Corpus ID: 210932380

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

@article{Kuznetsov2020FourNF,
  title={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},
  author={D. Kuznetsov},
  journal={arXiv: Probability},
  year={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 Taylor-Stratonovich expansions are transformed to the four new representations, which includes the minimal sets of different types of iterated Ito and Stratonovich stochastic integrals. Therefore, these representations (the so-called unified Taylor-Ito and Taylor-Stratonovich expansions) are more… Expand

Tables from this paper

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

References

SHOWING 1-10 OF 71 REFERENCES
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
Stratonovich and Ito Stochastic Taylor Expansions
The Stratonovich stochastic Taylor formula for diffusion processes is stated and proved. It has a simpler structure and is a more natural generalization of the deterministic Taylor formula than theExpand
Numerical Solution of Stochastic Differential Equations
1. Probability and Statistics.- 2. Probability and Stochastic Processes.- 3. Ito Stochastic Calculus.- 4. Stochastic Differential Equations.- 5. Stochastic Taylor Expansions.- 6. Modelling withExpand
Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MATLAB Programs, 6th Edition. [In Russian]. Electronic Journal ”Differential Equations and Control Processes
  • 2018
Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab programs. 4th Edition. [In Russian
  • Polytechnical University Publishing House: St.-Petersburg,
  • 2010
Stochastic Differential Equations: Theory and Practice of Numerical Solution. With MatLab programs, 2nd Edition. [In Russian
  • Polytechnical University Publishing House, Saint-Petersburg,
  • 2007
Integration order replacement in iterated stochastic integrals with respect to martingale
  • Preprint. St.-Petersburg: SPbGTU Publ.,
  • 1999
Problems of the numerical analysis of Ito stochastic differential equations. [In Russian]. Electronic Journal ”Differential Equations and Control Processes
  • ISSN 1817-2172 (online),
  • 1998
Theorems about integration order replacement in iterated stochastic integrals
  • Dep. VINITI. 3607-V97,
  • 1997
Stochastic Differential Equations
In the present section we introduce the notion of a stochastic differential equation and prove some general theorems concerning the existence and uniqueness of solutions of these equations. For thisExpand
...
1
2
3
4
5
...