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 approximations 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 with the same strong convergence rate as the strong convergence rate of the numerical scheme for the corresponding RSDE. 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（SDEs）提出了一种新颖的人工壁垒方法，并利用该方法构建了用于标量SDE强逼近的保边界数值格式。所考虑的SDE可能具有非全局Lipschitz的漂移项和扩散系数，其状态空间为有界或半有界。人工壁垒方法的核心思想是在状态空间外部引入人工壁垒，从而在不改变原解过程的前提下，将原SDE转化为带反射边界的SDE（RSDE），然后对RSDE应用保边界数值格式。这使得我们能够构建保边界数值格式，其强收敛阶与对应RSDE数值格式的强收敛阶相同。基于人工壁垒方法，我们构建了两种保边界格式，分别称为人工壁垒欧拉-丸山（ABEM）格式和人工壁垒欧拉-皮亚诺（ABEP）格式。我们对ABEM格式进行了数值实验，实验结果与理论分析结果一致。