Mathematics

VITTORIA SILVESTRI
VITTORIA SILVESTRI

General Goals: rigorous knowledge of probabilistic models from
applications to the relationship with other part of mathematic.

Specific goals:

Knowledge and understanding: at the end of the course the student will
have acquired the basic notions and results related to probability
spaces, random variables, independence, laws of large numbers,
characteristic functions, weak convergence, limit theorems.

Apply knowledge and understanding: at the end of the
course the student will be able to solve simple problems that require
the use of probabilistic techniques both in applications and in
problems of pure mathematics.

Critical and judgmental skills: the student will have the basis to
analyze the analogies and the relationships between the topics covered
and topics of analysis or mathematical physics; will also acquire the
tools that have historically led to the solution of classical
problems.

Communication skills: ability to expose the contents in the oral part
of the assessment and in any theoretical questions present in the
written test.

Learning skills: the acquired knowledge will allow a study, individual
or given in an LM course, related to more specialized aspects of
probability.

Probability I

1. Topics from Probability I and measure theory
2. Expectation and conditional expectation of a random variable
3. Discrete time martingales
4. Continuous time martingales
5. Brownian motion

R. Durrett Probability: Theory and Examples. Wadsworth 1991
O. Kallenberg Foundations of Modern Probability. Springer 1997
J.F.C. Kingman Poisson Processes. Oxford Studies in Probability
P. M¨orters and Y. Peres Brownian motion. Cambridge University Press 2010
L.C.G. Rogers and D. Williams Diffusions, Markov processes, and Martingales Vol. I (2nd edition). Wiley 1994
D.W. Stroock Probability Theory – An analytic view. Cambridge University Press 1993
D. Williams Probability with Martingales. Cambridge University Press 1991

Lectures and example classes

Attendance is not compulsory.

There will be a written exam and an oral exam.

Exam simulations will be made available on the course webpage.