Equazioni differenziali stocastiche ed applicazioni. P. Baldi, Pitagora editore (2001)
Notes on Mean Field Games. P. Cardaliaguet, https://www.ceremade.dauphine.fr/~cardaliaguet/
An Introduction to Stochastic Differential Equations, L.C.Evans, American Mathematical Society (2014);
Introduction to differential stochastic equations, G. Da Prato, Scuola Normale Superiore (1995)
Prerequisites
Basic calculus (Analisi Matematica I and II in the local department), Basic Measure theory (Analisi Matematica III e Calcolo delle Probabilità in the local department)
Teaching Methods
Standard classroom lectures and selected lectures.
Type of Assessment
The student will give a lecture on a topic related but not explored in the classroom. During the presentations the teachers will ask questions about how this relate to the other topics and will propose basic exercises.
Course program
1)Introduction: Problems and mathematical models involving random elements.
2) Elements of probability theory.
3) Brownian motion, Wiener process and white noise.
4) Stochastic integral and Ito's formula.
5) Ordinary differential stochastic equations (existence, uniqueness, well posedness, the Markovian case).
6)Applications to financial mathematics: derivation of the Black and Scholes equation, introduction to backward stochastic differential equations and application to contingent claims (a kind of derivative).
7) Applications to partial differential equations: Feymann-Kac and Girsanov theorems .
8) Games theory and tug of war games, connection with the p-Laplacian.
9) Second order mean field games: Introduction to mean field games, stochastic differential equation and Fokker-Planck equation, existence and uniqueness for second order mean field games. Variational second order mean field games.