Insegnamento a.a. 2024-2025

1. Probability spaces, random variables and random vectors. Expectations. Independence.
2. Convergence of sequences of random variables. Limit Theorems.
3. Conditional expectations in L1 and in L2 spaces. Conditional probabilities.
4. Constructions of stochastic processes. Some examples: Markov chains and Gaussian processes.
5. Filtrations and stopping times. Martingales.
6. Poisson random measures.
7. Lévy processes.
8. Brownian motion.

• Deal with advanced mathematical tools that lie at the foundations of modern probabilistic applications in Data Science.
• Gain a deep understanding of stochastic models that are used to describe complex dependence structures occurring in real world applications.
• Profitably attend graduate courses on advanced topics in Statistics and Machine Learning.

• Address modeling of the time evolution of random phenomena in a rigorous probabilistic framework.
• Master the interplay between probability theory and advanced modeling tools used in Statistics and Machine Learning.

• Lectures

The assessment consists of a written exam and may be structured as follows

1) Two partial exams

• 1st partial exam on the first part of the course, which coincides with topics 1-4 in the course content summary
• 2nd partial exam on the remaining topics 5-8 in the content summary, at the end of the course,

or as a 

2) General exam at the end of the course, on the whole program. This is split into two parts related to the 1st and the 2nd part of the course, respectively.

• In either case, the weights applied to marks of the 1st part and of the 2nd part are equal to 0.4 and 0.6, respectively, for determining the final overall mark.
• If one passes the 1st partial exam, then they can take 2nd partial in any of the two dates of the winter exams session.
• In the September session only the general exam can be taken
• The exam contains open questions on the theory, and examples, presented during the lectures. It aims at ascertaining students’ understanding of the stochastic processes theory and of the specific examples which help illustrating the applications of the most relevant mathematical results.
• There are no different assessment methods or exam programs for attending and non-attending students.

• E. CINLAR, Probability and Stochastics. Springer, New York, 2011.
• Lecture notes by the instructors.