# Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 4, Based on Generalized Multiple Fourier Series

@article{Kuznetsov2017ExpansionsOI, title={Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 4, Based on Generalized Multiple Fourier Series}, author={D. F. Kuznetsov}, journal={arXiv: Probability}, year={2017} }

The article is devoted to the expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 4 on the basis of the method of generalized multiple Fourier series. Mean-square convergence of the expansions for the case of multiple Fourier-Legendre series and for the case of multiple trigonometric Fourier series is proven. 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… Expand

#### 32 Citations

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

- Mathematics
- 2018

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

Strong Numerical Methods of Orders 2.0, 2.5, and 3.0 for Ito Stochastic Differential Equations Based on the Unified Stochastic Taylor Expansions and Multiple Fourier-Legendre Series

- Mathematics
- 2018

Abstract. The article is devoted to the construction of explicit one-step strong numerical methods with the orders of convergence 2.0, 2,5, and 3.0 for Ito stochastic differential equations with… Expand

Four New Forms of the Taylor-Ito and Taylor-Stratonovich Expansions and its Application to the High-Order Strong Numerical Methods for Ito Stochastic Differential Equations

- Mathematics
- 2020

The problem of the Taylor-Ito and Taylor-Stratonovich expansions of the Ito stochastic processes in a neighborhood of a fixed moment of time is considered. The classical forms of the Taylor-Ito and… Expand

Implementation of Strong Numerical Methods of Orders 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 for Ito SDEs with Non-Commutative Noise Based on the Unified Taylor-Ito and Taylor-Stratonovich Expansions and Multiple Fourier-Legendre Series

- Mathematics
- 2020

The article is devoted to the implementation of strong numerical methods with convergence orders $0.5,$ $1.0,$ $1.5,$ $2.0,$ $2.5,$ and $3.0$ for Ito stochastic differential equations with… Expand

New Simple Method of Expansion of Iterated Ito Stochastic integrals of Multiplicity 2 Based on Expansion of the Brownian Motion Using Legendre Polynomials and Trigonometric Functions.

- Mathematics
- 2020

The atricle is devoted to the new simple method for obtainment an expansion of iterated Ito stochastic integrals of multiplicity 2 based on expansion of the Brownian motion (standard Wiener process)… Expand

Optimization of the Mean-Square Approximation Procedures for Iterated Ito Stochastic Integrals of Multiplicities 1 to 5 from the Unified Taylor-Ito Expansion Based on Multiple Fourier-Legendre Series

- Mathematics
- 2020

The article is devoted to optimization of the mean-square approximation procedures for iterated Ito stochastic integrals of multiplicities 1 to 5. The mentioned stochastic integrals are part of… Expand

Strong Approximation of Iterated Ito and Stratonovich Stochastic Integrals Based on Generalized Multiple Fourier Series. Application to Numerical Solution of Ito SDEs and Semilinear SPDEs

- Mathematics
- 2020

The book is devoted to the strong approximation of iterated stochastic integrals (ISIs) in the context of numerical integration of Ito SDEs and non-commutative semilinear SPDEs with nonlinear… Expand

A Comparative Analysis of Efficiency of Using the Legendre Polynomials and Trigonometric Functions for the Numerical Solution of Ito Stochastic Differential Equations

- Mathematics
- Computational Mathematics and Mathematical Physics
- 2019

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

Application of Multiple Fourier-Legendre Series to Implementation of Strong Exponential Milstein and Wagner-Platen Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations

- Mathematics
- 2019

The article is devoted to the application of multiple Fourier-Legendre series to implementation of strong exponential Milstein and Wagner-Platen methods for non-commutative semilinear stochastic… Expand

Application of the Method of Approximation of Iterated Stochastic Ito Integrals Based on Generalized Multiple Fourier Series to the High-Order Strong Numerical Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations

- Mathematics
- 2019

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

#### References

SHOWING 1-10 OF 77 REFERENCES

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

Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MATLAB Programs, 6th Edition. [In Russian

- Electronic Journal "Differential Equations and Control Processes"ISSN 1817-2172 (online),
- 2018

Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on MATLAB, 5th Edition. [In Russian

- Electronic Journal "Differential Equations and Control Processes"ISSN 1817-2172 (online),
- 2017

Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab programs. 4th Edition. [In Russian

- Polytechnical University Publishing House: St.-Petersburg,
- 2010

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

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

Problems of the numerical analysis of Ito stochastic differential equations. [In Russian

- Electronic Journal "Differential Equations and Control Processes"ISSN 1817-2172 (online),
- 1998

A method of expansion and approximation of repeated stochastic Stratonovich integrals based on multiple Fourier series on full orthonormal systems. [In Russian

- Electronic Journal "Differential Equations and Control Processes"ISSN 1817-2172 (online),
- 1997

A method of expansion and approximation of repeated stochastic Stratonovich integrals based on multiple Fourier series on full orthonormal systems. [In Russian

- Electronic Journal "Differential Equations and Control Processes"ISSN 1817-2172 (online),
- 1997

Stochastic Differential Equations and Diffusion Processes. 2nd Edition

- 1989