Explicit One-Step Strong Numerical Methods of Orders 2.0 and 2.5 for Ito Stochastic Differential Equations Based on the Unified Taylor-Ito and Taylor-Stratonovich Expansions

Abstract

The article is devoted to the construction of explicit one-step strong numerical methods with the orders 2.0 and 2.5 of convergence for Ito stochastic differential equations with multidimensional non-commutative noise. We consider the numerical methods based on the unified Taylor-Ito and Taylor-Stratonovich expansions. For the numerical modeling of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 5 we apply the method of multiple Fourier-Legendre series converging in the sense of norm in Hilbert space $L_2([t,T{]}^k),$ $k=1,\dots ,5$. The article is addressed to engineers who use numerical modeling in stochastic control and for solving the non-linear filtering problem. The article will be interesting to scientists who working in the field of numerical integration of stochastic differential equations.

37 pages. Minor changes. arXiv admin note: text overlap with arXiv:1801.00231, arXiv:1712.08991, arXiv:1802.00643, arXiv:1801.08862. text overlap with arXiv:1801.00784

References in corpus (10)

• To Numerical Modeling With Strong Orders 1.0, 1.5, and 2.0 of Convergence for Multidimensional Dynamical Systems With Random Disturbances
• Development and Application of the Fourier Method to the Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals
• Comparative Analysis of the Efficiency of Application of Legendre Polynomials and Trigonometric Functions to the Numerical Integration of Ito Stochastic Differential Equations
• Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on PYTHON and MATLAB
• Numerical Simulation of 2.5-Set of Iterated Ito Stochastic Integrals of Multiplicities 1 to 5 From the Taylor-Ito Expansion
• Explicit One-Step Strong Numerical Methods of Orders 2.0 and 2.5 for Ito Stochastic Differential Equations Based on the Unified Taylor-Ito and Taylor-Stratonovich Expansions
• Expansion of Iterated Stratonovich Stochastic Integrals of Fifth, Sixth, Seventh and Eighth Multiplicities Based on Generalized Multiple Fourier Series
• 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
• Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 2 Based on Double Fourier-Legendre Series Summarized by Pringsheim Method
• The Hypotheses on Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity and Their Partial Proof

Cited by in corpus (8)

• To Numerical Modeling With Strong Orders 1.0, 1.5, and 2.0 of Convergence for Multidimensional Dynamical Systems With Random Disturbances
• Development and Application of the Fourier Method to the Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals
• Comparative Analysis of the Efficiency of Application of Legendre Polynomials and Trigonometric Functions to the Numerical Integration of Ito Stochastic Differential Equations
• Numerical Simulation of 2.5-Set of Iterated Ito Stochastic Integrals of Multiplicities 1 to 5 From the Taylor-Ito Expansion
• Explicit One-Step Strong Numerical Methods of Orders 2.0 and 2.5 for Ito Stochastic Differential Equations Based on the Unified Taylor-Ito and Taylor-Stratonovich Expansions
• Numerical Simulation of 2.5-Set of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 5 From the Taylor-Stratonovich Expansion
• 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
• A new proof of the expansion of iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process based on generalized multiple Fourier series and Hermite polynomials