Course 2022-2023 a.y.

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.

• Face-to-face lectures

The assessment consists of two oral individual exams:

1. An oral exam on the first part of the course, which coincides with topics 1.-4. in the course content summary. The mark of this part receives a weight equal to 0.35 for determining the final overall mark.
2. An oral exam on the second part of the course, which covers topics 5.-8. in the course content summary. The mark of this part receives a weight equal to 0.65 for determining the final overall mark.

• The exam aims at ascertaining students’ understanding of the stochastic processes theory and of the specific examples that are developed during lectures for 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.