Discrete approximations to local times for reflected diffusions

TL;DR

This paper introduces a discrete approximation for the boundary local time of reflected diffusions, enabling practical simulation and establishing convergence results with applications to PDE problems.

Contribution

It presents a novel discrete local time for reflected diffusions, derived from lattice-based random walks, with proven weak convergence as lattice size diminishes.

Findings

Discrete local time can be effectively simulated from lattice random walks.

Weak convergence of the joint law of discrete local time and random walks is established.

Applications to PDE problems demonstrate the utility of the convergence results.

Abstract

We propose a discrete analogue for the boundary local time of reflected diffusions in bounded Lipschitz domains. This discrete analogue, called the discrete local time, can be effectively simulated in practice and is obtained pathwise from random walks on lattices. We establish weak convergence of the joint law of the discrete local time and the associated random walks as the lattice size decreases to zero. A cornerstone of the proof is the local central limit theorem for reflected diffusions developed in [7]. Applications of the join convergence result to PDE problems are illustrated.