Using Spectral Form of Mathematical Description to Represent Iterated Stratonovich Stochastic Integrals

Рыбаков, К. А.

doi:10.1007/978-981-33-4826-4_20

Cited by 5 publications

(5 citation statements)

References 23 publications

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“…To obtain the main result, it is proposed to apply the spectral method [33,34]. Here, it is described rather briefly, where only the notations and properties required below are given.…”

Section: Fundamentals Of the Spectral Methodsmentioning

confidence: 99%

“…In [33,34], the spectral characteristic W of the Wiener process W(•) is proposed in the form W = P −1 V.…”

Section: Fundamentals Of the Spectral Methodsmentioning

confidence: 99%

“…For iterated Itô stochastic integrals, it avoids relations with the indicators [20] or the Wick product [31] of random variables that correspond to Wiener processes. In this context, iterated Itô and Stratonovich stochastic integrals are considered earlier with the simplest weight functions [32][33][34]. Here, we propose a general case of weight functions and consider mixed-type iterated stochastic integrals of arbitrary multiplicity.…”

Section: Introductionmentioning

confidence: 99%

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Spectral Representations of Iterated Stochastic Integrals and Their Application for Modeling Nonlinear Stochastic Dynamics

Rybakov

2023

Mathematics

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Spectral representations of iterated Itô and Stratonovich stochastic integrals of arbitrary multiplicity, including integrals from Taylor–Itô and Taylor–Stratonovich expansions, are obtained by the spectral method. They are required for the implementation of numerical methods for solving Itô and Stratonovich stochastic differential equations with high orders of mean-square and strong convergence. The purpose of such numerical methods is the modeling of nonlinear stochastic dynamics in many fields. This paper contains necessary theoretical results, as well as the results of numerical experiments.

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“…To obtain the main result, it is proposed to apply the spectral method [33,34]. Here, it is described rather briefly, where only the notations and properties required below are given.…”

Section: Fundamentals Of the Spectral Methodsmentioning

confidence: 99%

“…In [33,34], the spectral characteristic W of the Wiener process W(•) is proposed in the form W = P −1 V.…”

Section: Fundamentals Of the Spectral Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Spectral Representations of Iterated Stochastic Integrals and Their Application for Modeling Nonlinear Stochastic Dynamics

Rybakov

2023

Mathematics

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“…Hereinafter in this section always s ∈ [t, T ]. Differentiating by the Itô formula the expression in parentheses on the right-hand side of equality (30) and combining the result of differentiation with (30), we obtain w. p. 1…”

Section: Modification and Generalization Of Itô's Theorem Proof On Th...mentioning

confidence: 99%

Optimization of the mean-square approximation procedures for iterated Stratonovich stochastic integrals of multiplicities 1 to 3 with respect to components of the multi-dimensional Wiener process based on Multiple Fourier–Legendre series

Kuznetsov

2022

MATEC Web of Conferences

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The article is devoted to approximation of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 3 by the method of multiple Fourier–Legendre series. The mentioned stochastic integrals are part of strong numerical methods with convergence order 1.5 for Ito stochastic differential equations with multidimensional noncommutative noise. These numerical methods are based on the so-called Taylor–Ito and Taylor–Stratonovich expansions. We calculate the exact lengths of sequences of independent standard Gaussian random variables required for the mean-square approximation of iterated Stratonovich stochastic integrals. Thus, the computational cost for the implementation of numerical methods can be significantly reduced.

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“…There are various approaches to solving the problem of the mean-square approximation of iterated stochastic integrals. Among them, we note the approach based on the Karhunen-Loeve expansion of the Brownian bridge process [1]- [4], [13], [18], [21], approach based on the expansion of the Wiener process using various basis systems of functions [6], [10], [30], [31], approach based on the conditional joint characteristic function of a stochastic integral of multiplicity 2 [11], [12] as well as an approach based on multiple integral sums [1], [19].…”

Section: Introductionmentioning

confidence: 99%

The Proof of Convergence with Probability 1 in the Method of Expansion of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series

Kuznetsov¹

2020

Preprint

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The article is devoted to the formulation and proof of the theorem on convergence with probability 1 of expansion of iterated Itô stochastic integrals of arbitrary multiplicity based on generalized multiple Fourier series converging in the sense of norm in Hilbert space. The cases of multiple Fourier-Legendre series and multiple trigonomertic Fourier series are considered in detail. The proof of the mentioned theorem is based on the general properties of multiple Fourier series as well as on the estimate for the fourth moment of approximation error in the method of expansion of iterated Itô stochastic integrals based on generalized multiple Fourier series.

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Using Spectral Form of Mathematical Description to Represent Iterated Stratonovich Stochastic Integrals

Cited by 5 publications

References 23 publications

Spectral Representations of Iterated Stochastic Integrals and Their Application for Modeling Nonlinear Stochastic Dynamics

Spectral Representations of Iterated Stochastic Integrals and Their Application for Modeling Nonlinear Stochastic Dynamics

Optimization of the mean-square approximation procedures for iterated Stratonovich stochastic integrals of multiplicities 1 to 3 with respect to components of the multi-dimensional Wiener process based on Multiple Fourier–Legendre series

The Proof of Convergence with Probability 1 in the Method of Expansion of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series

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