## 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, 2018 | Jan 31, 2019 |

II semestre | Mar 4, 2019 | Jun 14, 2019 |

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

Sessione invernale d'esame | Feb 1, 2019 | Feb 28, 2019 |

Sessione estiva d'esame | Jun 17, 2019 | Jul 31, 2019 |

Sessione autunnale d'esame | Sep 2, 2019 | Sep 30, 2019 |

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

Sessione di laurea estiva | Jul 22, 2019 | Jul 22, 2019 |

Sessione di laurea autunnale | Oct 15, 2019 | Oct 15, 2019 |

Sessione di laurea autunnale straordinaria | Nov 21, 2019 | Nov 21, 2019 |

Sessione di laurea invernale | Mar 19, 2020 | Mar 19, 2020 |

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

Sospensione attività didattica | Nov 2, 2018 | Nov 3, 2018 |

Vacanze di Natale | Dec 24, 2018 | Jan 6, 2019 |

Vacanze di Pasqua | Apr 19, 2019 | Apr 28, 2019 |

Vacanze estive | Aug 5, 2019 | Aug 18, 2019 |

## 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.

## Academic staff

Imperio Michele

## 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 (2020/2021)

Teaching code

4S00254

Academic staff

Coordinatore

Credits

6

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/06 - PROBABILITY AND STATISTICS

Period

I semestre dal Oct 1, 2020 al Jan 29, 2021.

## 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, 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.

## Program

The entire course will be available online. In addition, a number of the lessons/all the lessons (see the course

schedule) will be held in-class.

1. Discrete-time Markov chains. Markov properties and transition probability. Irreducibility, aperiodicity. Stationary distributions. Reversible distributions.

2. Hitting times. Convergence to the stationary distribution. Law of large numbers for Markov chains. MCMC: Metropolis algorithm and Gibbs sampler.

3. Reducible Markov chains. Transient and recurring states. Probability of absorption.

4 .. Markov chains with countable states. Recurrence and transience of the random walk on Z ^ d. Positive recurrent states and stationary distributions. Convergence theorem for irreducible Markov chains with countable states.

5. Continuous-time Markov chains. The Poisson process and its properties. Continuous-time Markov property. Semigroup associated with a Markov chain: continuity and differentiability; generator. Kolmogorov equations. Stationary distributions. Dynkin's formula. Probabilistic construction of a continuous-time Markov chain.

6. Erdos-Renyi random graphs. Model definition. Connected components.

7. Conditional Expectation and Conditional Distribution. Martingale. Stopping theorem and convergence theorem.

Author | Title | Publishing house | Year | ISBN | Notes |
---|---|---|---|---|---|

Levin, David A., and Yuval Peres | Markov chains and mixing times | American Mathematical Society | 2017 |

## Examination Methods

Written exam, with exercises and theoretical questions.

The assessment methods could change according to the academic rules

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## 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 |

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 |

Proposte Tesi A. Gnoatto | Various topics |

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.