Weak approximation of killed diffusion using Euler schemes

We study the weak approximation of a multidimensional diffusion (Xt)0[less-than-or-equals, slant]t[less-than-or-equals, slant]T killed as it leaves an open set D, when the diffusion is approximated by its continuous Euler scheme or by its discrete one , with discretization step T/N. If we set [tau] := inf{t>0: Xt[negated set membership]D} and , we prove that the discretization error can be expanded to the first order in N-1, provided support or regularity conditions on f. For the discrete scheme, if we set , the error 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.