Stopping Times: Martingales and Brownian Motion

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Explores transformations of Brownian motion, Markov properties, and scaling invariance.

Explores the proof of the martingale convergence theorem and the concept of stopping time in square-integrable martingales.

Explores sub- and supermartingales, stopping times, and their applications in stochastic processes.

Explores Markov chain convergence, emphasizing invariant distribution, Law of Large Numbers, and mean rewards computation.

Discusses convergence, martingales, Brownian motion, joint laws, testing procedures, and stop times.

Explores the generalization of martingales to sub- and supermartingales with a focus on convergence properties.

Covers the concept of Doob's martingale and its properties, including integrability and convergence theorem.

Covers the optional stopping theorem for martingales, providing a detailed proof and discussing its implications.

Explores three stopping theorems in martingales and Brownian motion.

Explores spectral gap and mixing time in Markov chains, including their definitions and behavior.