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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

Mikhail D. Kuznetsov, Dmitriy F. Kuznetsov

TL;DR

This work tackles high-accuracy, high-order simulation of Itô SDEs with multidimensional non-commutative noise by combining Unified Taylor--Itô and Taylor--Stratonovich expansions with generalized multiple Fourier series on Legendre bases. It provides explicit one-step strong numerical schemes of orders $0.5$ through $3.0$, supported by robust mean-square approximations of iterated stochastic integrals up to multiplicity six. The authors implement a comprehensive Python package, SDE-MATH, featuring symbolic-coefficient generation via SymPy, Legendre-based integral approximations, a spectral-decomposition algorithm for linear systems, and a GUI-enabled workflow with a SQLite database of ~270k Fourier--Legendre coefficients for fast, scalable simulations. The package is demonstrated with a nonlinear Itô SDE test, yielding accurate results and interactive visualization, highlighting practical impact for stochastic modeling in finance, physics, and engineering.

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.

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 work tackles high-accuracy, high-order simulation of Itô SDEs with multidimensional non-commutative noise by combining Unified Taylor--Itô and Taylor--Stratonovich expansions with generalized multiple Fourier series on Legendre bases. It provides explicit one-step strong numerical schemes of orders through , supported by robust mean-square approximations of iterated stochastic integrals up to multiplicity six. The authors implement a comprehensive Python package, SDE-MATH, featuring symbolic-coefficient generation via SymPy, Legendre-based integral approximations, a spectral-decomposition algorithm for linear systems, and a GUI-enabled workflow with a SQLite database of ~270k Fourier--Legendre coefficients for fast, scalable simulations. The package is demonstrated with a nonlinear Itô SDE test, yielding accurate results and interactive visualization, highlighting practical impact for stochastic modeling in finance, physics, and engineering.

Abstract

The article is devoted to the implementation of strong numerical methods with convergence orders and 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 for the mean-square approximation of iterated Ito and Stratonovich stochastic integrals.

Paper Structure

This paper contains 57 sections, 657 equations, 101 figures.

Table of Contents

  1. Introduction
  2. Theoretical Results Underlying the SDE-MATH Software Package
  3. Strong Numerical Methods with Convergence Orders $0.5,$ $1.0,$ $1.5,$ $2.0,$ $2.5,$ and $3.0$ for Itô SDEs Based on the Unified Taylor--Itô Expansion
  4. Strong Numerical Methods with Convergence Orders $1.0,$ $1.5,$ $2.0,$ $2.5,$ and $3.0$ for Itô SDEs Based on the Unified Taylor--Stra-to-no-vich Expansion
  5. Method of Expansion and Approximation of Iterated Itô and Stra-to-no-vich Stochastic Integrals Based on Generalized Multiple Fourier Series
  6. Approximations of Iterated Itô Stochastic Integrals from the Numerical Schemes (\ref{['sm1']})--(\ref{['al5']}) Using Legendre Polynomials
  7. Optimization of Approximations of Iterated Itô Stochastic Integrals from the Numerical Schemes (\ref{['al1']})--(\ref{['al5']})
  8. Approximations of Iterated Stratonovich Stochastic Integrals from the Numerical Schemes (\ref{['al1x']})--(\ref{['al5x']}) Using Legendre Polynomials
  9. Numerical Algorithm for Linear Stationary Systems of Itô SDEs Based on Spectral Decomposition
  10. The Structure of the SDE-MATH Software Package
  11. Development Tools
  12. Dependency Libraries
  13. Architecture
  14. Integration with SymPy
  15. Purpose of NumPy
  16. ...and 42 more sections

Figures (101)

  • Figure 1: The SDE-MATH software package architecture
  • Figure 2: Fourier--Legendre coefficients calculations explanation
  • Figure 3: Information model of graphical user interface
  • Figure 4: Fourier--Legendre coefficients database preparation screen
  • Figure 5: Greetings screen
  • ...and 96 more figures