Studying at the University of Verona

A.A. 2016/2017

Academic calendar

Il calendario accademico riporta le scadenze, gli adempimenti e i periodi rilevanti per la componente studentesca, personale docente e personale dell'Università. Sono inoltre indicate le festività e le chiusure ufficiali dell'Ateneo.
L’anno accademico inizia il 1° ottobre e termina il 30 settembre dell'anno successivo.

Academic calendar

Course calendar

The Academic Calendar sets out the degree programme lecture and exam timetables, as well as the relevant university closure dates..

Definition of lesson periods
Period From To
I sem. Oct 3, 2016 Jan 31, 2017
II sem. Mar 1, 2017 Jun 9, 2017
Exam sessions
Session From To
Sessione invernale Appelli d'esame Feb 1, 2017 Feb 28, 2017
Sessione estiva Appelli d'esame Jun 12, 2017 Jul 31, 2017
Sessione autunnale Appelli d'esame Sep 1, 2017 Sep 29, 2017
Degree sessions
Session From To
Sessione estiva Appelli di Laurea Jul 20, 2017 Jul 20, 2017
Sessione autunnale Appelli di laurea Nov 23, 2017 Nov 23, 2017
Sessione invernale Appelli di laurea Mar 22, 2018 Mar 22, 2018
Period From To
Festa di Ognissanti Nov 1, 2016 Nov 1, 2016
Festa dell'Immacolata Concezione Dec 8, 2016 Dec 8, 2016
Vacanze di Natale Dec 23, 2016 Jan 8, 2017
Vacanze di Pasqua Apr 14, 2017 Apr 18, 2017
Anniversario della Liberazione Apr 25, 2017 Apr 25, 2017
Festa del Lavoro May 1, 2017 May 1, 2017
Festa della Repubblica Jun 2, 2017 Jun 2, 2017
Vacanze estive Aug 8, 2017 Aug 20, 2017

Exam calendar

The exam roll calls are centrally administered by the operational unit  Science and Engineering Teaching and Student Services Unit
Exam Session Calendar and Roll call enrolment sistema ESSE3. If you forget your password to the online services, please contact the technical office in your Faculty or to the service credential recovery.

Exam calendar

Per dubbi o domande Read the answers to the more serious and frequent questions - F.A.Q. Examination enrolment

Academic staff


Albi Giacomo +39 045 802 7913

Angeleri Lidia 045 802 7911

Baldo Sisto 045 802 7935

Bos Leonard Peter +39 045 802 7987

Caliari Marco +39 045 802 7904

Chignola Roberto 045 802 7953

Cordoni Francesco Giuseppe

Daffara Claudia +39 045 802 7942

Daldosso Nicola +39 045 8027076 - 7828 (laboratorio)

De Sinopoli Francesco 045 842 5450

Di Persio Luca +39 045 802 7968

Gregorio Enrico 045 802 7937

Malachini Luigi 045 8054933

Marigonda Antonio +39 045 802 7809

Mariotto Gino +39 045 8027031

Mariutti Gianpaolo 045 802 8241

Mazzuoccolo Giuseppe +39 0458027838

Orlandi Giandomenico

giandomenico.orlandi at 045 802 7986
Foto,  September 29, 2016

Rinaldi Davide

Rizzi Romeo +39 045 8027088

Schuster Peter Michael +39 045 802 7029

Solitro Ugo +39 045 802 7977

Zuccher Simone

Study Plan

The Study Plan includes all modules, teaching and learning activities that each student will need to undertake during their time at the University. Please select your Study Plan based on your enrolment year.


2° Anno


3° Anno

Teachings Credits TAF SSD
Between the years: 1°- 2°- 3°
Between the years: 1°- 2°- 3°
Altre attività formative

Legend | Type of training activity (TTA)

TAF (Type of Educational Activity) All courses and activities are classified into different types of educational activities, indicated by a letter.

SPlacements in companies, public or private institutions and professional associations

Teaching code



Luca Di Persio



Scientific Disciplinary Sector (SSD)


Language of instruction



I sem. dal Oct 3, 2016 al Jan 31, 2017.

Learning outcomes

Stochastic Systems [ Applied Mathematics ]
AA 2016/2017

The Stochastic Systems course aims at giving an introduction to the basic concepts underlying the rigorous mathematical description of the temporal dynamics for random quantities.

The course prerequisites are those of a standard course in Probability, for Mathematics / Physics.

It is supposed that students are familiar with the basics Probability calculus, in the Kolmogorov assiomatisation setting, in particular with respect to the concepts of density function, probability distribution, conditional probability, conditional expectation for random variables, measure theory (basic ), characteristic functions of random variables, convrgence theorems (in measure, almost everywhere, etc.), central limit theorem and its (basic) applications, etc.

The Stochastic Systems course aims, in particular, to provide the basic concepts of: Filtered probability space, martingale processes, stopping times, Doob theorems, theory of Markov chains in discrete and continuous time (classification of states, invariant and limit,measures, ergodic theorems, etc.), basics on queues theory and an introduction to Brownian motion.

A part of the course is devoted to the computer implementation of operational concepts underlying the discussion of stochastic systems of the Markov chain type, both in discrete and continuous time.

A part of the course is dedicated to the introduction and the operational study, via computer simulations, to univariate time series.

It is important to emphasize how the Stochastic Systems course is organized in such a way that students can concretely complete and further develop their own:
° capacity of analysis, synthesis and abstraction;
° specific computational and computer skills;
° ability to understand texts, even advanced, of Mathematics in general and Applied Mathematics in particular;
• ability to develop mathematical models for physical and natural sciences, while being able to analyze its limits and actual applicability, even from a computational point of view;
° skills concerning how to develop mathematical and statistical models for the economy and financial markets;
° capacity to extract qualitative information from quantitative data;
° knowledge of programming languages or specific software.


Stochastic Systems [ Applied Mathematics ]
AA 2016/2017


• Conditional Expectation ( from Chap.1 of [BMP] )
• Definitions and basic properties
• Conditional expectations and conditional laws

• Introduction to stochastic processes ( From Chap.1 di [BMP] )
• Filtered probability space, filtrations
• Adapted stochastic process (wrt a given filtration)
• Martingale (first definitions and examples: Markov chains)
• Kolmogorov characterization theorem
• Stopping times

• Martingale ( From Chap.3 of [BMP]
• Definition of martingale process, resp. super, resp. lower, martingale
• Fundamental properties
• Stopping times for martingale processes
• Convergence theorems for martingales

• Markov chains (MC) ( From Chap.4 of [Beichelet] , Chap.5 of di [Baldi] )
• Transition matrix for a MC
• Construction and existence for MC
• Omogeneous MC (with respect to time and space)
• Canonical MC
• Classification of states for a given MC ( and associated classes )
• Chapman-Kolmogorov equation
• Recurrent, resp. transient, states ( classification criteria )
• Irriducible and recurrent chains
• Invariant (stationary) measures, ergodic measures, limit measures ( Ergodic theorem )
• Birth and death processes (discrete time)

• Continuous time MC ( From Chap.5 of [Beichelt] )
• Basic definitions
• Chapman-Kolmogorov equations
• Absolute and stationary distributions
• States classifications
• Probability and rates of transition
• Kolmogorov differential equations
• Stationary laws
• Birth and death processes ( first steps in continuous time )
• Queuing theory (first steps in continuous time)

• Point, Counting and Poisson Processes ( From Chap.3 of [Beichelt] )
• Basic definitions and properties
• Stochastic point processes (SPP) and Stochastic Counting Processes (SCP)
• Marked SPP
• Stationarity, intensity and composition for SPP and SCP
• Homogeneous Poisson Processes (HPP)
• Non Homogeneous Poisson Processes (nHPP)
• Mixed Poisson Processes (MPP)

• Birth and Death processes (B&D) ( From Chap.5 of [Beichelt] )
• Birth processes
• Death processes
• B&D processes
° Time-dependent state probabilities
° Stationary state probabilities
° Inhomogeneous B&D processes


[Baldi] P. Baldi, Calcolo delle Probabilità, McGraw-Hill Edizioni (Ed. 01/2007)

[Beichelt] F. Beichelt, Stochastic Processes in Science, Engineering and Finance, Chapman & Hall/CRC, Taylor & Francis group, (Ed. 2006)

[BPM] P. Baldi, L. Matzliak and P. Priouret, Martingales and Markov Chains – Solve Exercises and Elements of Theory, Chapman & Hall/CRC (English edition, 2002)

Further interesting books are:

N. Pintacuda, Catene di Markov, Edizioni ETS (ed. 2000)

Brémaud, P., Markov Chains. Gibbs Fields, Monte Carlo Simulation, and Queues, Texts in Applied Mathematics, 31. Springer-Verlag, New York, 1999

Duflo, M., Random Iterative Models, Applications of Mathematics, 34. SpringerVerlag, Berlin, 1997

Durrett, R., Probability: Theory and Examples, Wadsworth and Brooks, Pacific Grove CA, 1991

Grimmett, G. R. and Stirzaker, D. R., Probability and Random Processes. Solved Problems. Second edition. The Clarendon Press, Oxford University Press, New York, 1991

Hoel, P. G., Port, S. C. and Stone, C. J., Introduction to Stochastic Processes, Houghton Mifflin, Boston, 1972


Reference texts
Author Title Publishing house Year ISBN Notes
P. Baldi Calcolo delle Probabilità McGraw Hill 2007 9788838663659
N. Pintacuda Catene di Markov Edizioni ETS 2000
Hoel, P. G., Port, S. C. and Stone, C. J. Introduction to Stochastic Processes Houghton Mifflin, Boston 1972
Levin, David A., and Yuval Peres Markov chains and mixing times American Mathematical Society 2017
P. Brémaud Markov Chains. Gibbs Fields, Monte Carlo Simulation, and Queues Texts in Applied Mathematics, 31. Springer-Verlag, New York 1999
P. Baldi, L. Matzliak and P. Priouret Martingales and Markov Chains – Solve Exercises and Elements of Theory Chapman & Hall/CRC (English edition) 2002
G. R. Grimmett, D. R. Stirzaker Probability and Random Processes: Solved Problems (Edizione 2) The Clarendon Press, Oxford University Press, New York 1991
Durrett, R Probability: Theory and Examples Wadsworth and Brooks, Pacific Grove CA 1991
Duflo, M. Random Iterative Models, Applications of Mathematics, 34 SpringerVerlag, Berlin 1997

Examination Methods

Stochastic Systems [ Applied Mathematics ]
AA 2016/2017

The course is diveded into the following three parts

1) Theory of stochastic systems
2) Introduction to time-series analysis
3) Computer exercises ( mainly based on the theory of Markov Chains, in discrete as well in continuous time )

Part (2) will be mainly performed in laboratory mode, using computer equipped classrooms, with the possibility, for each student to use a computer in order to implement , real time, the models proposed during the lesson. This activity will be supported by a tutor for a total amount of 24 (frontal) hours.

Part (3) will be taught by Prof. Caliari in a computer equipped laboratory.

The exam will be subdivided into the following three parts

* a written exam concerning point (1)
* a project presented in agreement with the programme developed with prof. Marco Caliari (point 3)
* exercises and a project concerning point (2)

The programme concerning the written exam, with respect to point (1), is the one reported in the Program section.
The project to be presented with prof. Caliari has to be decided with him.
The project to be presented with respect to point (2), will be chosen, by each student, within the the following list

@Warning: Since the list of projects may vary during the year, Students are warmly invited to directly contact prof. Di @Persio in order to choose the right project to develop, within the list of arguments that will be actually developed @during laboratory hours

1-Compare the following methods of estimate and/or elimination of time series trends

*First order differences study
*Smoothing with moving average filter
*Fourier transform
*Exponential Smoothing
*Polynomial Data fitting

2- Describe and provide a numerical implementation of the one-step predictor for the following models


3- Compare the Prediction Error Minimization (PEM) and the Maximum Likelihood (ML) approach for the identification of the model parameters (it requires a personal effort in the homes ML)

4- Provide a concrete implementation for the k-fold cross-validation, e.g. using Matlab/Octave, following the example-test that has been given during the lessons

5-Detailed explanation of (at least) one of the following test

Practical implementation of the project chosen by the student can be realized exploiting one of the following software frameworks : R, Python, Matlab, Gnu Octave, Excel

The final grade, expressed in thirtieths, will result from the following formula
Rating = (5/6) * T + (1/6) * E + P
T is the mark out of 30 on the part of Theory (written exam with prof. Di Persio)
It is the mark out of 30 on the part of Exercises (oral exam with prof. Caliari)
P is a score within the range [0,2]

It is important to emphasize how the objectives of the exam are also centered on assessing the individual student's ability to:

° carry out technical tasks defined in the model-mathematical settings;
° extract qualitative information from quantitative data with particular reference to the analysis of historical series, the study and the realization of predictive models, the development of automatic processes in the analysis of random phenomena;
° use computer/software tools such as R, Matlab, Gnu Octave, etc. , to realize models analyzed in the course and / or implemented in laboratory hours.

Tipologia di Attività formativa D e F

Academic year

Course not yet included

Career prospects

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List of theses and work experience proposals

theses proposals Research area
Formule di rappresentazione per gradienti generalizzati Mathematics - Analysis
Formule di rappresentazione per gradienti generalizzati Mathematics - Mathematics
Mathematics Bachelor and Master thesis titles Various topics
Stage Research area
Internship proposals for students in mathematics Various topics

University Language Centre - CLA

Further services

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