Simplest random walk for approximating Robin boundary value problems and ergodic limits of reflected diffusions

TL;DR

This paper introduces a simple numerical method for approximating reflected stochastic differential equations with first-order weak convergence, enabling efficient solutions to PDEs with Robin boundary conditions and computation of ergodic limits.

Contribution

It proposes a novel, easy-to-implement weak-sense numerical scheme for RSDEs, with proven convergence and applications to PDEs and ergodic limit computations.

Findings

Method achieves first-order weak convergence.

Effective for solving PDEs with Robin boundary conditions.

Supports computation of ergodic limits inside and on domain boundaries.

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, 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.