Girsanov: Martingales and Brownian Motion

This lecture delves into the study of measures in the space of filtrations, focusing on martingales and their properties. It explores the concept of local martingales, the impact of changing measures on martingale properties, and the Girsanov theorem, which relates martingales under different measures. The lecture also covers the characterization of Brownian motion, the Ito formula, and the identification of parameters in Brownian motion processes. Through various examples and demonstrations, the instructor illustrates the importance of adaptation and the symmetries between different measures in probability theory.