Semi-implicit Euler--Maruyama scheme for polynomial diffusions on the unit ball

TL;DR

This paper introduces a semi-implicit Euler--Maruyama numerical scheme with projection for polynomial diffusions on the unit ball, providing convergence rates and leveraging transformation techniques for non-Lipschitz diffusions.

Contribution

The paper presents a novel semi-implicit Euler--Maruyama scheme with projection for polynomial diffusions on the unit ball, including convergence analysis and innovative use of transformation methods.

Findings

Established the $L^{2}$-rate of convergence for the scheme.

Proved the scheme's effectiveness for polynomial diffusions with non-Lipschitz coefficients.

Utilized transformation arguments to handle non-Lipschitz diffusion coefficients.

Abstract

In this article, we consider numerical schemes for polynomial diffusions on the unit ball, which are solutions of stochastic differential equations with a diffusion coefficient of the form $\sqrt{1-|x|^{2}}$. We introduce a semi-implicit Euler--Maruyama scheme with the projection onto the unit ball and provide the $L^{2}$-rate of convergence. The main idea to consider the numerical scheme is the transformation argument introduced by Swart for proving the pathwise uniqueness for some stochastic differential equation with a non-Lipschitz diffusion coefficient.