Optimal Approximation of the Second Iterated Integral of Brownian Motion

Abstract

In this article, a theorem is proved that describes the optimal approximation (in the L ²(?)-sense) of the second iterated integral of a standard two-dimensional Wiener process, W, by a function of finitely many elements of the Gaussian Hilbert space generated by W. This theorem has some interesting corollaries: First of all, it implies that Euler's method has the optimal rate of strong convergence among all algorithms that depend solely on linear functionals of the Wiener process, W; second, it shows that the approximation of the second iterated integral based on Karhunen–Loève expansion of the Brownian bridge is asymptotically optimal.