Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Four New Forms of the Taylor-Ito and Taylor-Stratonovich Expansions and its Application to the High-Order Strong Numerical Methods for Ito Stochastic Differential Equations

Dmitriy F. Kuznetsov

TL;DR

The paper introduces four unified Taylor expansions—two for Ito and two for Stratonovich forms—designed to minimize the number of distinct iterated stochastic integrals needed for high-order strong methods solving Ito SDEs. It establishes integration-order replacement techniques and presents explicit one-step schemes achieving convergence orders up to $3.0$, using mean-square approximations of iterated integrals via generalized multiple Fourier (notably Legendre) series. By contrasting the unified expansions with classical forms, the work demonstrates reduced stochastic bases and provides detailed rank analyses, informing method design for non-commutative noise. The practical contribution is a rigorous, computable framework for high-order strong numerical methods, with concrete schemes and provable mean-square error control for the approximated stochastic integrals.

Abstract

The problem of the Taylor-Ito and Taylor-Stratonovich expansions of the Ito stochastic processes in a neighborhood of a fixed moment of time is considered. The classical forms of the Taylor-Ito and Taylor-Stratonovich expansions are transformed to the four new representations, which includes the minimal sets of different types of iterated Ito and Stratonovich stochastic integrals. Therefore, these representations (the so-called unified Taylor-Ito and Taylor-Stratonovich expansions) are more convenient for constructing of high-order strong numerical methods for Ito stochastic differential equations. Explicit one-step strong numerical schemes with the orders of convergence 1.0, 1.5, 2.0, 2.5, and 3.0 based on the unified Taylor-Ito and Taylor-Stratonovich expansions are derived. Effective mean-square approximations of iterated Ito and Stratonovich stochastic integrals from these numerical schemes are constructed on the base of the multiple Fourier-Legendre series with multiplicities 1 to 6.

Four New Forms of the Taylor-Ito and Taylor-Stratonovich Expansions and its Application to the High-Order Strong Numerical Methods for Ito Stochastic Differential Equations

TL;DR

The paper introduces four unified Taylor expansions—two for Ito and two for Stratonovich forms—designed to minimize the number of distinct iterated stochastic integrals needed for high-order strong methods solving Ito SDEs. It establishes integration-order replacement techniques and presents explicit one-step schemes achieving convergence orders up to , using mean-square approximations of iterated integrals via generalized multiple Fourier (notably Legendre) series. By contrasting the unified expansions with classical forms, the work demonstrates reduced stochastic bases and provides detailed rank analyses, informing method design for non-commutative noise. The practical contribution is a rigorous, computable framework for high-order strong numerical methods, with concrete schemes and provable mean-square error control for the approximated stochastic integrals.

Abstract

The problem of the Taylor-Ito and Taylor-Stratonovich expansions of the Ito stochastic processes in a neighborhood of a fixed moment of time is considered. The classical forms of the Taylor-Ito and Taylor-Stratonovich expansions are transformed to the four new representations, which includes the minimal sets of different types of iterated Ito and Stratonovich stochastic integrals. Therefore, these representations (the so-called unified Taylor-Ito and Taylor-Stratonovich expansions) are more convenient for constructing of high-order strong numerical methods for Ito stochastic differential equations. Explicit one-step strong numerical schemes with the orders of convergence 1.0, 1.5, 2.0, 2.5, and 3.0 based on the unified Taylor-Ito and Taylor-Stratonovich expansions are derived. Effective mean-square approximations of iterated Ito and Stratonovich stochastic integrals from these numerical schemes are constructed on the base of the multiple Fourier-Legendre series with multiplicities 1 to 6.

Paper Structure

This paper contains 14 sections, 912 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Integration Order Replacement Technique for Iterated Ito Stochastic Integrals
  3. The Taylor--Ito Expansion
  4. The First Form of the Unified Taylor--Ito Expansion
  5. The Second Form of the Unified Taylor--Ito Expansion
  6. The Taylor--Stratonovich Expansion
  7. The First Form of the Unified Taylor--Stratonovich Expansion
  8. The Second Form of the Unified Taylor--Stratonovich Expansion
  9. A Remark on Theorems 4 and 5
  10. Comparison of the Unified Taylor--Ito and Taylor--Stra-to-no-vich Expansions With the Classical Taylor--Ito and Taylor--Stratonovich Expansions
  11. Application of First Form of the Unified Taylor--Ito Expansion to the High-Order Strong Numerical Methods for Ito SDEs
  12. Application of First Form of the Unified Taylor--Stratonovich Expansion to the High-Order Strong Numerical Methods for Ito SDEs
  13. Method of the Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals Based on Generalized Multiple Fourier Series
  14. Theorems 7--16 from Point of View of the Wong--Zakai Approximation