El moviment brownià i la seva aplicació al càlcul estocàstic

Abstract

[en] The main goal of this work is to rigorously define Brownian motion and understand its relevance when doing mathematical models of different phenomena from the reality. In addition, going from the properties of nowhere differentiability and finite quadratic variation of the sample paths, we illustrate the necessity of a new calculus in order to solve stochastic differential equations (SDE), which model dynamical systems with random perturbations, using the definition of the stochastic (or Itô) integral and its lemma. Finally, we apply the developed Mathematics theory to the problem of the motion of a Brownian particle in suspension in a fluid, being able to correctly describe the velocity distribution that follows the particle and recovering some important Physics’ results.