Note on Martingale Theory and Applications
This paper provides a foundational review of martingale theory and its application to a classic problem in probability, primarily for students or researchers seeking a concise overview.
This note explores fundamental properties of martingales, including conditional expectation, the martingale transform, and the upcrossing lemma. These concepts are then used to derive the Martingale Convergence Theorem, which is applied to analyze extinction in Galton--Watson branching processes.
This note investigates core properties of martingales, emphasizing the measure-theoretic formulation of conditional expectation, the martingale transform, and the upcrossing lemma. These results lead to the Martingale Convergence Theorem, which we then apply to study the extinction behavior in Galton--Watson branching processes.