Weak approximation of killed di usion using Euler schemes
نویسنده
چکیده
We study the weak approximation of a multidimensional di usion (Xt)06t6T killed as it leaves an open set D, when the di usion is approximated by its continuous Euler scheme (X̃t)06t6T or by its discrete one (X̃ti )06i6N , with discretization step T=N . If we set := inf{t ¿ 0: Xt 6∈ D} and ̃c := inf{t ¿ 0: X̃t 6∈ D}, we prove that the discretization error Ex[5T¡ ̃c f(X̃T )]−Ex[5T¡ f(XT )] can be expanded to the rst order in N−1, provided support or regularity conditions on f. For the discrete scheme, if we set ̃d := inf{ti ¿ 0: X̃ti 6∈ D}, the error Ex[5T¡ ̃d f(X̃T )]−Ex[5T¡ 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. c © 2000 Elsevier Science B.V. All rights reserved.
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