[Paper Review] Application of the Fourier Method to the Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals

Dmitriy F. Kuznetsov|arXiv (Cornell University)|Dec 25, 2017

Stochastic processes and financial applications23 references15 citations

TL;DR

This paper presents a Fourier-based mean-square approximation method for multiple Ito and Stratonovich stochastic integrals of arbitrary multiplicity, using multiple Fourier-Legendre and trigonometric series expansions. The approach achieves exact and approximate expressions for the mean-square error, enabling accurate numerical integration of Ito stochastic differential equations with controlled approximation error.

ABSTRACT

The article is devoted to the mean-square approximation of multiple Ito and Stratonovich stochastic integrals in the context of numerical integration of Ito stochastic differential equations. The expansion of multiple Ito stochastic integrals of any arbitrary multiplicity $k$ and expansions of multiple Stratonovich stochastic integrals of 1-4 multiplicities using multiple Fourier-Legendre and multiple trigonometric Fourier series are obtained. The exact and approximate expressions for the mean-square error of approximation are derived.

Motivation & Objective

• To develop a systematic method for mean-square approximation of multiple Ito and Stratonovich stochastic integrals in stochastic differential equation (SDE) numerical integration.
• To extend existing approximation techniques to handle integrals of arbitrary multiplicity k for Ito integrals and multiplicities 1–4 for Stratonovich integrals.
• To derive exact and approximate expressions for the mean-square error of the approximation, enabling error control in numerical schemes.

Proposed method

• Utilizes multiple Fourier-Legendre series expansions to represent multiple Ito stochastic integrals of any multiplicity k.
• Employs multiple trigonometric Fourier series for approximating multiple Stratonovich stochastic integrals of multiplicities 1 to 4.
• Derives exact expressions for the mean-square error of approximation based on the series expansions.
• Applies convergence analysis to ensure the validity and accuracy of the approximations across different multiplicities.
• Integrates the Fourier series expansions into numerical schemes for Ito SDEs, enabling higher-order weak approximation methods.

Experimental results

Research questions

• RQ1 How can multiple Ito stochastic integrals of arbitrary multiplicity k be approximated with controlled mean-square error using Fourier series?
• RQ2 What is the structure and convergence behavior of multiple Stratonovich stochastic integrals of multiplicities 1 to 4 when expanded via trigonometric Fourier series?
• RQ3 What are the exact and approximate expressions for the mean-square error in the Fourier-based approximation of these stochastic integrals?
• RQ4 How do the Fourier-Legendre and trigonometric series compare in terms of approximation accuracy and computational efficiency for SDE numerical integration?

Key findings

• The method provides exact expressions for the mean-square error of approximation, enabling precise error control in numerical solutions of Ito SDEs.
• Multiple Ito stochastic integrals of any multiplicity k are successfully expanded using multiple Fourier-Legendre series with analytically derived error bounds.
• Multiple Stratonovich stochastic integrals of multiplicities 1 to 4 are accurately approximated using multiple trigonometric Fourier series.
• The derived mean-square error expressions allow for the construction of higher-order weak numerical methods for SDEs with guaranteed convergence rates.
• The approach supports the development of efficient and accurate numerical schemes for stochastic systems requiring high-precision integration.