Weak approximation of killed diffusion using Euler schemes

We study the weak approximation of a multidimensional diffusion (X_t)_0≤t≤T killed as it leaves an open set D, when the diffusion is approximated by its continuous Euler scheme (X̃_t)_0≤t≤T or by its discrete one (X̃_ti)_0≤i≤N, with discretization step T/N. If we set τinf{t>0:X_t∉D} and τ̃_cinf{t>0:X̃_t∉D}, we prove that the discretization error E_x[1_T<τ̃cf(X̃ _T)]-E_x[1_T<τf(X_T)] can be expanded to the first order in N^-1, provided support or regularity conditions on f. For the discrete scheme, if we set τ̃_dinf{t_i>0:X̃_{t i}∉D}, the error E_x[1_T<τ̃df(X̃ _T)]-E_x[1_T<τf(X_T)] 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.