Weak approximation of killed diffusion using Euler schemes

We study the weak approximation of a multidimensional diffusion (X-t)(0 les s than or equal to t less than or equal to T) killed as it leaves an open s et 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 i ts 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 condi tions 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 orde r N-1/2, under analogous assumptions on f. This rate of convergence is actu ally exact and intrinsic to the problem of discrete killing time. (C) 2000 Elsevier Science B.V. All rights reserved.