Objectives and Contextualisation

We introduce the students into the theory of the stochastic processes, with a special emphasis on how they can be used to mathematically model several examples and real situations. More precisely, the main part of the course is devoted to deal with Markov chains, which provide one of the more important tools in stochastic modeling, with a lot of applications, for example in biology, medicine or queuing theory. We introduce discrete-time and continuous-time Markov chains and, due to its highly level of applicability, we specially treat the Poisson process, as a example of birth and death processes, and branching processes. During the course mathematical proofs will be mostly avoided, although we give a reference for them, and we focus on the applications of the methods and techniques to particular examples. Indeed, one of the principal aims consists that students learn to choose the convenient method in order to model some phenomenon, and also to implement it as well as to extract conclusions.

Another major objective is to introduce the Brownian motion, which represents the paradigmatic example of stochastic process with non-countable state space. Its definition will be motivated, emphasizing all variety of fields where Brownian motion can be applied, and some of its properties will be studied. We also introduce some processes related to Brownian motion, such as the Brownian bridge and the geometric Brownian motion.

Finally, the last aim of the course is that the students learn to use the software R in order to simulate different types of stochastic processes and extract the pertinent conclusions in view of the phenomenon that is being modeled.