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

TL;DR

This paper introduces a new, more efficient Fourier series-based algorithm for approximating iterated stochastic integrals and Lévy areas driven by multidimensional Brownian motion, improving accuracy and reducing computational cost.

Contribution

The paper presents a novel algorithm that uses a diagonal covariance matrix for better approximation of remainder terms, enhancing efficiency over previous methods.

Findings

Proves convergence in L^p-norm for the new and existing algorithms.

Demonstrates higher accuracy and lower computational cost of the new algorithm.

Provides theoretical analysis of the algorithm's efficiency.

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.