Corpus ID: 221995470

# 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

@article{Kuznetsov2020ImplementationOS,
title={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},
author={M. D. Kuznetsov and D. Kuznetsov},
journal={arXiv: Probability},
year={2020}
}
• Published 2020
• Mathematics
• arXiv: Probability
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 multidimensional non-commutative noise based on the unified Taylor--Ito and Taylor-Stratonovich expansions and multiple Fourier-Legendre series. Algorithms for the implementation of these methods are constructed and a package of programs on the Python programming language is presented. An important part of… Expand
10 Citations
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 ofExpand
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.
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
A Comparative Analysis of Efficiency of Using the Legendre Polynomials and Trigonometric Functions for the Numerical Solution of Ito Stochastic Differential Equations
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 differentialExpand
Exact Calculation of the Mean-Square Error in the Method of Approximation of Iterated Ito Stochastic integrals, Based on Generalized Multiple Fourier Series
The article is devoted to the developement of the method of expansion and mean-square approximation of iterated Ito stochastic integrals based on generalized multiple Fourier series converging in theExpand
Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Iterated Fourier Series Converging Pointwise
The article is devoted to the expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k (k ∈ N) based on generalized iterated Fourier series converging pointwise. The caseExpand
Expansion of Iterated Stratonovich Stochastic Integrals of Fifth Multiplicity, Based on Generalized Multiple Fourier Series
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 expansionExpand
Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 2. Combined Approach Based on Generalized Multiple and Iterated Fourier Series
Abstract. The article is devoted to the expansion of iterated Stratonovich stochastic integrals of multiplicity 2 on the base of the combined approach of generalized multiple and repeated FourierExpand
Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3, Based on Generalized Multiple Fourier Series, Converging in the Mean: General Case of Series Summation
The article is devoted to the development of the method of expansion and mean-square approximation of iterated Ito stochastic integrals, based on generalized multiple Fourier series, converging inExpand
Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 4. Combained Approach Based on Generalized Multiple and Repeated Fourier series
The article is devoted to the expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 4 on the base of the combined approach of generalized multiple and repeated FourierExpand
Expansions of Multiple Stratonovich Stochastic Integrals From the Taylor-Stratonovich Expansion, Based on Multiple Trigonometric Fourier Series. Comparison With the Milstein Expansion
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 FourierExpand

#### References

SHOWING 1-10 OF 82 REFERENCES
Strong Numerical Methods of Order 3.0 for Ito Stochastic Differential Equations, Based on the Unified Stochastic Taylor Expansions and Multiple Fourier-Legendre Series
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 theExpand
Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals of Multiplicities 1 to 6 from the Taylor-Ito and Taylor-Stratonovich Expansions Using Legendre Polynomials.
The article is devoted to the practical material on expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 with respect toExpand
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
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 andExpand
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
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 nonlinearExpand
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
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 theExpand
Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures and with Respect to Martingales Based on Generalized Multiple Fourier Series
In the article we consider some versions and generalizations of the approach to expansion of iterated Ito stochastic integrals of arbitrary multiplicity based on generalized multiple Fourier series.Expand
Expansion of iterated Stratonovich stochastic integrals based on generalized multiple Fourier series
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-squareExpand
On Numerical Modeling of the Multidimensional Dynamic Systems under Random Perturbations with the 1.5 and 2.0 Orders of Strong Convergence
• D. Kuznetsov
• Mathematics, Computer Science
• Autom. Remote. Control.
• 2018
The paper was devoted to developing numerical methods with the orders 1.5 and 2.0 of strong convergence for the multidimensional dynamic systems under random perturbations obeying stochasticExpand
Strong approximation of multiple Ito and Stratonovich stochastic integrals: multiple Fourier series approach
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 toExpand
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