Academic Year 2023-2024



Paolo Vidoni
Unit Credits
Teaching Period
Second Period
Course Type
Prerequisites. No formal prerequisite is set; nevertheless, the frequency of a basic course in probability and measure theory is suggested to properly follow the lectures
Teaching Methods. Frontal lectures and exercises.
Verification of Learning. The examination consists in an oral discussion on thetopics presented during the course. The general evaluation criteria are available at
More Information. For each topic, additional references and complementary material will be provided on the dedicated university web page (
Knowledge and understanding

Knowledge and understanding of the fundamental elements of probability theory for the description of univariate and multivariate random phenomena, of the basics in the theory of stochastic processes, of the usefulness of stochastic processes for the description of random phenomena in biology, finance and engineering.

Applying knowledge and understanding

Understanding of the probabilistic models as useful instruments for research in applied sciences an ability to use stochastic processes in order to describe random phenomena.

Making judgements

Making judgements on the appropriate probabilistic models and methods to be used for analyzing a specific dataset and on the interpretation of the experimental results.

Communication skills

Communication skills in order to present a probabilistic model, including both the methodology and the expected results, in a consistent and convincing way.

Learning skills

Learning skills based on the prerequisites that are required for understanding autonomously a report on the application of the theory of stochastic processes and for learning more advanced probabilistic procedures.

More information on the degree course are available on

This course gives some complements on basic probability theory and introduces the theory of stochastic processes, with a view toward applications.

Course contents

1) Complements on elementary probability; 2) Introduction to stochastic processes; 3) Discrete-time Markov chains; 4) Poisson processes; 5) Discrete-time martingales; 6) Brownian motion; 7) Hidden Markov models.

Supplementary readings:

1) Bosq, D., Nguyen, H.T. (1996). A course in stochastic processes: stochastic models and statistical inference. Kluwer;

2) Cınlar, E. (2011). Probability and stochastics. Springer;

3) Kopp, E. (2011). From measures to Itô integrals. Cambridge University Press;

4) Ross, S. M. (1996). Stochastic processes. Wiley.