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

Abstract

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.

arXiv comment95 pages. Minor changes in Sect. 13, 14. arXiv admin note: text overlap with arXiv:1807.02190, arXiv:1801.00231, arXiv:1801.01564, arXiv:1712.08991, arXiv:1801.01962

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Cited by in corpus (2)

• Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on PYTHON and MATLAB
• Application of Multiple Fourier-Legendre Series to Implementation of Strong Exponential Milstein and Wagner-Platen Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations