Students are given the opportunity to deepen their knowledge into well-known concepts from Probability Theory and Stochastic Processes, and to understand new ones such as e.g. those of the conditional expectations with respect to a σ-algebra, the martingales and the Brown motion, which are useful for Stochastic Analysis. The aim of the course is the understanding of the basic concepts of Stochastic Analysis, in such a way that students will be able to apply them in modern Financial Mathematics and especially in the pricing of derivative products.

Upon successful completion of the course, students will be able to:

• prove that a given family of sets is a σ-algebra;
• prove that a given set-function is a measure;
• solve integrals on probability spaces;
• prove that a given sequence of random variables is a martingale;
• prove that a stochastic process is a Brownian motion;
• solve stochastic integrals by using Itô’s formula.