The goal of this course is to introduce the student to the formal way to deal with uncertainty. The course aims at being rigorous, although the technicalities will be reduced to the barest minimum. The course will cover the basic chapters of probability theory, with an introduction to martingales and their properties.

The main goal of this course is to endow the student with the mathematical language and competence necessary for the modeling of uncertain phenomena. This will be achieved by introducing the main tools of rigorous probability theory. These tools find a plethora of applications in different fields, such as finance, econometrics, economic theory, data science, statistical learning, etc. Knowledge and understanding: The student who will actively take part in the course activities will be able to tackle a vast number of situations that require the usage of sophisticated probabilistic tools. Applying knowledge and understanding: The student--acquiring notions and methods--will be able to interpret and apply the appropriate reference models. Making judgements: The successful student will acquire analytical skills and the ability to select the suitable probabilistic model for problem-solving. Specifically, critical thinking, problem-solving and self-management will be adequately developed. Communication skills: At the end of the course the student will be able to understand and use the language of probability. Through the various activities that will take place during the course, the student will be able to put these communication skills into practice in various contexts, by adapting the concepts used to the interlocutor in the specific case. Learning skills: The mathematical-probabilistic knowledge acquired during the course will allow the student to autonomously adapt the various problems to the specific reference context. The student will develop a solid knowledge of the fundamental aspects of the subject so as to be able to pursue further studies or to undertake post-graduate professional training courses.

Events and sigma-fields Axioms of probability Conditional probability and independence Random variables Distribution functions Integration with respect to a probability measure Generating functions Sums of independent random variables Random vectors Gaussian random vectors Convergence of random variables Weak convergence Laws of large numbers Central limit theorem Conditional expectation Martingales Stopping times

Jacod, J. & Protter, P. Probability Essentials Springer 2004 Brémaud, P. Probability Theory and Stochastic Processes Springer 2020 Williams, D. A. (1991). Probability with martingales. Cambridge University press. Link https://tinyurl.com/ydrd8n8g

The whole course will be interactive and will require the students' active participation. Students will be strongly encouraged to form groups and engage in discussions so as to find themselves the solutions to the problems, rather than passively absorbing what is taught to them.

To guarantee a continuous assessment, a written midterm will be open to attending students, on November 6th 2023. The midterm and all continuous assessment during the course period will count for 50% of the final grade. It will be then necessary to sit for a second written test, after the end of the course, aimed at assessing the second half of the course and it will therefore count for the remaining 50% of the grade. In alternative, students may opt to sit for a total written exam. Non attending students will have solely the option of the final written exam.

An interview to verify understanding and motivation.