论文标题
最简单的随机步行，用于近似罗宾边界价值问题和反射扩散的千古限制
Simplest random walk for approximating Robin boundary value problems and ergodic limits of reflected diffusions
论文作者
论文摘要
提出并分析了一种简单的弱态数值方法，用于近似反射的随机微分方程（RSD）。证明该方法具有弱收敛的一阶。与蒙特卡洛技术一起，它可用于用罗宾边界条件求解线性抛物线和椭圆形PDE。本文的关键结果之一是使用所提出的方法计算千古限制，即对RSDE的不变定律的期望，两者都在$ \ Mathbb {r}^{d} $中的一个域内，并且在其边界上。这允许从具有紧凑的支持的分布中有效采样。考虑和分析时间平时和合奏平均估计量。考虑了许多扩展，包括二阶弱近似，任意倾斜方向的情况以及一种新的自适应弱方案，以解决具有Neumann边界条件的Poisson PDE。几个数值实验支持了所提出的理论结果。
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.
