On the global error of Ito-Taylor schemes for strong approximation of scalar stochastic differential equations

We analyze the L-2([0, 1])-error of general numerical methods based on multiple Ito-integrals for pathwise approximation of scalar stochastic differential equations on the interval [0, 1]. We show that the minimal error that can be obtained is at most of order N-1/2, where N is the number of multiple Ito-integrals that are evaluated. As a consequence, there are no Ito-Taylor methods of higher order with respect to the global L-2-error on [0, 1], which is in sharp contrast to the well-known fact that arbitrary high orders can be achieved by these methods with respect to the error at the discretization points. In particular, it turns out that the asymptotic performance of piecewise linear interpolated Ito Taylor schemes gets worse the more multiple Ito-integrals are involved. (C) 2003 Elsevier Inc. All rights reserved.