Weak and strong approximations of reflected diffusions via penalization methods

We study approximations of reflected Itô diffusions on convex subsets D of R d by solutions of stochastic differential equations with penalization terms. We assume that the diffusion coefficients are merely measurable functions. In the case of Lipschitz continuous coefficients we give the rate of L p approximation for every p ≥ 1 . We prove that if D is a convex polyhedron then the rate is O ( ( ln n n ) 1 / 2 ) , and in the general case the rate is O ( ( ln n n ) 1 / 4 ) .