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

@article{Kuznetsov2019ACA, title={A Comparative Analysis of Efficiency of Using the Legendre Polynomials and Trigonometric Functions for the Numerical Solution of Ito Stochastic Differential Equations}, author={D. Kuznetsov}, journal={Computational Mathematics and Mathematical Physics}, year={2019}, volume={59}, pages={1236-1250} }

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 equations in the framework of the method of approximation of iterated Ito and Stratonovich stochastic integrals based on generalized multiple Fourier series. On the example of iterated Ito stochastic integrals of multiplicities 1 to 3, included in the Taylor-Ito expansion, it is shown that expansions of… Expand

#### 27 Citations

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

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- 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

Expansion of Iterated Ito Stochastic Integrals of Arbitrary Multiplicity, Based on Generalized Multiple Fourier Series, Converging in the Mean

- Mathematics
- 2017

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

Numerical Simulation of 2.5-Set of Iterated Ito Stochastic Integrals of Multiplicities 1 to 5 From the Taylor-Ito Expansion.

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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

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

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- 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

Exact Calculation of Mean-Square Error of Approximation of Multiple Ito Stochastic integrals for the Method, Based on the Multiple Fourier Series

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- 2018

The article is devoted to the obtainment of exact and approximate expressions for mean-square error of approximation of multiple Ito stochastic integrals from the stochastic Taylor-Ito expansion for… Expand

New Simple Method for Obtainment an Expansion of Double Stochastic Ito integrals, Based on the Expansion of Brownian Motion Using Legendre polynomials and Trigonometric Functions.

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- 2019

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

Expansion of Multiple Stratonovich Stochastic Integrals of Arbitrary Multiplicity, Based on Generalized Repeated Fourier Series, Converging Pointwise

- Mathematics
- 2018

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

Numerical Simulation of 2.5-Set of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 5 From the Taylor-Stratonovich Expansion

- Mathematics
- 2018

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

Expansions of Multiple Stratonovich Stochastic Integrals From the Taylor-Stratonovich Expansion, Based on Multiple Trigonometric Fourier Series. Comparison With the Milstein Expansion

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The article is devoted to comparison of the Milstein expansion of multiple stochastic integrals with the method of expansion of multiple stochastic integrals, based on generalized multiple Fourier… Expand

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

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The article is devoted to explicit one-step numerical methods with strong order of convergence 3.0 for Ito stochastic differential equations with multidimensional non-additive noise. We consider the… Expand

#### References

SHOWING 1-10 OF 76 REFERENCES

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

- Mathematics
- 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… Expand

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

Expansions of multiple Stratonovich stochastic integrals, based on generalized multiple Fourier series

- Mathematics
- 2017

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

Expansion of Iterated Ito Stochastic Integrals of Arbitrary Multiplicity, Based on Generalized Multiple Fourier Series, Converging in the Mean

- Mathematics
- 2017

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

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

- Mathematics
- 2020

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

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

Exact Calculation of Mean-Square Error of Approximation of Multiple Ito Stochastic integrals for the Method, Based on the Multiple Fourier Series

- Mathematics
- 2018

The article is devoted to the obtainment of exact and approximate expressions for mean-square error of approximation of multiple Ito stochastic integrals from the stochastic Taylor-Ito expansion for… Expand

Expansion of Multiple Stratonovich Stochastic Integrals of Arbitrary Multiplicity, Based on Generalized Repeated Fourier Series, Converging Pointwise

- Mathematics
- 2018

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

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

- Mathematics
- 2018

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

Strong approximation of multiple Ito and Stratonovich stochastic integrals: multiple Fourier series approach

- Mathematics
- 2011

It is well known, that Ito stochastic differential equations (SDE) are adequate mathematical models of dynamic systems under the influence of random disturbances. One of the effective approaches to… Expand