## Studying at the University of Verona

Here you can find information on the organisational aspects of the Programme, lecture timetables, learning activities and useful contact details for your time at the University, from enrolment to graduation.

## Academic calendar

The academic calendar shows the deadlines and scheduled events that are relevant to students, teaching and technical-administrative staff of the University. Public holidays and University closures are also indicated. The academic year normally begins on 1 October each year and ends on 30 September of the following year.

## Course calendar

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

Period | From | To |
---|---|---|

I semestre | Oct 1, 2013 | Jan 31, 2014 |

II semestre | Mar 3, 2014 | Jun 13, 2014 |

Session | From | To |
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Sessione straordinaria | Feb 3, 2014 | Feb 28, 2014 |

Sessione estiva | Jun 16, 2014 | Jul 31, 2014 |

Sessione autunnale | Sep 1, 2014 | Sep 30, 2014 |

Session | From | To |
---|---|---|

Sessione autunnale | Oct 15, 2013 | Oct 15, 2013 |

Sessione straordinaria | Dec 9, 2013 | Dec 9, 2013 |

Sessione invernale | Mar 18, 2014 | Mar 18, 2014 |

Sessione estiva | Jul 21, 2014 | Jul 21, 2014 |

Period | From | To |
---|---|---|

Vacanze Natalizie | Dec 22, 2013 | Jan 6, 2014 |

Vacanze di Pasqua | Apr 17, 2014 | Apr 22, 2014 |

Festa del S. Patrono S. Zeno | May 21, 2014 | May 21, 2014 |

Vacanze Estive | Aug 11, 2014 | Aug 15, 2014 |

## Exam calendar

Exam dates and rounds are managed by the relevant Science and Engineering Teaching and Student Services Unit.

To view all the exam sessions available, please use the Exam dashboard on ESSE3.

If you forgot your login details or have problems logging in, please contact the relevant IT HelpDesk, or check the login details recovery web page.

Should you have any doubts or questions, please check the Enrolment FAQs

## Academic staff

Dos Santos Vitoria Jorge Nuno

jorge.vitoria@univr.itSquassina Marco

marco.squassina@univr.it +39 045 802 7913## 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.

Modules | Credits | TAF | SSD |
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Modules | Credits | TAF | SSD |
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Modules | Credits | TAF | SSD |
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1° Year

Modules | Credits | TAF | SSD |
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2° Year

Modules | Credits | TAF | SSD |
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3° Year

Modules | Credits | TAF | SSD |
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Modules | Credits | TAF | SSD |
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#### 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.

### Stochastic systems (2015/2016)

Teaching code

4S00254

Credits

6

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/06 - PROBABILITY AND STATISTICS

The teaching is organized as follows:

##### Catene di Markov in tempo discreto

##### Analisi di serie temporali

##### esercitazioni

## Learning outcomes

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, 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.

## Program

Stochastic Systems [ Applied Mathematics ]

AA 2015/2016

Syllabus

• Conditional Expectetion ( 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 )

• Queque 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

• An introduction to quequing theory (From Chap.5 of [Beichelt] )

• Basic concepts

• Classification A/B/s/m by Kendall

• Explicitly studied examples:

° M/M/+\infty

° M/M/s/0

# partial results for M/M/+\infty e M/G/+\infty

° M/M/s/+\infty

• Erlang's loss formula

• Little's formula

• Brownian Motion (BM) ( From Chap.7 of [Beichelt] )

• Definitions and basic properties

• Transformations of 1-dimensional BM

° exponential martingale

° variance martingale

Bibliography

Text used in the course are:

[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

## Examination Methods

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 )

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

========

Projects

========

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

FIR(4)

ARX(3,1)

OE(3,1)

ARMA(2,3)

ARMAX(2,1,2)

Box-Jenkins(nb,nc,nd,nf)

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

*Shapiro-Wilk

*Kolmogorov-Smirnov

*Lilliefors

## Type D and Type F activities

**Modules not yet included**

## Career prospects

## Module/Programme news

##### News for students

There you will find information, resources and services useful during your time at the University (Student’s exam record, your study plan on ESSE3, Distance Learning courses, university email account, office forms, administrative procedures, etc.). You can log into MyUnivr with your GIA login details.

## Further services

I servizi e le attività di orientamento sono pensati per fornire alle future matricole gli strumenti e le informazioni che consentano loro di compiere una scelta consapevole del corso di studi universitario.

## Graduation

## Attachments

Title | Info File |
---|---|

1. Come scrivere una tesi | 31 KB, 29/07/21 |

2. How to write a thesis | 31 KB, 29/07/21 |

4. Regolamento tesi (valido da luglio 2020) | 259 KB, 29/07/21 |

5. Regolamento tesi (valido da luglio 2022) | 171 KB, 17/02/22 |

## 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 |

## Attendance

As stated in point 25 of the Teaching Regulations for the A.Y. 2021/2022, except for specific practical or lab activities, attendance is not mandatory. Regarding these activities, please see the web page of each module for information on the number of hours that must be attended on-site.Please refer to the Crisis Unit's latest updates for the mode of teaching.