Prof.ssa L.  Beghin
Semester I
6 credits

Info (classes, topics, textbook)

The purpose of this course is to present a comprehensive treatment of advanced probabilistic techniques for applications to stochastic modelling and data analysis. Preliminary background material will be quickly reviewed and some basic tools in the thoery of stochastic processes (i.e., conditional expectation, filtrations, stopping times) will be introduced. This will allow to cover discrete and continuous time model, with particular reference to the theory of martingales, random walks and Markov chains. A number of applications will also be covered, such as Markov chains Monte Carlo methods.


  1. Preliminaries and basic probability concepts: random variables, expectation, variance.

  2. Conditional expectation. Filtrations. Stopping times.

  3. Martingales. Optional stopping theorem.

  4. Random walks: ballot theorem, first passage times.

  5. Tail bounds and concentration.

  6. Markov chains: classifications of the states, invariant distribution, ergodic theorem.

  7. Markov chain Monte Carlo

  8. Probabilistic methods