Strong convergence rate of positivity-preserving truncated Euler--Maruyama method for multi-dimensional stochastic differential equations with positive solutions

TL;DR

This paper introduces a new positivity-preserving numerical scheme for multi-dimensional SDEs with positive solutions, achieving an optimal strong convergence rate of 1/2 without transformation techniques.

Contribution

The paper proposes a novel truncated Euler–Maruyama method that preserves positivity and attains the optimal convergence rate for multi-dimensional SDEs.

Findings

Method achieves strong convergence order of 1/2.

Numerical experiments confirm positivity preservation.

The approach works under reasonable conditions.

Abstract

To construct positivity-preserving numerical methods, a vast majority of existing works employ transformation techniques such as the Lamperti transformation or logarithmic transformation. However, using these techniques often leads to the transformed stochastic differential equations (SDEs) not meeting the global monotonicity condition, particularly in multi-dimension case. This condition is essential for achieving strong convergence rates of numerical schemes. A pertinent question arises from this issue regarding the existence of an effective method with a convergence rate for solving multi-dimensional SDEs with positive solutions. This paper presents a positivity-preserving method that combines a novel truncated mapping with a truncated Euler--Maruyama discretization. We investigate both the strong convergence of the numerical method under some reasonable conditions. Furthermore, we demonstrate that this method achieves the optimal strong convergence order of 1/2 under certain additional assumptions. Numerical experiments are conducted to validate these theoretical results and demonstrate the positivity of the numerical solutions.