Numerical Integration of stochastic differential equations: The Heun Algorithm Revisited and It\^o-Stratonovich Calculus

TL;DR

This paper critically re-examines the Heun algorithm for stochastic differential equations, comparing its performance with alternatives and clarifying its convergence properties through extensive simulations.

Contribution

It provides a comprehensive evaluation of the Heun algorithm and alternatives, clarifying its robustness and correcting previous misconceptions about its convergence.

Findings

The standard Heun scheme remains a robust benchmark for SDE integration.

Alternative implementations are evaluated but do not outperform the standard Heun scheme.

The paper disproves previous claims about the strong convergence of the Heun scheme.

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.