An Analysis of Approximation Algorithms for Iterated Stochastic Integrals and a Julia and MATLAB Simulation Toolbox

TL;DR

This paper reviews four Fourier-based algorithms for approximating iterated stochastic integrals, analyzes their theoretical and practical performance, and introduces an open-source Julia and MATLAB toolbox that significantly speeds up simulations.

Contribution

It provides a unified analysis of four algorithms, compares their convergence properties, and offers an efficient, open-source simulation toolbox for stochastic differential equations.

Findings

Wiktorsson's algorithm is highly efficient.

The new algorithm by Mrongowius and Rössler improves speed.

The toolbox achieves significant speed-up over existing implementations.

Abstract

For the approximation and simulation of twofold iterated stochastic integrals and the corresponding L\'{e}vy areas w.r.t. a multi-dimensional Wiener process, we review four algorithms based on a Fourier series approach. Especially, the very efficient algorithm due to Wiktorsson and a newly proposed algorithm due to Mrongowius and R\"ossler are considered. To put recent advances into context, we analyse the four Fourier-based algorithms in a unified framework to highlight differences and similarities in their derivation. A comparison of theoretical properties is complemented by a numerical simulation that reveals the order of convergence for each algorithm. Further, concrete instructions for the choice of the optimal algorithm and parameters for the simulation of solutions for stochastic (partial) differential equations are given. Additionally, we provide advice for an efficient implementation of the considered algorithms and incorporated these insights into an open source toolbox that is freely available for both Julia and MATLAB programming languages. The performance of this toolbox is analysed by comparing it to some existing implementations, where we observe a significant speed-up.