# Expansion of iterated Stratonovich stochastic integrals based on generalized multiple Fourier series

@inproceedings{Kuznetsov2019ExpansionOI, title={Expansion of iterated Stratonovich stochastic integrals based on generalized multiple Fourier series}, author={D. Kuznetsov}, year={2019} }

The article is devoted to expansions of iterated Stratonovich stochastic integrals of multiplicities 1-4 on the base of the method of generalized multiple Fourier series. We prove the mean-square convergence of expansions in the case of Legendre polynomials as well as in the case of trigonometric functions. The considered expansions contain only one passage to the limit in contrast to its existing analogues. This property is very convenient for the mean-square approximation of iterated… Expand

#### 32 Citations

Expansion of Iterated Stratonovich Stochastic Integrals of Fifth Multiplicity, Based on Generalized Multiple Fourier Series

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The article is devoted to the construction of expansion of iterated Stratonovich stochastic integrals of fifth multiplicity, based on the method of generalized multiple Fourier series. This expansion… Expand

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The article is devoted to construction of effective procedures of the mean-square approximation for iterated Stratonovich stochastic integrals of multiplicities 1 to 5. We apply the method of… Expand

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The article is devoted to the expansion of multiple Stratonovich stochastic integrals of arbitrary multiplicity $k$, based on the generalized repeated Fourier series. The case of Fourier-Legendre… Expand

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The article is devoted to expansions of multiple Ito stochastic integrals, based on generalized multiple Fourier series converging in the mean. The method of generalized multiple Fourier series for… Expand

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- Mathematics
- 2018

Abstract. The article is devoted to the expansion of iterated Stratonovich stochastic integrals of second multiplicity into the double series of products of standard Gaussian random variables. The… Expand

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- Mathematics
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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 Ito stochastic integrals of arbitrary multiplicity based on generalized… Expand

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The article is devoted to comparative analysis of the efficiency of application of Legendre polynomials and trigonometric functions to the numerical integration of Ito stochastic differential… Expand

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We consider a method for the approximation of iterated stochastic Ito integrals of arbitrary multiplicity $k$ $(k\in \mathbb{N})$ with respect to the infinite-dimensional Wiener process using the… Expand

Expansion of Multiple Stratonovich Stochastic Integrals of Multiplicity 2, Based on Double Fourier-Legendre Series, Summarized by Prinsheim Method

- Mathematics
- 2018

The article is devoted to the expansion of multiple Stratonovich stochastic integrals of multiplicity 2 into double series of standard Gaussian random variables. The proof of the expansion is based… Expand

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- 2019

The article is devoted to the construction of effective procedures of the mean-square approximation of iterated Ito stochastic integrals of multiplicities 1 to 5 from the Taylor-Ito expansion, based… Expand

#### References

SHOWING 1-10 OF 18 REFERENCES

New Representations of the Taylor–Stratonovich Expansion

- Mathematics
- 2003

The problem of the Taylor–Stratonovich expansion of the Itô random processes in a neighborhood of a point is considered. The usual form of the Taylor–Stratonovich expansion is transformed to a new… Expand

Numerical Solution of Stochastic Differential Equations

- Mathematics
- 2015

This paper provides an introduction to the main concepts and techniques necessary for someone who wishes to carryout numerical experiments involving Stochastic Differential Equation (SDEs). As SDEs… Expand

New representations of explicit one-step numerical methods for jump-diffusion stochastic differential equations

- Mathematics
- 2001

Numerical integration methods for jump-diffusion stochastic differential equations (SDEs) are considered. The numerical methods are constructed by using a special time discretization adapted to the… Expand

Numerical integration of stochastic differential equations — ii

- Mathematics
- The Bell System Technical Journal
- 1981

In a previous paper, a method was presented to integrate numerically nonlinear stochastic differential equations (SDEs) with additive, Gaussian, white noise. The method, a generalization of the Range… Expand

Stochastic Numerics for Mathematical Physics

- Mathematics
- 2004

1 Mean-square approximation for stochastic differential equations.- 2 Weak approximation for stochastic differential equations.- 3 Numerical methods for SDEs with small noise.- 4 Stochastic… Expand

The approximation of multiple stochastic integrals

- Mathematics
- 1992

A method for approximating the multiple stochastic integrals appearing in stochaslic Taylor expansions is proposed. It is based on a series expansion of the Brownian bridge process. Some higher order… Expand

The Theory of Spherical and Ellipsoidal Harmonics

- Mathematics
- 1955

Preface 1. The transformation of Laplaces's equation 2. The solution of Laplace's equation in polar coordinates 3. The Legendres associated functions 4. Spherical harmonics 5. Spherical harmonics of… Expand

Numerical Integration of Stochastic Differential Equations

- Physics
- 2004

This chapter provides an introduction into the numerical integration of stochastic differential equations (SDEs). Again X t denotes a stochastic process and solution of an SDE,
$$\frac{{\partial… Expand

Numerical integration of stochastic differential equations

- The Bell System Technical Journal
- 1979

A procedure for numerical integration of a stochastic differential equation, by extension of the Runge-Kutta method, is presented. The technique produces results which are statistically correct to a… Expand

Classical orthogonal polynomials. Fizmatlit

- (in Russian). Dmitriy Feliksovich Kuznetsov, Peter the Great St.Petersburg Polytechnic University, Polytechnicheskaya str
- 2005