Weak approximation of killed diffusion using Euler schemes

We study the weak approximation of a multidimensional diffusion (Xt)0⩽t⩽T killed as it leaves an open set D, when the diffusion is approximated by its continuous Euler scheme (X̃t)0⩽t⩽T or by its discrete one (X̃ti)0⩽i⩽N, with discretization step T/N. If we set τ ≔ inf{t>0: Xt∉D} and τ̃c ≔ inf{t>0: X̃t∉D}, we prove that the discretization error Ex[1T<τ̃c f(X̃T)]−Ex[1T<τ f(XT)] can be expanded to the first order in N−1, provided support or regularity conditions on f. For the discrete scheme, if we set τ̃d ≔ inf{ti>0: X̃ti∉D}, the error Ex[1T<τ̃d f(X̃T)]−Ex[1T<τ f(XT)] is of order N−1/2, under analogous assumptions on f. This rate of convergence is actually exact and intrinsic to the problem of discrete killing time.