Weak and strong approximations of reflected diffusions via penalization methods

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 $O((\frac{\ln n}n)^{1/4})$.