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

This paper is concerned with the problem of simulation of ${\left({X}_{t}\right)}_{0\le t\le T}$, 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 $\partial D$ 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...