Random walks for solving Robin boundary value problems and sampling in a bounded domain - Sharma, Akash (2022)
A weak-sense numerical method to approximate reflected stochastic differential equations (RSDEs) is proposed and analysed. The method is simple to implement. 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 Rd 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 new second-order weak approximation method is also presented and investigated. The case of arbitrary oblique direction of reflection is also considered. Further, a new adaptive weak scheme to solve a Poisson PDE with Neumann boundary condition is proposed and is analysed with regards to convergence and cost. The presented theoretical results are supported by several numerical experiments.