An Approximation Scheme for Reflected Stochastic Differential Equations

TL;DR

This paper introduces an approximation scheme for reflected stochastic differential equations, demonstrating weak convergence of solutions and applying the method to analyze geometric properties of reflected Brownian motion.

Contribution

It provides a novel approximation approach for reflected SDEs using dyadic interpolation and establishes weak convergence results, extending Wong and Zakai's classical theorem.

Findings

Weak convergence of the approximation scheme to the true reflected SDE solutions

Representation of the approximated derivative as a projection of the interpolated Brownian motion

Application to geometric properties of reflected Brownian motion in specific domains

Abstract

In this paper we consider the Stratonovich reflected stochastic differential equation $dX_t=\sigma(X_t)\circ dW_t+b(X_t)dt+dL_t$ in a bounded domain $\O$ which satisfies conditions, introduced by Lions and Sznitman, which are specified below. Letting $W^N_t$ be the $N$-dyadic piecewise linear interpolation of $W_t$ what we show is that one can solve the reflected ordinary differential equation $\dot X^N_t=\sigma(X^N_t)\dot W^N_t+b(X^N_t)+\dot L^N_t$ and that the distribution of the pair $(X^N_t,L^N_t)$ converges weakly to that of $(X_t,L_t)$. Hence, what we prove is a distributional version for reflected diffusions of the famous result of Wong and Zakai. Perhaps the most valuable contribution made by our procedure derives from the representation of $\dot X^N_t$ in terms of a projection of $\dot W_t^N$. In particular, we apply our result in hand to derive some geometric properties of coupled reflected Brownian motion in certain domains, especially those properties which have been used in recent work on the "hot spots" conjecture for special domain.