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

Abstract

The article is devoted to optimization of the mean-square approximation procedures for iterated Ito stochastic integrals of multiplicities 1 to 5. The mentioned stochastic integrals are part of strong numerical methods with convergence orders 1.0, 1.5, 2.0, and 2.5 for Ito stochastic differential equations with multidimensional non-commutative noise based on the unified Taylor-Ito expansion and multiple Fourier-Legendre series converging in the sense of norm in Hilbert space $L_2([t,T{]}^k)$ $(k=1,\dots ,5).$ In this article we use multiple Fourier-Legendre series within the framework of the method of expansion and mean-square approximation of iterated Ito stochastic integrals based on generalized multiple Fourier series. We show that the lengths of sequences of independent standard Gaussian random variables required for the mean-square approximation of iterated Ito stochastic integrals of multiplicities 1 to 5 can be significantly reduced without the loss of the mean-square accuracy of approximation for these stochastic integrals.

arXiv comment63 pages. Minor changes. arXiv admin note: substantial text overlap with arXiv:2009.14011, arXiv:2003.14184, arXiv:2001.10192, arXiv:1801.03195, arXiv:1901.02345, arXiv:1806.10705, arXiv:1802.00888, arXiv:1801.00231, arXiv:1712.09516, arXiv:1801.01962, arXiv:1712.09746

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• Expansion of Iterated Stratonovich Stochastic Integrals of Fifth, Sixth, Seventh and Eighth Multiplicities Based on Generalized Multiple Fourier Series
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Cited by in corpus (4)

• To Numerical Modeling With Strong Orders 1.0, 1.5, and 2.0 of Convergence for Multidimensional Dynamical Systems With Random Disturbances
• Development and Application of the Fourier Method to the Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals
• Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on PYTHON and MATLAB
• 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