A simple-to-implement weak-sense numerical method to approximate reflected stochastic differential equations (RSDEs) is proposed and analysed. It is proved that the method has the first order of weak convergence. Together with the Monte Carlo technique, it can be used to numerically solve linear parabolic and elliptic PDEs with Robin boundary condition. One of the key results of this paper is the use of the proposed method for computing ergodic limits, i.e. expectations with respect to the invariant law of RSDEs, both inside a domain in $\mathbb{R}^{d}$ and on its boundary. This allows to efficiently sample from distributions with compact support. Both time-averaging and ensemble-averaging estimators are considered and analysed. A number of extensions are considered including a second-order weak approximation, the case of arbitrary oblique direction of reflection, and a new adaptive weak scheme to solve a Poisson PDE with Neumann boundary condition. The presented theoretical results are supported by several numerical experiments.
翻译:提出并分析一种简单到执行的弱智数字方法,以近似反映的随机差分方程(RSDEs),证明该方法具有第一级弱化趋同。与蒙特卡洛技术一起,该方法可用于在数字上解决线性抛物线和椭圆式 PDEs与Robin边界条件。本文件的主要结果之一是使用拟议方法计算ERgodic限制值,即对RSDEs的惰性法律的期望,该方法在$\mathbb{R ⁇ d}$的域内及其边界上都是如此。这可以有效地从分布上抽样,并获得紧凑支持。审议和分析的是时间-寿命和共振-动态估计器。一些扩展考虑包括二级弱近似、任意偏斜的反省方向以及用Neumann边界条件解决Poisson PDE的新的适应性弱计划。提出的理论结果得到了若干数字实验的支持。