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

## TL;DR

This paper compares the efficiency of Legendre polynomials and trigonometric functions in numerically integrating Ito stochastic differential equations, showing Legendre polynomials are more computationally efficient for high-order methods.

## Contribution

It provides a comparative analysis demonstrating that Legendre polynomial expansions are more efficient than trigonometric functions for stochastic integral approximation in Ito SDEs.

## Key findings

- Legendre polynomial expansions are easier to implement.
- Legendre methods require less computational cost.
- Results aid in developing high-order numerical methods for Ito SDEs.

## 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.