Studying at the University of Verona
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.
Course calendar
The Academic Calendar sets out the degree programme lecture and exam timetables, as well as the relevant university closure dates. For further information, please get in touch with Operational unit: Science and Engineering Teaching and Student Services Unit
| 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 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
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.
Per dubbi o domande Read the answers to the more serious and frequent questions - F.A.Q. Examination enrolment
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.
| Teachings | Credits | TAF | SSD |
|---|
1° Anno
| Teachings | Credits | TAF | SSD |
|---|
| Teachings | 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 differential equations (2018/2019)
Teaching code
4S001444
Teacher
Coordinatore
Credits
6
Scientific Disciplinary Sector (SSD)
MAT/06 - PROBABILITY AND STATISTICS
Language of instruction
English
Period
II semestre dal Mar 4, 2019 al Jun 14, 2019.
Learning outcomes
This course will provide an introduction to the theory of Stochastic Differential Equations (SDEs), mainly based on the Brownian motion type of noise. The purpose of this course is to introduce and analyse probability models that capture the stochastic features of the system under study to predict the short and long term effects that this randomness will have on the systems under consideration. The study of probability models for continuous-time stochastic processes involves a broad range of mathematical and computational tools. This course will strike a balance between the mathematics and the applications. The main applications will be mathematical finance, biology and populations evolution, also with respect to their descriptions in terms of the associated SDEs. Topics include: construction of Brownian motion; martingales in continuous time; stochastic integral; Ito calculus; stochastic differential equations; Girsanov theorem; martingale representation; the Feynman-Kac formula and Lévy processes.
Program
Course Plan
I) BACKGROUND: sigma-algebras, filtrations, conditional expectation, martingale property, variations of a function, quadratic variation.
II) RANDOM WALK: random walk, re-scaled random walk, martingale property.
III) BROWNIAN MOTION: definition of Brownian motion, function of a Brownian motion, martingale property, exponential martingale, applications in biology and finance, examples and exercises.
IIIa) SHORT INTRODUCTION TO JUMP PROCESSES: motivation, Poisson processes, properties, discrete time case, introduction to Galton Watson model.
III) WIENER INTEGRAL: motivation, case of step function, definition of Wiener integral, properties, law, martingale, quadratic variation, applications in biology and finance, examples and exercises.
IV) STOCHASTIC INTEGRALS: motivation, case of step function, definition of stochastic integral, properties, martingale, quadratic variation, variance, finite variation processes, Ito processes, applications in biology and finance, examples and exercises.
V) ITO CALCULUS: motivation, Itō-Doeblin formula for Brownian motion, Itō-Doeblin formula for function depending on time,Itō-Doeblin formula for Ito processes, applications in biology and finance, examples and exercises.
VI) SDEs: motivations, definition, existence and uniqueness result, applications in biology and finance, examples and exercises.
VII) MULTI-DIMENSIONAL CASE: multi-dim Brownian motion, correlation, multi-dim Ito-formula,
SDE, applications in biology and finance, examples and exercises.
VIII) CHANGE OF PROBABILITY: motivations, Cameron-Martin theorem, Girsanov theorem, representation of martingale theorem, applications in biology and finance, examples and exercises.
IX) FEYNMAN KAC FORMULA: motivation, Feynman Kac formula, link PDE/SDE, Monte-Carlo methods.
X) JUMPS PROCESSES: Levy processes, characterization and properties.
Bibliografia
| Author | Title | Publishing house | Year | ISBN | Notes |
|---|---|---|---|---|---|
| I. Karatzas and S. Shreve | Brownian motion and stochastic calculus | ||||
| D. Revuz and M. Yor | Continuous martingales and Brownian motion | ||||
| M.Yor et al | Exponential Functionals of Brownian Motion and related Processes | Springer | 2010 | ||
| D. Williams | Probability with martingales | ||||
| B. Øksendal | Stochastic Differential Equations | ||||
| N. Ikeda and S. Watanabe | Stochastic Differential Equations and Diffusion Processes | ||||
| P. Protter | Stochastic integration and differential equations |
Examination Methods
Oral exam with written exercises:
the exam is based on open-ended questions as well as on the discussion of written exercises to be carried out during the test itself. Open questions and exercises are aimed at verifying the knowledge related to the topics developed in the course program, hence concerning problems of both the basic theory of stochastic processes, and of stochastic differential equations.
Tipologia di Attività formativa D e F
Course not yet included
Career prospects
Avvisi degli insegnamenti e del corso di studio
Per la comunità studentesca
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Graduation
List of theses and work experience proposals
| theses proposals | Research area |
|---|---|
| Controllo di sistemi multiagente | Calculus of variations and optimal control; optimization - Hamilton-Jacobi theories, including dynamic programming |
| Controllo di sistemi multiagente | Calculus of variations and optimal control; optimization - Manifolds |
| Controllo di sistemi multiagente | Calculus of variations and optimal control; optimization - Optimality conditions |
| 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 |
Double degree
The University of Verona, through a network of agreements with foreign universities, offers international courses that enable students to gain a Double/Joint degree at the time of graduation. Indeed, students enrolled in a Double/Joint degree programme will be able to obtain both the degree of the University of Verona and the degree issued by the Partner University abroad - where they are expected to attend part of the programme -, in the time it normally takes to gain a common Master’s degree. The institutions concerned shall ensure that both degrees are recognised in the two countries.
Places on these programmes are limited, and admissions and any applicable grants are subject to applicants being selected in a specific Call for applications.
The latest Call for applications for Double/Joint Degrees at the University of Verona is available now!
University Language Centre - CLA
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.