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

B. Leimkuhler, A. Sharma, M. V. Tretyakov

TL;DR

This work develops a simple, implementable weak-sense scheme for reflected SDEs in bounded domains and demonstrates its effectiveness for solving Robin-boundary PDEs via Monte Carlo. The core contribution is a weak Euler-type method augmented with a symmetric boundary reflection that preserves the domain and yields first-order weak convergence; extensions include second-order schemes and oblique reflection. The authors establish probabilistic representations for both parabolic and elliptic problems, introduce time- and ensemble-averaging estimators for ergodic limits, and provide rigorous error analyses with numerical validation. The framework enables efficient sampling from distributions with compact support and practical numerical solutions to Robin/Neumann boundary problems in higher dimensions, supported by comprehensive experiments.

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.

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

TL;DR

This work develops a simple, implementable weak-sense scheme for reflected SDEs in bounded domains and demonstrates its effectiveness for solving Robin-boundary PDEs via Monte Carlo. The core contribution is a weak Euler-type method augmented with a symmetric boundary reflection that preserves the domain and yields first-order weak convergence; extensions include second-order schemes and oblique reflection. The authors establish probabilistic representations for both parabolic and elliptic problems, introduce time- and ensemble-averaging estimators for ergodic limits, and provide rigorous error analyses with numerical validation. The framework enables efficient sampling from distributions with compact support and practical numerical solutions to Robin/Neumann boundary problems in higher dimensions, supported by comprehensive experiments.

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 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.

Paper Structure

This paper contains 37 sections, 23 theorems, 251 equations, 7 figures, 5 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Numerical method to approximate reflected SDEs
  3. Solving parabolic PDEs with Robin boundary condition
  4. Probabilistic representations
  5. Numerical method
  6. Finite-time convergence theorem
  7. Proof of Theorem \ref{['conp']}
  8. One-step approximation
  9. Lemma on the number of steps when $X^{'}_{k} \notin \bar{G}$
  10. Convergence theorem
  11. Computing ergodic limits
  12. Ergodic RSDEs and Poisson PDE
  13. Poisson PDE with Neumann boundary condition
  14. Ergodic limits in $G$
  15. Ergodic limits on $\partial G$
  16. ...and 22 more sections

Key Result

Theorem 3.1

The weak order of accuracy of Algorithm algorithm3.1 is $\mathcal{O}(h)$ under Assumptions as:3-as:2 and a01-ass4, i.e., for sufficiently small $h>0$ where $u(t,x)$ is solution of (eq:2)-(eq:4) and $C$ is a positive constant independent of $h$.

Figures (7)

  • Figure 2.1: Four possible realizations ${}_{\footnotesize(i)}X_{k+1}^{'}$ of $X_{k+1}^{'}$ given $X_{k}$ in two dimensions.
  • Figure 2.2: One step transition in two dimensions from $X_{k+1}^{'}$ to $X_{k+1}$ using projection $X_{k+1}^{\pi}$ of $X_{k+1}^{'}$ on $\partial G$.
  • Figure 7.1: Plot to show the first order of convergence of Algorithm \ref{['algorithm3.1']} applied to (\ref{['expeq1']})-(\ref{['expeq3']}). Error bars correspond to the Monte Carlo error.
  • Figure 7.2: Plot to show linear dependence of the errors in computing the ergodic limit, $error_{\rm{ta}}$ and $error_{\rm{ea}}$, on $h$.
  • Figure 7.3: Plot to show the dependence of the errors $e_{1}$ and $e_{2}$ on $h$.
  • ...and 2 more figures

Theorems & Definitions (53)

  • Remark 2.1
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Corollary 3.1
  • proof
  • ...and 43 more