Martingales and Brownian Motion

This lecture introduces a new approach to understanding Brownian motion through discrete objects, specifically symmetric simple random walks. By studying the probabilities of these walks, the lecture explores the concept of martingales and their relation to Brownian motion. The instructor explains the Chapman-Kolmogorov equation and the properties of Brownian motion, emphasizing the continuity of functions and the invariance by translation in space. The lecture concludes with a discussion on the potential positive outcomes that may arise from the current crisis, drawing parallels to historical events and predicting future cycles of highs and lows.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related lectures (34)

Related concepts (15)

MATH-330: Martingales et mouvement brownien

Introduction à la théorie des martingales à temps discret, en particulier aux théorèmes de convergence et d'arrêt. Application aux processus de branchement. Introduction au mouvement brownien et étude

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).

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.

Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function (and is related to the modulus of uniform continuity).

Continuous function

In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is .

Wiener process

In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is often also called Brownian motion due to its historical connection with the physical process of the same name originally observed by Scottish botanist Robert Brown.

Martingales and Brownian Motion: Construction and Properties MATH-330: Martingales et mouvement brownien

Explores the construction and properties of a Brownian motion with continuous trajectories using Paul Lévy's method.

Martingales and Brownian Motion: Drift and Exit Time MATH-330: Martingales et mouvement brownien

Explores Brownian motion with drift, exit probabilities, and average exit times from intervals.

Martingales and Brownian Motion MATH-330: Martingales et mouvement brownien

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

Martingales and Brownian Motion: Global Behavior and Zero Set Length MATH-330: Martingales et mouvement brownien

Explores the behavior of Brownian motion and its zero set length.

Limits and Continuity

Covers limits, continuity, and the intermediate value theorem in functions.