Numerical Integration of stochastic differential equations: The Heun Algorithm Revisited and It\^o-Stratonovich Calculus
Abstract
The widely used Heun algorithm for the numerical integration of stochastic differential equations (SDEs) is critically re-examined. We discuss and evaluate several alternative implementations, motivated by the fact that the standard Heun scheme is constructed from a low-order integrator. The convergence, stability, and equilibrium properties of these alternatives are assessed through extensive numerical simulations. Our results confirm that the standard Heun scheme remains a benchmark integration algorithm for SDEs due to its robust performance. As a byproduct of this analysis, we also disprove a previous claim in the literature regarding the strong convergence of the Heun scheme.
Keywords
Cite
@article{arxiv.2508.19040,
title = {Numerical Integration of stochastic differential equations: The Heun Algorithm Revisited and It\^o-Stratonovich Calculus},
author = {Riccardo Mannella},
journal= {arXiv preprint arXiv:2508.19040},
year = {2025}
}
Comments
To appear on Entropy special issue: From Order to Disorder: Superfluidity, Stochastic Processes, and the Dynamics of Life, Dedicated to Professor Peter McClintock on the Occasion of His 85th Birthday