To Numerical Modeling With Strong Orders 1.0, 1.5, and 2.0 of Convergence for Multidimensional Dynamical Systems With Random Disturbances

Abstract

The article is devoted to explicit one-step numerical methods with strong orders 1.0, 1.5, and 2.0 of convergence for Ito stochastic differential equations with multidimensional and non-commutative noise. For numerical modeling of iterated Ito stochastic integrals with multiplicities 1 to 4 we use the method of multiple Fourier-Legendre series converging in the sense of norm in Hilbert space $L_2([t,T{]}^k),$ $k=1,2,3,4.$ The article is addressed to engineers who use numerical modeling in stochastic control and for solving the nonlinear filtering problem.

29 pages. Minor changes. arXiv admin note: text overlap with arXiv:1801.01564, arXiv:1802.00643, arXiv:1801.01962, arXiv:1801.08862, arXiv:1712.09516, arXiv:1801.03195, arXiv:1712.09746

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

Cited by in corpus (20)

• 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
• Expansion of Iterated Ito Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Multiple Fourier Series Converging in the Mean
• 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
• Application of the Method of Approximation of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series to the High-Order Strong Numerical Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations
• 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
• Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Iterated Fourier Series Converging Pointwise
• Comparative Analysis of the Efficiency of Application of Legendre Polynomials and Trigonometric Functions to the Numerical Integration of Ito Stochastic Differential Equations
• Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures and with Respect to Martingales Based on Generalized Multiple Fourier Series
• Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on PYTHON and MATLAB
• Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 Based on Generalized Multiple Fourier Series Converging in the Mean: General Case of Series Summation
• Application of Multiple Fourier-Legendre Series to Implementation of Strong Exponential Milstein and Wagner-Platen Methods for Non-Commutative Semilinear Stochastic Partial 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
• 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
• 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
• 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 Proof of Convergence with Probability 1 in the Method of Expansion of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier 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