Brownian motion

Abstract

[en] The aim of this work is to study the Brownian motion from a theoretical approach. Brownian motion (also named Wiener process) is one of the best known stochastic processes and plays an important role in both pure and applied Mathematics. In the first chapter, we present the basic concepts of the theory of stochastic processes such as filtrations, stopping times and martingales which are needed to develop further sections of the project. In the second chapter, we define the Brownian motion itself. Furthermore, two different constructions of Brownian motion are provided. The first one presents theorems of existence and continuity of stochastic processes from which we end up building the Brownian motion. The second construction provides another proof for the existence of Brownian motion based on the idea of the weak limit of a sequence of random walks. In the third chapter, we present a discussion of some properties of Brownian motion paths, also called sample path properties. These include characterizations of bad behaviour such as the nondifferentiability, as well as characterizations of good behaviour like the law of the iterated logarithm. Moreover, we study the zero sets, the quadratic variation and the lack of monotonicity of the Brownian paths. Finally, we show some Python simulations of one dimensional Brownian paths.