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.
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 enrollment year.
1° Year
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2° Year It will be activated in the A.Y. 2025/2026
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| Modules | Credits | TAF | SSD |
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1 module between the following:
- A.A. 2024/2025 Computational algebra not activated;
- A.A. 2025/2026 Homological Algebra not activated.1 module between the following 3 modules among the followingLegend | 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 Calculus (2024/2025)
Teaching code
4S008268
Teacher
Coordinator
Credits
6
Language
English
Scientific Disciplinary Sector (SSD)
MAT/06 - PROBABILITY AND STATISTICS
Period
Semester 2 dal Mar 3, 2025 al Jun 13, 2025.
Courses Single
Authorized
Learning objectives
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.
Prerequisites and basic notions
Basic tools of probability calculus, eg: definition of discrete / continuous random variables,, central limit theorem, discrete / continuous time Markov chains.
Program
* Probability essential recalls
* SP: definitions/main properties recall ; Martingales ; Option Sampling Theorem ; Quadratic Variation ;
* Stochastic processes at discrete time: recalls and emphasis on random walk (starting from the binomial model, also in more than 1 dimension);
* Different constructions of the Brownian motion: Kolmogorov Consistency Theorem / Kolmogorov-
Cénstor Th.eorem;
* Properties of the Brownian motion
* Derivation/construction of the Stochastic Integral(s) notion(s)
* Ito-Doeoblin rule: Levy's Criteria / Martingale Representation
* Stratonovich approach / Ito representation Theorem (applications/examples)
* Markov processes and relation(s) with the Brownian motion sp [further Bm's properties]
* Girsanov formula / Cameron-Martin (Girsanov) Theorem and Exponential Martingales
* Construction and rigorous derivation of Stochastic Differential Equations
* Strong solutions / Gronwall Lemma / Weak solutions (for SDEs)
* Diffusions / Semi-group approach / Markov property(ies)
* Dynkin's formula / Kolmogorov equation(s) / Feynman-Kac theorem
* Interplay between PDEs and SPDEs (via F-K theorem)
* SDEs application w.r.t. the Financial framework
Bibliography
Didactic methods
Lectures with sharing of slides, lecture notes, specific bibliographic references for particular theoretical insights.
Learning assessment procedures
Oral exam with written exercise:
the exam is based on open questions and/or on the presentation of a project agreed with the course professor and or on the resolution of written exercises to be solved during the test itself. Questions, open-ended and exercises, aim at verify the knowledge about arguments developed within the course, the solution of related concrete problems as well as to verify the acquired acquaintance using stochastic analysis tools.
Evaluation criteria
Evaluation of the understanding and ability to use of the main tools of stochastic analysis, with emphasis on their rigorous definition / analytical derivation.
Criteria for the composition of the final grade
The final grade is the result of the evaluation of the final oral exam, with the possible inclusion of ongoing assessments relating to projects (optional) carried out by the student.
Exam language
Inglese / English
