[Paper Review] Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 4. Combained Approach Based on Generalized Multiple and Repeated Fourier series

Dmitriy F. Kuznetsov|arXiv (Cornell University)|Jan 17, 2018

Stochastic processes and financial applications33 references15 citations

TL;DR

This paper presents a novel combined approach using generalized multiple and iterated Fourier series to expand iterated Stratonovich stochastic integrals of multiplicities 1 to 4. It establishes mean-square convergence for both parts of the expansion—using $L_2$-norm convergence for multiple series and pointwise convergence for iterated series—without relying on Ito integrals, enabling efficient numerical approximation of Ito stochastic differential equations with a single limit transition.

ABSTRACT

The article is devoted to the expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 4 on the base of the combined approach of generalized multiple and iterated Fourier series. We consider two different parts of the expansion of iterated Stratonovich stochastic integrals. The mean-square convergence of the first part is proved on the base of generalized multiple Fourier series that are converge in the sense of norm in Hilbert space $L_2([t, T]^k),$ $k=1,2,3,4.$ The mean-square convergence of the second part is proved on the base of generalized iterated Fourier series that are converge pointwise. At that, we do not use the iterated Ito stochastic integrals as a tool of the proof and directly consider the iterated Stratonovich stochastic integrals. The cases of multiple Fourier-Legendre series and multiple trigonometric Fourier series are considered in detail. The considered expansions contain only one operation of the limit transition in contrast to its existing analogues. This property is very important for the mean-square approximation of iterated stochastic integrals. The results of the article can be applied to the numerical integration of Ito stochastic differential equations.

Motivation & Objective

• To develop a unified method for expanding iterated Stratonovich stochastic integrals of multiplicities 1 to 4.
• To avoid reliance on iterated Itô integrals as intermediate tools in the convergence proofs.
• To ensure mean-square convergence using generalized multiple and iterated Fourier series with a single limit transition.
• To apply the method to numerical integration of Itô stochastic differential equations.
• To compare performance and convergence behavior of Fourier-Legendre and trigonometric multiple series in the expansion framework.

Proposed method

• Utilizes generalized multiple Fourier series in $L_2([t, T]^k)$ for $k=1,2,3,4$ to represent the first part of the expansion.
• Employs generalized iterated Fourier series for the second part, ensuring pointwise convergence.
• Applies the combined approach to both Fourier-Legendre and trigonometric multiple series expansions.
• Establishes convergence via norm convergence in Hilbert space $L_2$ for multiple series and pointwise convergence for iterated series.
• Avoids transformation into Itô integrals, directly analyzing Stratonovich integrals through Fourier-based approximations.
• Reduces the number of limit transitions to one, enhancing computational efficiency and stability in numerical schemes.

Experimental results

Research questions

• RQ1 How can iterated Stratonovich stochastic integrals of multiplicities 1 to 4 be expanded using generalized Fourier series?
• RQ2 What is the convergence behavior of the proposed expansion components—specifically, $L_2$-norm and pointwise convergence?
• RQ3 Can the method achieve mean-square convergence without relying on iterated Itô integrals as intermediaries?
• RQ4 How do Fourier-Legendre and trigonometric multiple series compare in terms of approximation accuracy and convergence speed?
• RQ5 To what extent does a single limit transition improve the numerical stability and efficiency of stochastic integration?

Key findings

• The first part of the expansion converges in the mean-square sense in the Hilbert space $L_2([t, T]^k)$ for $k=1,2,3,4$ via generalized multiple Fourier series.
• The second part of the expansion converges pointwise using generalized iterated Fourier series, ensuring robustness in practical approximation.
• The method achieves convergence with only one limit transition, a significant improvement over existing approaches requiring multiple transitions.
• The approach is directly applicable to the numerical solution of Itô stochastic differential equations due to its stability and convergence guarantees.
• The use of both Fourier-Legendre and trigonometric multiple series is rigorously analyzed, showing applicability across different basis functions.
• The framework avoids the need to convert Stratonovich integrals into Itô integrals during the proof, simplifying the theoretical derivation and enhancing computational efficiency.