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ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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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

Author(s):

Mikhail Dmitrievich Kuznetsov

Saint-Petersburg Electrotechnical University,
ul. Professora Popova, 5, 197376, Saint-Petersburg, Russia,
Faculty of Computer Technologies and Informatics,
Department of Information Systems,
Master student

Dmitriy Feliksovich Kuznetsov

Peter the Great Saint-Petersburg Polytechnic University,
Polytechnicheskaya ul., 29, 195251, Saint-Petersburg, Russia,
Institute of Applied Mathematics and Mechanics,
Department of Higher Mathematics,
Doctor of physico-mathematical sciences, Professor

sde_kuznetsov@inbox.ru

Abstract:

The article is devoted to the implementation of strong numerical methods with convergence orders 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 for Ito stochastic differential equations with multidimensional non-commutative noise based on multiple Fourier-Legendre series and unified Taylor-Ito and Taylor-Stratonovich expansions. Algorithms for the implementation of these methods are constructed and a package of programs in the Python programming language is presented. An important part of this software package concerning the mean-square approximation of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 with respect to components of the multidimensional Wiener process is based on the method of generalized multiple Fourier series. More precisely, we used multiple Fourier-Legendre series converging in the sense of norm in Hilbert space for the mean-square approximation of iterated Ito and Stratonovich stochastic integrals.

Keywords

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