Weak approximation of killed diffusion using Euler schemes

We study the weak approximation of a multidimensional diffusion (X-t)(0 less than or equal to t less than or equal to T) killed as it leaves an open set D, when the diffusion is approximated by its continuous Euler scheme ((X) over tilde(t))(0 less than or equal to t less than or equal to T) or by its discrete one ((X) over tilde(ti))(0 less than or equal to i less than or equal to N), with discretization step T/N. If we set tau : = inf{t > 0: X-t 5 is not an element of D} and <(tau)over tilde>c : = inf{t > 0: (X) over tilde(t) is not an element of D}, we prove that the discretization error E-x[1(T<<(tau)over tilde>c) f ((X) over tilde(T)] - E-x[1(T < t) f(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 <(tau)over tilde>(d) := inf{t(i) > 0: (X) over tilde(ti) is not an element of D}, the error E-x[1T < (<(tau)over tilde>d) f ((X) over tilde(T))] - E-x[1T < tau f(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. (C) 2000 Elsevier Science B.V. All rights reserved.