Understanding the mathematical architecture of martingales and stochastic integration
In this third-year lecture from Oxford Mathematics, Jan Obloj explores the fundamental properties of martingales. Discover how these processes are constructed and how complex sequences can be decomposed into predictable and unpredictable components.
The lecture introduces the concept of martingales, focusing on their core definitions and essential properties. It demonstrates how new martingales can be generated through methods such as discrete stochastic integration, also known as the martingale transform.
A central feature of the session is the presentation and proof of Doob’s decomposition theorem in discrete time. This theorem provides a way to represent an adapted process as the sum of two distinct parts: a martingale and a predictable process.
Source: Probability, Measure and Martingales - Martingales: definition and first properties - 3rd Yr Lecture
