Comparative Analysis of the Efficiency of Application of Legendre Polynomials and Trigonometric Functions to the Numerical Integration of Ito Stochastic Differential Equations

Abstract

The article is devoted to comparative analysis of the efficiency of application of Legendre polynomials and trigonometric functions to the numerical integration of Ito stochastic differential equations in the framework of the method of approximation of iterated Ito and Stratonovich stochastic integrals based on generalized multiple Fourier series. On the example of iterated Ito stochastic integrals of multiplicities 1 to 3, included in the Taylor-Ito expansion, it is shown that expansions of stochastic integrals based on Legendre polynomials are much easier and require significantly less computational costs compared to their analogues obtained using the trigonometric system of functions. The results of the article can be useful for construction of high-order strong numerical methods for Ito stochastic differential equations.

arXiv comment41 pages. Minor changes in Sect. 3, 4, 11. arXiv admin note: substantial text overlap with arXiv:1801.08862, arXiv:1801.00231, arXiv:1807.02190, arXiv:1802.00643, arXiv:1806.10705, arXiv:1712.09516, arXiv:1801.05654, arXiv:1712.08991, arXiv:1801.03195, arXiv:1712.09746, arXiv:1801.01564, arXiv:1801.01962, arXiv:1801.07248

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• 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
• Numerical Simulation of 2.5-Set of Iterated Ito Stochastic Integrals of Multiplicities 1 to 5 From the Taylor-Ito Expansion
• 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

Cited by in corpus (11)

• 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
• 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
• Numerical Simulation of 2.5-Set of Iterated Ito Stochastic Integrals of Multiplicities 1 to 5 From the Taylor-Ito Expansion
• Numerical Simulation of 2.5-Set of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 5 From the Taylor-Stratonovich Expansion
• An Analysis of Approximation Algorithms for Iterated Stochastic Integrals and a Julia and MATLAB Simulation Toolbox
• 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
• Spectral Representation and Simulation of Fractional Brownian Motion
• Formulae for mixed moments of Wiener processes and a stochastic area integral
• A new proof of the expansion of iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process based on generalized multiple Fourier series and Hermite polynomials
• On Approximate Representation of Fractional Brownian Motion