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Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 Based on Generalized Multiple Fourier Series Converging in the Mean: General Case of Series Summation

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Stochastic processes and financial applications Mathematical functions and polynomials Numerical methods for differential equations

Abstract

The article is devoted to the development of the method of expansion and mean-square approximation of iterated Ito stochastic integrals based on generalized multiple Fourier series converging in the mean. We adapt this method for iterated Stratonovich stochastic integrals of multiplicity 3 from the Taylor-Stratonovich expansion. The main result of the article has been derived with using the triple Fourier-Legendre series and triple trigonometric Fourier series for the general case of series summation. Some recent results on the expansion of iterated Stratonovich stochastic integrals of multiplicities 3 to 6 are given. The results of the article can be applied to the numerical integration of Ito stochastic differential equations in accordance with the strong criterion of convergence.

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References

2016Numerical solution of stochastic differential equations 1983Stochastic Differential Equations and Diffusion Processes 1965On the relation between ordinary and stochastic differential equations 1965On the Convergence of Ordinary Integrals to Stochastic Integrals 2010Numerical Solution of Stochastic Differential Equations with Jumps in Finance 1994Numerical solution of SDE through computer experiments 2006Wiener Chaos Expansion and Numerical Solutions of Stochastic Partial Differential Equations 1992The approximation of multiple stochastic integrals

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2018The Hypotheses on Expansions of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity and Their Partial Proof 2020Explicit One-Step Numerical Method with the Strong Convergence Order of 2.5 for Ito Stochastic Differential Equations with a Multi-Dimensional Nonadditive Noise Based on the Taylor–Stratonovich Expansion 2018Expansion of Multiple Stratonovich Stochastic Integrals of Multiplicity 2, Based on Double Fourier-Legendre Series, Summarized by Prinsheim Method

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