SIMPLEST RANDOM WALK FOR APPROXIMATING ROBIN BOUNDARY VALUE PROBLEMS AND ERGODIC LIMITS OF REFLECTED DIFFUSIONS

• 成果类型: Article
• 署名作者: Leimkuhler, Benedict; Sharma, Akash; V. Tretyakov, Michael
• 署名单位: University of Edinburgh; University of Edinburgh; Heriot Watt University; University of Nottingham
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1856
• 发表日期: 2023
• 页码: 1904-1960
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; parabolic equations; langevin; simulation; BEHAVIOR

摘要

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, that is, 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 number of extensions are considered including a second-order weak approximation, the case of ar-bitrary oblique direction of reflection, and a new adaptive weak scheme to solve a Poisson PDE with Neumann boundary condition. The presented the-oretical results are supported by several numerical experiments.