Abstract
We analyze the complexity of nonlinear Lebesgue integration problems in the average case setting for continuous functions with the Wiener measure and the complexity of approximating the Itô stochastic integral. G. W. Wasilkowski and H. Woźniakowski (2001, Math. Comp., 685–698) studied these problems, observed that their complexities are closely related, and showed that for certain classes of smooth functions with boundedness conditions on derivatives the complexity is proportional to ε−1. Here ε>0 is the desired precision with which the integral is to be approximated. They showed also that for certain natural function classes with weaker smoothness conditions the complexity is at most of order ε−2 and conjectured that this bound is sharp. We show that this conjecture is true.
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