Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Martingales and Brownian Motion

Description

This lecture covers the convergence of stochastic processes, martingales, Brownian motion, Cauchy criterion, convergence theorems, and joint laws. It also discusses testing procedures and stop times in the context of martingales and Brownian motion.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Official source

Related concepts (23)

Doob's martingale convergence theorems

In mathematics specifically, in the theory of stochastic processes Doob's martingale convergence theorems are a collection of results on the limits of supermartingales, named after the American mathematician Joseph L. Doob. Informally, the martingale convergence theorem typically refers to the result that any supermartingale satisfying a certain boundedness condition must converge.

Geometric Brownian motion

A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It is an important example of stochastic processes satisfying a stochastic differential equation (SDE); in particular, it is used in mathematical finance to model stock prices in the Black–Scholes model.

Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). This motion pattern typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena).

Stochastic calculus

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created and started by the Japanese mathematician Kiyosi Itô during World War II. The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces.

Stochastic process

In probability theory and related fields, a stochastic (stəˈkæstɪk) or random process is a mathematical object usually defined as a sequence of random variables, where the index of the sequence has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.

Related lectures (38)

Fourier Transform and Spectral Densities

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

Maximum Entropy Principle: Stochastic Differential Equations

Explores the application of randomness in physical models, focusing on Brownian motion and diffusion.

Stochastic Processes: Brownian Motion

Explores Brownian motion, Langevin equations, and stochastic processes in physics.

White Noise Form of the Langevin Equation

Covers the white noise form of the Langevin equation and its applications.

Sub- and Supermartingales: Theory and Applications