On the Approximation and Simulation of Iterated Stochastic Integrals and the Corresponding L\'{e}vy Areas in Terms of a Multidimensional Brownian Motion

Abstract

A new algorithm for the approximation and simulation of twofold iterated stochastic integrals together with the corresponding L\'{e}vy areas driven by a multidimensional Brownian motion is proposed. The algorithm is based on a truncated Fourier series approach. However, the approximation of the remainder terms differs from the approach considered by Wiktrosson (2001). As the main advantage, the presented algorithm makes use of a diagonal covariance matrix for the approximation of one part of the remainder term and has a higher accuracy due to an exact approximation of the other part of the remainder. This results in a significant reduction of the computational cost compared to, e.g., the algorithm introduced by Wiktorsson. Convergence in $L^p(\Omega)$-norm with $p \geq 2$ for the approximations calculated with the new algorithm as well as for approximations calculated by the basic truncated Fourier series algorithm is proved and the efficiency of the new algorithm is analyzed.