ABSTRACT The standard error propagation formula assumes linearity and small uncertainties. When applied to multiplicative processes it can drastically fail. This work provides both theoretical insight and practical implementation guidance for correctly quantifying uncertainty in multiplicative processes using geometric Brownian motion (GBM). We derive the exact solution, implement the Euler–Maruyama scheme, validate convergence, and provide a complete reproducible Python script. Beyond tutorial exposition, we introduce original practical diagnostics: a criterion for when simulation is necessary, tail risk measures (VaR, CVaR) connecting statistical moments to practical risk assessment, and a flexible parameterization framework. To demonstrate generality, along with “modest” failure of “only” 30% (like future lithium price affecting battery replacement cost), we present examples of drastic failure by nearly 200% in ecological modeling (population growths subject to environmental variability). Designed for researchers and practitioners, this guide bridges textbook formulas and real‐world decision making while providing novel implementation guidance.
Ioan Bâldea (Fri,) studied this question.