Weak and strong approximations of reflected diffusions via penalization methods

TL;DR

This paper investigates how well penalized stochastic differential equations approximate reflected diffusions in convex sets, providing explicit convergence rates even with discontinuous coefficients and specific geometric conditions.

Contribution

It introduces new approximation rates for reflected diffusions using penalization methods, including cases with discontinuous coefficients and convex polyhedral domains.

Findings

Rate of $O((rac{ ext{ln} n}{n})^{1/2})$ for convex polyhedra

Rate of $O((rac{ ext{ln} n}{n})^{1/4})$ in general convex sets

Effective approximation with measurable, possibly discontinuous coefficients

Abstract

We study approximations of reflected It\^o diffusions on convex subsets $D$ of $\Rd$ by solutions of stochastic differential equations with penalization terms. We assume that the diffusion coefficients are merely measurable (possibly discontinuous) functions. In the case of Lipschitz continuous coefficients we give the rate of $L^p$ approximation for every $p\geq1$. We prove that if $D$ is a convex polyhedron then the rate is $O((\frac{\ln n}n)^{1/2})$, and in the general case the rate is $O((\frac{\ln n}n)^{1/4})$.