Euler schemes and half–space approximation for the simulation of diffusion in a domain

This paper is concerned with the problem of simulation of (X_t)0≤t≤T, the solution of a stochastic differential equation constrained by some boundary conditions in a smooth domain D: namely, we consider the case where the boundary ∂D is killing, or where it is instantaneously reflecting in an oblique direction. Given N discretization times equally spaced on the interval [0; T], we propose new discretization schemes: they are fully implementable and provide a weak error of order N^-1 under some conditions. The construction of these schemes is based on a natural principle of local approximation of the domain into a half space, for which efficient simulations are available.