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

TL;DR

This paper develops explicit one-step numerical methods with strong convergence orders 1.0, 1.5, and 2.0 for multidimensional Ito stochastic differential equations, enabling more accurate stochastic simulations.

Contribution

It introduces new explicit numerical schemes with high strong orders for multidimensional stochastic differential equations with non-commutative noise.

Findings

Methods achieve strong orders 1.0, 1.5, and 2.0 of convergence.

Uses Fourier-Legendre series for iterated Ito integrals.

Applicable to stochastic control and nonlinear filtering problems.

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.