## 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 |
---|---|---|

primo semestre | Sep 15, 2014 | Jan 9, 2015 |

secondo semestre | Feb 19, 2015 | May 29, 2015 |

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

sessione invernale | Jan 12, 2015 | Feb 18, 2015 |

sessione estiva | Jun 4, 2015 | Jul 11, 2015 |

sessione autunnale | Aug 24, 2015 | Sep 9, 2015 |

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

sessione autunnale | Dec 12, 2014 | Dec 19, 2014 |

sessione invernale | Apr 8, 2015 | Apr 10, 2015 |

sessione estiva | Sep 10, 2015 | Sep 11, 2015 |

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

festività natalizie | Dec 22, 2014 | Jan 5, 2015 |

festività pasquali | Apr 3, 2015 | Apr 7, 2015 |

vacanze estive | Aug 10, 2015 | Aug 22, 2015 |

## Exam calendar

Exam dates and rounds are managed by the relevant Economics 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

## 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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1° Year

Modules | Credits | TAF | SSD |
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2° Year activated in the A.Y. 2015/2016

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 Models for Finance (2014/2015)

Teaching code

4S02482

Teacher

Coordinatore

Credits

9

Language

Italian

Scientific Disciplinary Sector (SSD)

SECS-S/01 - STATISTICS

Period

primo semestre dal Sep 15, 2014 al Jan 9, 2015.

## Learning outcomes

The course provides to students in economics and finance an overview of the theory of probability at an intermediate level.

Prerequisite to the course is an elementary knowledge of probability at the level of an undergraduate first or second year introductory course in probability and statistics.

In particular, a basic knowledge of the following topics is recommended: most common univariate discrete and continuous distributions; weak law of large numbers; central limit theorem.

The final objective of the course is to give an introduction to the advanced theory of conditional expectation, of stochastic processes in the discrete and continuous time domains and to stochastic integration.

## Program

Probability spaces and Kolmogorov’s axioms: sigma-algebras; event trees; elementary conditional probability; Bayes theorem; independence.

Random variables: discrete, absolutely continuous and singular random variables; expectation; Tchebycheff inequality; Jensen inequality; moment generating function.

Multidimensional random variables: multidimensional discrete and continuous random variables; joint distribution function; joint density function; marginal and conditional distributions; marginal and conditional densities; independence; covariance; coefficient of correlation of Bravais; Cauchy-Schwarz inequality; joint moment generating function.

Distributions of functions of random variables: transformations of random variables; method of the distribution function; distribution of the minimum and the maximum; method of the moment generating function; log-normal distribution; probability integral transform; transformations of vectors of random variables.

Limits of random variables: infinite series of random variables; convergence in probability, in distribution, with probability one (almost surely) and in mean; weak law of large numbers and law of large numbers of Bernoulli for relative frequencies; central limit theorem; Borel’s lemma and Borel’s strong law of large numbers; order statistics; empirical distribution function.

Conditional expectation: conditional probability and conditional expectation with respect to a finite partition; conditional expectation with respect to a sigma-algebra.

Discrete time martingales: filtrations; martingales on finite probability spaces; martingales and stopping times; betting strategies and impossibility of a winning betting strategy.

Continuous time stochastic processes: definitions and finite-dimensional distributions; filtrations; adapted processes; filtrations generated by a stochastic process; stationary processes; processes with stationary increments and with independent increments; counting processes and Poisson processes; Gaussian processes and Wiener processes (Brownian motions); Wiener process as a limit of a random walk; properties and irregularities of the sample trajectories (non derivability and infinite variation); Markov processes, transition probabilities and Chapman-Kolmogorov equations; continuous time martingales.

Stochastic integrals: overview of Riemann-Stiltjes integral; definition and properties of Itô’s integral; Itô’s formula, properties and applications; martingales associated to a Wiener process; diffusions; geometric Brownian motion; Radom-Nikodym derivative; Girsanov's theorem.

The course consists of a series of lectures (54 hours).

All classes are essential to a proper understanding of the topics of the course.

The working language is Italian.

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

W. Feller | An Introduction to Probability Theory and Its Applications, Volume 1 (Edizione 3) | Wiley | 1968 | ||

S. Lipschutz | Calcolo delle Probabilità, Collana Schaum | ETAS Libri | 1975 | ||

P. Baldi | Calcolo delle Probabilità e Statistica (Edizione 2) | Mc Graw-Hill | 1998 | 8838607370 | |

T. Mikosch | Elementary Stochastic Calculus With Finance in View | World Scientific, Singapore | 1999 | ||

R. V. Hogg, A. T. Craig | Introduction to Mathematical Statistics (Edizione 5) | Macmillan | 1994 | ||

D. M. Cifarelli | Introduzione al Calcolo delle Probabilità | McGraw-Hill, Milano | 1998 | ||

A. M. Mood, F. A. Graybill, D. C. Boes | Introduzione alla Statistica | McGraw-Hill, Milano | 1991 | ||

G. R. Grimmett, D. R. Stirzaker | One Thousand Exercises in Probability | Oxford University Press | 2001 | 0198572212 | |

A. N. Shiryaev | Probability (Edizione 2) | Springer, New York | 1996 | ||

G. R. Grimmett, D. R. Stirzaker | Probability and Random Processes (Edizione 3) | Oxford University Press | 2001 | 0198572220 | |

J. Jacod, P. Protter | Probability Essentials | Springer, New York | 2000 | ||

S. E. Shreve | Stochastic Calculus for Finance II: Continuous-Time Models | Springer, New York | 2004 | ||

S. E. Shreve | Stochastic Calculus for Finance I: The Binomial Asset Pricing Model | Springer, New York | 2004 | ||

B. V. Gnedenko | Teoria della Probabilità | Editori Riuniti Roma | 1979 |

## Examination Methods

For the official examination both written and oral sessions are mandatory.

The course is considered completed if the candidate has done the written test and passed the oral exam.

Students that has received at least 15 out of 30 in the written exam are allowed to attend the oral exam.

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## Teaching materials e documents

- 01) Informazioni sul corso (pdf, it, 87 KB, 04/10/14)

## 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

## List of theses and work experience proposals

theses proposals | Research area |
---|---|

Tesi di laurea magistrale - Tecniche e problemi aperti nel credit scoring | Statistics - Foundational and philosophical topics |

Il metodo Monte Carlo per la valutazione di opzioni americane | Various topics |

L'acquisto di azioni proprie | Various topics |

Proposte Tesi A. Gnoatto | Various topics |