Abstract
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 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.
Funding Statement
BL was supported by EPSRC grant no. EP/P006175/1 and by the Alan Turing Institute (EPSRC EP/N510129/1) as a Turing Fellow. AS was supported by the University of Nottingham Vice-Chancellor’s Scholarship for Research Excellence (International).
Acknowledgments
BL and MVT thank the Institute for Computational and Experimental Research in Mathematics (ICERM, Providence, RI) where this work was nucleated during a workshop hosted by ICERM. The authors thank anonymous referees for useful suggestions.
Citation
Benedict Leimkuhler. Akash Sharma. Michael V. Tretyakov. "Simplest random walk for approximating Robin boundary value problems and ergodic limits of reflected diffusions." Ann. Appl. Probab. 33 (3) 1904 - 1960, June 2023. https://doi.org/10.1214/22-AAP1856
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