A positivity-preserving truncated Euler--Maruyama method for stochastic differential equations with positive solutions: multi-dimensional case

Abstract: Many positivity-preserving numerical methods have been developed to solve stochastic differential equations (SDEs) with positive solutions in recent years. A common technique in these methods is transformation, such as the Lamperti or logarithmic transformations. It is widely used in one-dimensional cases. However, an effective method for solving multi-dimensional general SDEs with positive solutions has yet to be established. To fill this gap, we propose a positivity-preserving method combining a novel truncated mapping and a truncated Euler--Maruyama discretization in this paper. The strong and weak convergence of the numerical method is studied under local Lipschitz and integrability conditions. Moreover, we show that this method has the optimal strong convergence order 1/2 and the weak convergence order close to 1 under additional assumptions. Numerical experiments are presented to validate theoretical results.