# 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
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#### References

SHOWING 1-10 OF 18 REFERENCES
New Representations of the Taylor–Stratonovich Expansion
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 newExpand
Numerical Solution of Stochastic Differential Equations
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 SDEsExpand
New representations of explicit one-step numerical methods for jump-diffusion stochastic differential equations
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 theExpand
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 RangeExpand
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 StochasticExpand
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 orderExpand
The Theory of Spherical and Ellipsoidal Harmonics
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 ofExpand
Numerical Integration of Stochastic Differential Equations
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{{\partialExpand
Numerical integration of stochastic differential equations
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 aExpand
Classical orthogonal polynomials. Fizmatlit
• (in Russian). Dmitriy Feliksovich Kuznetsov, Peter the Great St.Petersburg Polytechnic University, Polytechnicheskaya str
• 2005