Lectures related to Sub- and Supermartingales: Theory and Applications

Quadratic Variation: Martingales and Stochastic Integrals

Explores quadratic variation in martingales and stochastic integrals, emphasizing their properties and extensions.

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

Martingales and Stochastic Integration

Covers martingales, stochastic integration, and localizing processes using stopping times.

Stochastic Integration: First Steps

Covers stochastic integration, process bracket, martingales, and variations in submartingales.

Martingale Convergence Theorem: Proof and Stopping Time

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

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.

Martingale Convergence Theorem

Covers the proof of the martingale convergence theorem and the convergence of the martingale sequence almost surely.

Stochastic Calculus: Brownian Motion

Explores stochastic processes in continuous time, emphasizing Brownian motion and related concepts.

Optional Stopping Theorem

Explores stopping times, the optional stopping theorem, F-measurable random variables, and martingales.

Stochastic Integral: Isometry Continuity

Covers stochastic integrals, emphasizing isometry and continuity properties in martingales and different spaces.

Optional Stopping Theorem: Proof and Applications

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

Martingale Convergence

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

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.

Stochastic Calculus: Integrals and Processes

Explores stochastic calculus, emphasizing integrals, processes, martingales, and Brownian motion.

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.

Optional Stopping Theorem: Martingales and Stepping Times

Explores the optional stopping theorem for martingales and stepping times, emphasizing its applications and implications.

Martingales and Brownian Motion Construction

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

Martingales and Brownian Motion

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

Fourier Transform and Spectral Densities

Covers the Fourier transform, spectral densities, Wiener-Khinchin theorem, and stochastic processes.