Martingales and Brownian Motion

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Martingale Convergence

Explores martingale convergence, discussing the conditions for convergence and variance in martingales.

Martingales and Brownian Motion

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

Martingale Convergence Theorem: Proof and Recap

Covers the proof and recap of the martingale convergence theorem, focusing on the conditions for the existence of a random variable.

Doob's Martingale

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

Martingale Convergence Theorem

Explains the martingale convergence theorem and its applications in probability theory.

Doob's Decomposition Theorem

Covers Doob's decomposition theorem for submartingales and explores Brownian motion properties, quadratic variation, and continuous martingales.

Girsanov's Theorem: Numerical Simulation of SDEs

Covers Girsanov's Theorem, absolutely continuous measures, and numerical simulation of Stochastic Differential Equations (SDEs) with applications in finance.

Martingales and Brownian Motion Construction

Explores the construction of Brownian motion with continuous trajectories and the dimension of its zero set.

Stochastic Calculus: Itô's Formula

Covers Stochastic Calculus, focusing on Itô's Formula, Stochastic Differential Equations, martingale properties, and option pricing.

Generalization of Martingales: Sub- & Supermartingales

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