A comparative analysis of efficiency of using the legendre polynomials and trigonometric functions for the numerical solution of Ito stochastic differential equations

This paper is devoted to the comparative analysis of the efficiency of using the Legendre polynomials and trigonometric functions for the numerical solution of Ito stochastic differential equations under the method of approximating multiple Ito and Stratonovich stochastic integrals based on generalized multiple Fourier series. Using the multiple stochastic integrals of multiplicity 1–3 appearing in the Ito–Taylor expansion as an example, it is shown that their expansions obtained using multiple Fourier–Legendre series are significantly simpler and less computationally costly than their analogs obtained on the basis of multiple trigonometric Fourier series. The results obtained in this paper can be useful for constructing and implementing strong numerical methods for solving Ito stochastic differential equations with multidimensional nonlinear noise.