Advanced Topics in Probability Theory
A.Y. 2024/2025
Learning objectives
The aim of the course is to introduce recent research topics in the field of Stochastic Analysis. In particular, we introduce some mathematical tools from Optimal Transport and Wasserstein spaces. We study stochastic differential equations of McKean-Vlasov type, presenting possible interpretations and applications. We investigate mean field games, from both a theoretical and an applicative point of view; we establish existence and uniqueness of solutions, characterizing them by both probabilistic and analytical tools (PDEs).
Expected learning outcomes
Students attending the course will become familiar with recent research topics in the field of Stochastic Analysis, as for instance differential calculus for functions defined on the Wasserstein space, stochastic differential equations of McKean-Vlasov type, mean field games.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
Part I
- Elements of optimal transport
- Wasserstein spaces
- Derivatives for measure functions
- McKean-Vlasov stocastic differential equations
- Applications
Part II
- Elements of game theory
- Stochastic differential games
- Mean field games: existence and uniqueness of equilibria, resolution methods
- Applications to economics and finance
- Elements of optimal transport
- Wasserstein spaces
- Derivatives for measure functions
- McKean-Vlasov stocastic differential equations
- Applications
Part II
- Elements of game theory
- Stochastic differential games
- Mean field games: existence and uniqueness of equilibria, resolution methods
- Applications to economics and finance
Prerequisites for admission
Students attending the course are expected to have a good knowledge of stochastic calculus.
Teaching methods
Classroom lectures. Attendance is not compulsory, but it is strongly recommended.
Teaching Resources
- R. Carmona, Lectures on BSDEs, Stochastic Control, and Stochastic Differential Games with Financial Applications, SIAM.
- R. Carmona, F. Delarue, Probabilistic Theory of Mean Field Games with Applications I-II, Springer
N.B. These books cover most of the program, other references will be given during the lectures if needed.
- R. Carmona, F. Delarue, Probabilistic Theory of Mean Field Games with Applications I-II, Springer
N.B. These books cover most of the program, other references will be given during the lectures if needed.
Assessment methods and Criteria
The final examination consists of an oral exam.
During the exam, students will be asked to illustrate results presented during the course, in order to evaluate her/his knowledge and understanding of topics as well as the ability to apply them.
The final mark is on a 30-point scale.
During the exam, students will be asked to illustrate results presented during the course, in order to evaluate her/his knowledge and understanding of topics as well as the ability to apply them.
The final mark is on a 30-point scale.
MAT/06 - PROBABILITY AND STATISTICS - University credits: 6
Practicals: 12 hours
Lessons: 35 hours
Lessons: 35 hours
Professors:
Campi Luciano, Cosso Andrea
Educational website(s)
Professor(s)
Reception:
Upon appointment by email
Department of Mathematics, via Saldini 50, office 1027 or on Microsoft Teams