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(ℙ)-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.

Keywords:

• Lévy area
• Numerical approximation
• Stochastic differential equations
• Wiener chaos decomposition

Mathematics Subject Classification:

• 65C30
• 65C05
• 68Q17

Acknowledgments

This research was carried out by the author at the University of Oxford under funding from the EPSRC. Any views expressed are the author's own and are not necessarily shared by Merrill Lynch. The author would like to thank Prof. Terry Lyons for the suggestion of this problem.