A nonstandard Euler-Maruyama scheme

TL;DR

This paper introduces a novel nonstandard finite difference scheme for stochastic differential equations, demonstrating improved stability and convergence properties over traditional Euler-Maruyama methods.

Contribution

The authors develop a new nonstandard scheme based on weighed steps, proving its strong convergence and domain invariance preservation under minimal conditions.

Findings

Scheme outperforms Euler-Maruyama in stability and accuracy

Proven strong convergence under Lipschitz and growth conditions

Maintains domain invariance unconditionally for expected solutions

Abstract

We construct a nonstandard finite difference numerical scheme to approximate stochastic differential equations (SDEs) using the idea of weighed step introduced by R.E. Mickens. We prove the strong convergence of our scheme under locally Lipschitz conditions of a SDE and linear growth condition. We prove the preservation of domain invariance by our scheme under a minimal condition depending on a discretization parameter and unconditionally for the expectation of the approximate solution. The results are illustrated through the geometric Brownian motion. The new scheme shows a greater behavior compared to the Euler-Maruyama scheme and balanced implicit methods which are widely used in the literature and applications.