Development and Application of the Fourier Method to the Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals

Abstract

The article is devoted to the mean-square approximation of iterated Ito and Stratonovich stochastic integrals in the context of the numerical integration of Ito stochastic differential equations. The expansion of iterated Ito stochastic integrals of arbitrary multiplicity $k$ $(k\in \mathbb{N})$ and expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 6 have been obtained. Considerable attention is paid to expansions based on multiple Fourier-Legendre series. The exact and approximate expressions for the mean-square error of approximation of iterated Ito stochastic integrals are derived. The results of the article will be useful for numerical integration of Ito stochastic differential equations with non-commutative noise.

arXiv comment58 pages. Some changes in Sect. 2, 4, 7. Bibliography has been updated

References in corpus (9)

• To Numerical Modeling With Strong Orders 1.0, 1.5, and 2.0 of Convergence for Multidimensional Dynamical Systems With Random Disturbances
• Development and Application of the Fourier Method to the Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals
• Comparative Analysis of the Efficiency of Application of Legendre Polynomials and Trigonometric Functions to the Numerical Integration of Ito Stochastic Differential Equations
• Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on PYTHON and MATLAB
• Numerical Simulation of 2.5-Set of Iterated Ito Stochastic Integrals of Multiplicities 1 to 5 From the Taylor-Ito Expansion
• Explicit One-Step Strong Numerical Methods of Orders 2.0 and 2.5 for Ito Stochastic Differential Equations Based on the Unified Taylor-Ito and Taylor-Stratonovich Expansions
• Numerical Simulation of 2.5-Set of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 5 From the Taylor-Stratonovich Expansion
• Expansion of Iterated Stratonovich Stochastic Integrals of Fifth, Sixth, Seventh and Eighth Multiplicities Based on Generalized Multiple Fourier Series
• Optimization of the Mean-Square Approximation Procedures for Iterated Ito Stochastic Integrals of Multiplicities 1 to 5 from the Unified Taylor-Ito Expansion Based on Multiple Fourier-Legendre Series

Cited by in corpus (22)

• To Numerical Modeling With Strong Orders 1.0, 1.5, and 2.0 of Convergence for Multidimensional Dynamical Systems With Random Disturbances
• Development and Application of the Fourier Method to the Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals
• Expansion of Iterated Ito Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Multiple Fourier Series Converging in the Mean
• Strong Numerical Methods of Orders 2.0, 2.5, and 3.0 for Ito Stochastic Differential Equations Based on the Unified Stochastic Taylor Expansions and Multiple Fourier-Legendre Series
• Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals of Multiplicities 1 to 6 from the Taylor-Ito and Taylor-Stratonovich Expansions Using Legendre Polynomials
• Application of the Method of Approximation of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series to the High-Order Strong Numerical Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations
• Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Iterated Fourier Series Converging Pointwise
• Comparative Analysis of the Efficiency of Application of Legendre Polynomials and Trigonometric Functions to the Numerical Integration of Ito Stochastic Differential Equations
• Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures and with Respect to Martingales Based on Generalized Multiple Fourier Series
• Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on PYTHON and MATLAB
• Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 Based on Generalized Multiple Fourier Series Converging in the Mean: General Case of Series Summation
• Application of Multiple Fourier-Legendre Series to Implementation of Strong Exponential Milstein and Wagner-Platen Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations
• Numerical Simulation of 2.5-Set of Iterated Ito Stochastic Integrals of Multiplicities 1 to 5 From the Taylor-Ito Expansion
• Explicit One-Step Strong Numerical Methods of Orders 2.0 and 2.5 for Ito Stochastic Differential Equations Based on the Unified Taylor-Ito and Taylor-Stratonovich Expansions
• Numerical Simulation of 2.5-Set of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 5 From the Taylor-Stratonovich Expansion
• New Simple Method of Expansion of Iterated Ito Stochastic integrals of Multiplicity 2 Based on Expansion of the Brownian Motion Using Legendre Polynomials and Trigonometric Functions
• An Analysis of Approximation Algorithms for Iterated Stochastic Integrals and a Julia and MATLAB Simulation Toolbox
• Optimization of the Mean-Square Approximation Procedures for Iterated Ito Stochastic Integrals of Multiplicities 1 to 5 from the Unified Taylor-Ito Expansion Based on Multiple Fourier-Legendre Series
• 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
• Formulae for mixed moments of Wiener processes and a stochastic area integral
• The Proof of Convergence with Probability 1 in the Method of Expansion of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series
• A new proof of the expansion of iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process based on generalized multiple Fourier series and Hermite polynomials