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
| Modules | Credits | TAF | SSD |
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2° Year It will be activated in the A.Y. 2026/2027
| Modules | Credits | TAF | SSD |
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| Modules | Credits | TAF | SSD |
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| Modules | Credits | TAF | SSD |
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2 modules among the following: area algebra and geometry + analysis
- A.A. 2025/2026 Homological Algebra not activated
- A.A. 2026/2027 Applied algebra not activated
3 modules among the following: area modeling and computational mathematics24 credits among the following courses
- A.A. 2025/2026 Homological Algebra not activated.
- A.A. 2025/2026 Physics education laboratory not activated
- A.A. 2026/2027 Applied algebra not activatedLegend | 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 (2025/2026)
Teaching code
4S008268
Teacher
Coordinator
Credits
6
Language
English
Scientific Disciplinary Sector (SSD)
MAT/06 - PROBABILITY AND STATISTICS
Period
2nd semester dal Mar 2, 2026 al Jun 12, 2026.
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, e.g.: definition of discrete/continuous random variables, central limit theorem, discrete/continuous-time Markov chains.
Program
Probability: Review of fundamental concepts and basic results, e.g., definition of the concept of a random variable, including in multiple dimensions, and limit theorems.
Stochastic processes: Review, definitions, and main properties, e.g., Martingale processes, Optional Sampling Theorem, Quadratic variation for stochastic processes in general and martingales in particular
Discrete-time stochastic processes: Review with emphasis on random walk, starting from the binomial model, including in more than one dimension
Various constructions of the Brownian motion: Kolmogorov's consistency theorem
Properties of Brownian motion
Derivation, construction, and basic notions of stochastic integrals: Ito's and Stratonovich's
Ito-Doeoblin theorem, Lévy's criteria, and Martingale representation theorem
Stratonovich's approach and Ito's representation theorem Markov processes and their relation to Brownian motion Girsanov's formula, Cameron-Martin (Girsanov) theorem, and exponential Martingale
Rigorous construction and derivation of Stochastic Differential Equations (SDEs)
Weak and strong solutions for SDEs and the Gronwall's lemma Diffusions with a semigroup-theoretic approach, and related Markov properties
Dynkin's formula, Kolmogorov equations, and the Feynman-Kac theorem with applications, e.g., in mathematical finance
Bibliography
Didactic methods
Lectures will be held, with the subsequent sharing of the teaching materials produced during the individual lessons. To complement the work completed in class, further readings may be shared, e.g., scientific articles, lecture notes, and specific bibliographical references for further study and the development of ongoing projects.
Learning assessment procedures
The exam will be an oral one, during which questions will be asked, the answers of which may require the student to write down definitions, theorems, and their proofs, analytical derivations, etc.
The exam will be based on the syllabus actually covered during lectures and will consist of open-ended questions.
Alternatively, individual students may arrange with the instructor to prepare a seminar, focusing on specific topics in depth.
During the seminar, the professor may ask questions both about the material presented and about concepts at the intersection between it and the scientific corpus characterizing the entire course.
Evaluation criteria
Assessment of the understanding of mathematical tools, with particular reference to the theory of stochastic calculus, for the treatment of continuous-time stochastic processes with both Brownian and Lévy noise and subsequent derivation and application to the theory of stochastic integration, including the introduction and study of stochastic differential equations.
Criteria for the composition of the final grade
The final grade will be the result of the evaluation of the final oral exam, with possible inclusion of evaluations based on the discussion of optional in-depth projects that the individual student wishes to discuss with the instructor.
Exam language
Inglese - English
