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Treballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Curs: 2019-2020.  Director: Marta Sanz-Solé

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[en] The objective of this project is to give an introduction to the theory of large deviations (LDP), a topic in stochastic analysis that can be described as the asymptotic evaluation of small probabilities at exponential scale. We start with the fundamental and initial result by Cramér (1938) and then, we formulate general LDP principles. A basic result in the field of large deviations for stochastic processes is Schilder’s Theorem regarding Brownian motion. A proof of this result is given in Chapter 4. Finally, we develop part of the Freidlin-Wentzell theory and give an application to LDPs for stochastic differential equations. Large deviations is a very active research area with many applications namely, in statistics, finance, engineering, statistical mechanics and applied probability. Nevertheless, because of time and space constrains applications are not considered in this work.