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
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| Format: | Preprint |
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2020
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| _version_ | 1866910019403907072 |
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| author | Kuznetsov, Dmitriy F. |
| author_facet | Kuznetsov, Dmitriy F. |
| contents | The problem of the Taylor-Ito and Taylor-Stratonovich expansions of the Ito stochastic processes in a neighborhood of a fixed moment of time is considered. The classical forms of the Taylor-Ito and Taylor-Stratonovich expansions are transformed to the four new representations, which includes the minimal sets of different types of iterated Ito and Stratonovich stochastic integrals. Therefore, these representations (the so-called unified Taylor-Ito and Taylor-Stratonovich expansions) are more convenient for constructing of high-order strong numerical methods for Ito stochastic differential equations. Explicit one-step strong numerical schemes with the orders of convergence 1.0, 1.5, 2.0, 2.5, and 3.0 based on the unified Taylor-Ito and Taylor-Stratonovich expansions are derived. Effective mean-square approximations of iterated Ito and Stratonovich stochastic integrals from these numerical schemes are constructed on the base of the multiple Fourier-Legendre series with multiplicities 1 to 6. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2001_10192 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | 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 Kuznetsov, Dmitriy F. Probability The problem of the Taylor-Ito and Taylor-Stratonovich expansions of the Ito stochastic processes in a neighborhood of a fixed moment of time is considered. The classical forms of the Taylor-Ito and Taylor-Stratonovich expansions are transformed to the four new representations, which includes the minimal sets of different types of iterated Ito and Stratonovich stochastic integrals. Therefore, these representations (the so-called unified Taylor-Ito and Taylor-Stratonovich expansions) are more convenient for constructing of high-order strong numerical methods for Ito stochastic differential equations. Explicit one-step strong numerical schemes with the orders of convergence 1.0, 1.5, 2.0, 2.5, and 3.0 based on the unified Taylor-Ito and Taylor-Stratonovich expansions are derived. Effective mean-square approximations of iterated Ito and Stratonovich stochastic integrals from these numerical schemes are constructed on the base of the multiple Fourier-Legendre series with multiplicities 1 to 6. |
| title | 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 |
| topic | Probability |
| url | https://arxiv.org/abs/2001.10192 |