishing House, Saint-Petersburg, 2011, 250 pp. DOI: http://doi.org/10.18720/SPBPU/2/s17-232
2011
Cited alongside, same era.
Kuznetsov D.F. Strong Approximation of Multiple Ito and Stratonovich Stochastic Integrals: Multiple Fourier Series Approach. 2nd Edition. [In English]. Polytechnical University Publishing House, Saint-Petersburg, 2011, 284 pp. DOI: http://doi.org/10.18720/SPBPU/2/s17-233
2011
Cited alongside, same era.
Kuznetsov D.F. Multiple Ito and Stratonovich Stochastic Integrals: Approximations, Properties, Formulas. [In English]. Polytechnical University Publishing House, Saint-Petersburg, 2013, 382 pp. DOI: http://doi.org/10.18720/SPBPU/2/s17-234
2013
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Allen E. Approximation of triple stochastic integrals through region subdivision. Communications in Applied Analysis (Special Tribute Issue to Professor V. Lakshmikantham), 17 (2013), 355-366
2013
Cited alongside, same era.
Kuznetsov D.F. Multiple Ito and Stratonovich Stochastic Integrals: Fourier-Legendre and Trigonometric Expansions, Approximations, Formulas. [In English]. Electronic Journal ”Differential Equations and Control Processes”. ISSN 1817-2172 (online), 1 (2017), A.1-A.385. DOI: http://doi.org/10.18720/SPBPU/2/z17-3
2017
Cited alongside, same era.
Kuznetsov D.F. Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on MATLAB, 5th Edition. [In Russian]. Electronic Journal ”Differential Equations and Control Processes”. ISSN 1817-2172 (online), 2 (2017), A.1-A.1000. DOI: http://doi.org/10.18720/SPBPU/2/z17-4
2017
Cited alongside, same era.
Kuznetsov D.F. Development and application of the Fourier method for the numerical solution of Ito stochastic differential equations. [In English]. Computational Mathematics and Mathematical Physics, 58, 7 (2018), 1058-1070. DOI: http://doi.org/10.1134/S0965542518070096
2018
Cited alongside, same era.
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 [In English]. Automation and Remote Control, 79, 7 (2018), 1240-1254. DOI: http://doi.org/10.1134/S0005117918070056
2018
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Kuznetsov D.F. Mean-Square Approximation of Iterated Itô and Stratonovich Stochastic Integrals Based on Generalized Multiple Fourier Series. Application to Numerical Integration of Itô SDEs and Semilinear SPDEs. [In English]. Electronic Journal ”Differential Equations and Control Processes”. ISSN 1817-2172 (online), 4 (2021), A.1-A.788. Available at: http://diffjournal.spbu.ru/EN/numbers/2021.4/article.1.9.html
2021
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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, Eds. Shiryaev, A.N., Samouylov, K.E., Kozyrev, D.V. Springer, Cham, 2021, pp. 17-32. DOI: http://doi.org/10.1007/978-3-030-83266-7_2
2021
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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. [In English]. Electronic Journal ”Differential Equations and Control Processes”. ISSN 1817-2172 (online), 1 (2021), 93-422. Available at: http://diffjournal.spbu.ru/EN/numbers/2021.1/article.1.5.html
2021
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Rybakov, K.A. Orthogonal expansion of multiple Itô stochastic integrals. Electronic Journal ”Differential Equations and Control Processes”. ISSN 1817-2172 (online), 3 (2021), 109-140. Available at: http://diffjournal.spbu.ru/EN/numbers/2021.3/article.1.8.html
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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. [In English]. Electronic Journal ”Differential Equations and Control Processes” ISSN 1817-2172 (online), 2 (2022), 83-186. Available at: http://diffjournal.spbu.ru/EN/numbers/2022.2/article.1.6.html
2022
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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. [In English]. Electronic Journal ”Differential Equations and Control Processes” ISSN 1817-2172 (online), 4 (2022), 135-194. Available at: http://diffjournal.spbu.ru/EN/numbers/2022.4/article.1.9.html
2022
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Kuznetsov D.F., Kuznetsov M.D. Optimization of the mean-square approximation procedures for iterated Stratonovich stochastic integrals of multiplicities 1 to 3 with respect to components of the multi-dimensional Wiener process based on Multiple Fourier-Legendre series. [In English]. MATEC Web of Conferences, Vol. 362 (2022), article id: 01014, 10 pp. DOI: http://doi.org/10.1051/matecconf/202236201014
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Kuznetsov, D.F. A new approach to the series expansion of iterated Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. The case of arbitrary complete orthonormal systems in Hilbert space. [In English]. Electronic Journal ”Differential Equations and Control Processes” ISSN 1817-2172 (online), 2 (2024), 73–170. Available at: http://diffjournal.spbu.ru/EN/numbers/2024.2/article.1.6.html
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Kuznetsov, D.F. A new approach to the series expansion of iterated Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. The case of arbitrary complete orthonormal systems in Hilbert space. II [In English]. Electronic Journal ”Differential Equations and Control Processes” ISSN 1817-2172 (online), 4 (2024), 104–190. Available at: http://diffjournal.spbu.ru/EN/numbers/2024.4/article.1.6.html
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Kuznetsov, D.F. 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. III [In English]. Electronic Journal ”Differential Equations and Control Processes” ISSN 1817-2172 (online), 3 (2025), 118–153. Available at: http://diffjournal.spbu.ru/EN/numbers/2025.3/article.1.8.html
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