Martingales: Definition and First Properties - Probability, Measure and Martingales

This lecture from the University of Oxford's 'Probability, Measure and Martingales' third-year course features Professor Jan Obloj introducing the fundamental concept of martingales. Learn about the definition and basic properties of martingales, explore motivating examples, and discover methods for constructing new martingales through techniques like discrete stochastic integration (martingale transform). The 54-minute lecture culminates with a statement and proof of Doob's decomposition theorem in discrete time, which represents an adapted process as a sum of a martingale and a predictable process. This lecture is part of a five-part series available on YouTube, with Oxford's mathematics program featuring tutorials for first and second-year students and classes for third and fourth-year students.