We develop the novel method of artificial barriers for scalar stochastic differential equations (SDEs) and use it to construct boundary-preserving numerical schemes for strong approximation of scalar SDEs, possibly with non-globally Lipschitz drift and diffusion coefficients, whose state-space is either bounded or half-bounded. The idea of artificial barriers is to augment the SDE with artificial barriers outside the state-space to not change the solution process, and then apply a boundary-preserving numerical scheme to the resulting reflected SDE (RSDE). This enables us to construct boundary-preserving numerical schemes that achieve the same strong convergence rate as the corresponding RSDE scheme. Based on the method of artificial barriers, we construct two boundary-preserving schemes that we call the Artificial Barriers Euler--Maruyama (ABEM) scheme and the Artificial Barriers Euler--Peano (ABEP) scheme, respectively. We provide numerical experiments for the ABEM scheme and the numerical results agree with the obtained theoretical results.

翻译：本文针对标量随机微分方程（SDE）提出了一种新颖的人工屏障方法，并利用该方法构建了适用于状态空间有界或半有界的标量SDE强逼近的保边界数值格式，这些方程可能具有非全局Lipschitz的漂移项和扩散系数。人工屏障方法的核心思想是在状态空间外部引入人工屏障，在不改变原解过程的前提下，将原SDE转化为反射随机微分方程（RSDE），然后对RSDE应用保边界数值格式。这使得我们能够构建出与相应RSDE格式具有相同强收敛阶的保边界数值格式。基于人工屏障方法，我们构建了两种保边界格式，分别称为人工屏障欧拉-丸山（ABEM）格式和人工屏障欧拉-佩亚诺（ABEP）格式。我们对ABEM格式进行了数值实验，所得数值结果与理论分析一致。