Numerical Integration of stochastic differential equations: The Heun Algorithm Revisited and Itô-Stratonovich Calculus

2025/08/26 by Mannella, Riccardo
#Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Mathematics #FOS: Physical sciences #Numerical Analysis (math.NA) #Statistical Mechanics (cond-mat.stat-mech)

paper · doi:10.48550/arxiv.2508.19040

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.

Citations

• Mean Field Forward-Backward Stochastic Differential Equations
• An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations

Related

• A comparison of a few numerical schemes for the integration of stochastic differential equations in the Stratonovich interpretation
• Modify the Improved Euler scheme to integrate stochastic differential equations
• Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on PYTHON and MATLAB
• Numerical Stability Revisited: A Family of Benchmark Problems for the Analysis of Explicit Stochastic Differential Equation integrators
• An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations