When Error Propagation Fails: A PhD Mini-Tutorial on Geometric Brownian Motion Simulation

The standard error propagation formula assumes linearity and small uncertain- ties. When applied to multiplicative processes—like future lithium price affecting battery replacement cost—it fails by nearly 30%. This mini-tutorial shows, step by step, how to use geometric Brownian motion (GBM) and numerical simulation to compute uncertainty correctly. We derive the exact solution, implement the Euler– Maruyama scheme, validate convergence, and provide a reproducible Python script. Designed for PhD students entering computational modeling, this guide bridges textbook formulas and real-world risk assessment.