Optional Stopping Theorem: Martingales and Stepping Times

This lecture by the instructor covers the consequences and applications of the optional stopping theorem for martingales with square-integrable filtrations. The theorem is presented in the context of stepping times, discussing the conditions under which the theorem holds and its implications. The proof of the theorem is revisited, emphasizing the use of the Monotone Convergence Theorem. Various definitions related to stopped martingales are introduced, and examples with classical random walks are provided to illustrate the concepts. The lecture concludes with a discussion on the behavior of martingales under different stopping conditions and the application of the theorem in practical scenarios.