Modify the Improved Euler scheme to integrate stochastic differential equations

TL;DR

This paper introduces a modified Improved Euler scheme for stochastic differential equations, demonstrating strong convergence and providing an accessible method for numerical integration in stochastic contexts.

Contribution

It presents a new Runge--Kutta scheme based on the deterministic Improved Euler method, adapted for stochastic differential equations, with proven strong convergence.

Findings

The scheme exhibits strong convergence in numerical experiments.

First order strong convergence is proved for both Ito and Stratonovich integrals.

The method is suitable as an entry-level scheme for stochastic differential equations.

Abstract

A practical and new Runge--Kutta numerical scheme for stochastic differential equations is explored. Numerical examples demonstrate the strong convergence of the method. The first order strong convergence is then proved using Ito integrals for both Ito and Stratonovich interpretations. As a straightforward modification of the deterministic Improved Euler/Heun method, the method is a good entry level scheme for stochastic differential equations, especially in conjunction with Higham's introduction [SIAM Review, 43:525--546, 2001].