Sanz-Solé, MartaTorre i Estévez, Víctor de la2018-04-252018-04-252017-06-29https://hdl.handle.net/2445/121869Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2017, Director: Marta Sanz[en] We start by defining the stochastic integral with respect continuous semimartingales. We then derive Itô’s formula and we show two important applications of this formula: Lévy’s characterization of Brownian motion and the Burkholder-Davis-Gundy inequalities. We extend Itô’s formula for convex functions by using local times. Finally, we apply the theory of local times to the case of Brownian motion: we proof the classical Trotter theorem and we identify the law of the Brownian local time at level 0.58 p.application/pdfcatcc-by-nc-nd (c) Vı́ctor de la Torre i Estévez, 2017http://creativecommons.org/licenses/by-nc-nd/3.0/esAnàlisi estocàsticaTreballs de fi de grauSemimartingales (Matemàtica)Moviment browniàIntegrals estocàstiquesProcessos de LévyAnalyse stochastiqueBachelor's thesesSemimartingales (Mathematics)Brownian movementsStochastic integralsLévy processesinfo:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/openAccess