Implementation of Strong Numerical Methods of Orders 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 for Ito SDEs with Non-Commutative Noise Based on the Unified Taylor-Ito and Taylor-Stratonovich Expansions and Multiple Fourier-Legendre Series

TL;DR

This paper develops and implements high-order strong numerical methods for Ito SDEs with non-commutative noise, utilizing unified Taylor expansions and Fourier-Legendre series, and provides a Python software package for their application.

Contribution

It introduces a set of strong numerical methods of orders 0.5 to 3.0 for Ito SDEs with non-commutative noise, based on unified Taylor expansions and Fourier-Legendre series, along with a Python implementation.

Findings

Algorithms for high-order methods are constructed.

A Python package for implementation is provided.

Mean-square approximation of stochastic integrals is achieved using Fourier-Legendre series.

Abstract

The article is devoted to the implementation of strong numerical methods with convergence orders $0.5,$ $1.0,$ $1.5,$ $2.0,$ $2.5,$ and $3.0$ for Ito stochastic differential equations with multidimensional non-commutative noise based on the unified Taylor--Ito and Taylor-Stratonovich expansions and multiple Fourier-Legendre series. Algorithms for the implementation of these methods are constructed and a package of programs in the Python programming language is presented. An important part of this software package, concerning the mean-square approximation of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 with respect to components of the multidimensional Wiener process is based on the method of generalized multiple Fourier series. More precisely, we used the multiple Fourier-Legendre series converging in the sense of norm in Hilbert space $L_2([t, T]^k)$ $(k=1,\ldots,6)$ for the mean-square approximation of iterated Ito and Stratonovich stochastic integrals.