Doob's Decomposition Theorem

This lecture covers Doob's decomposition theorem for submartingales, which states that a submartingale can be decomposed into two processes: a martingale and a non-decreasing predictable process. The proof involves defining a natural candidate process A and showing its predictability through induction. Additionally, the lecture discusses Brownian motion as a continuous-time process with specific properties, such as starting at 0 and having certain distributions. It also explores the connection between Brownian motion and classical random walks, emphasizing the quadratic variation of functions and their variations. The lecture concludes with insights into continuous martingales and Levy's theorem, highlighting the relationship between martingales and Brownian motion.