Martingales in Practice, Part I: When Is a Process Really a Martingale?

Martingales sit at the centre of derivative pricing, stochastic modelling, measure changes and quantitative model validation, yet they are often recognized through incomplete shortcuts. This first article in a three-part series develops a practical three-route framework: direct conditional expectation, inspection of the finite-variation term in an Ito decomposition, and the square-integrable Brownian stochastic-integral theorem. It separates true martingales from local martingales and applies the framework to Brownian motion, squared Brownian motion and compensated squared Brownian motion. The examples show why a flat simulated mean does not establish a martingale and why zero Ito drift initially yields only a local-martingale conclusion. The central message is practical: the probability measure, filtration and integrability argument must be made explicit before a martingale claim is used in pricing or model validation.