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

TL;DR

This paper develops explicit high-order numerical methods for Ito stochastic differential equations using unified stochastic Taylor expansions and Fourier-Legendre series, improving accuracy for complex stochastic systems.

Contribution

It introduces new strong order methods (2.0, 2.5, 3.0) based on unified Taylor expansions and Fourier-Legendre series for multidimensional Ito SDEs with non-commutative noise.

Findings

Methods achieve strong convergence orders 2.0, 2.5, and 3.0.

Utilizes Fourier-Legendre series for efficient approximation of stochastic integrals.

Applicable to stochastic control and nonlinear filtering problems.

Abstract

The article is devoted to the construction of explicit one-step numerical methods with the strong orders of convergence 2.0, 2,5, and 3.0 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 numerical modeling of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 we appling the method of multiple Fourier-Legendre series converging in the sense of norm in Hilbert space $L_2([t, T]^k),$ $k=1,\ldots,6$. The article is addressed to engineers who use numerical modeling in stochastic control and for solving the non-linear filtering problem. The article can be interesting for the mathematicians who working in the field of high-order strong numerical methods for Ito stochastic differential equations.