An approximation scheme for reflected stochastic differential equations

In this paper, we consider the Stratonovich reflected SDE dXt=σ(Xt)∘dWt+b(Xt)dt+dLt in a bounded domain OO. Letting WtN be the NN-dyadic piecewise linear interpolation of WtWt, we show that the distribution of the solution (XtN,LtN) to the reflected ODE ẊtN=σ(XtN)ẆtN+b(XtN)+L̇tN converges weakly to that of (Xt,Lt)(Xt,Lt). Hence, we prove a distributional version for reflected diffusions of the famous result of Wong and Zakai.In particular, we apply our result to derive some geometric properties of coupled reflected Brownian motion, especially those properties which have been used in the recent work on the “hot spots” conjecture for special domains.