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.
Highlights
Let D ⊂ Rd be a bounded Lipschitz domain where d ≥ 1
A reflected Brownian motion (RBM) in D is a continuous Markov process which behaves like a standard Brownian motion in the interior of D and which is instantaneously pushed back by the inward normal vector n when it visits the boundary ∂D of D
Question: What is a discrete analogue to the boundary local time of a reflected diffusion?
Summary
Let D ⊂ Rd be a bounded Lipschitz domain where d ≥ 1. Question: What is a discrete analogue to the boundary local time of a reflected diffusion? To the best of our knowledge, the question of discrete approximation to boundary local time of reflected diffusions has not even been rigorously addressed before. This paper is organized as follows: In Section 2, we construct the discrete local time L(k) for RBM. This candidate is defined pathwise explicitly in (2.2) (equivalently (2.4)) and is amenable to computer simulations.
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