On the approximation and simulation of iterated stochastic integrals and the corresponding Lévy areas in terms of a multidimensional Brownian motion

• Published in: Stochastic analysis and applications Vol. 40; no. 3; pp. 397 - 425
• Main Authors: Mrongowius, Jan, Rößler, Andreas
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 04-05-2022; Taylor & Francis Ltd
• Subjects: Algorithms; Approximation; Brownian motion; Covariance matrix; Fourier series; Integrals; Iterated stochastic integral; Lévy area; Mathematical analysis; stochastic simulation
• ISSN: 0736-2994; 1532-9356
• DOI: 10.1080/07362994.2021.1922291

A new algorithm for the approximation and simulation of twofold iterated stochastic integrals together with the corresponding Lévy 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 Wiktorsson (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
-norm with
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.