Linear information for approximation of the Itô integrals

Abstract

We study optimal approximation of stochastic integrals in the Ito sense when linear information, consisting of certain integrals of trajectories of Brownian motion, is available. Upper bounds on the nth minimal error, where n is the fixed cardinality of information, are obtained by the Wagner–Platen algorithm and are O(n − 3/2) or O(n − 2), depending on considered class of integrands. We also show that Ω(n − 2) is a lower bound which holds even for very smooth integrands.