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
This information is intended exclusively for students already enrolled in this course.If you are a new student interested in enrolling, you can find information about the course of study on the course page:
Laurea in Matematica applicata - Enrollment from 2025/2026The 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 activated in the A.Y. 2020/2021
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3° Year activated in the A.Y. 2021/2022
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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 (2021/2022)
Teaching code
4S00254
Academic staff
Coordinator
Credits
6
Language
Italian
Scientific Disciplinary Sector (SSD)
MAT/06 - PROBABILITY AND STATISTICS
Period
Primo semestre dal Oct 4, 2021 al Jan 28, 2022.
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. Conditional Expectation and Conditional Distribution. Martingale. Stopping theorem and convergence theorem.
2. Discrete-time Markov chains. Markov properties and transition probability. Irreducibility, aperiodicity. Stationary distributions. Reversible distributions.
3. Hitting times. One step analysis. Convergence to the stationary distribution. Law of large numbers for Markov chains. MCMC: Metropolis algorithm and Gibbs sampler.
4. Reducible Markov chains. Transient and recurring states. Probability of absorption.
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.
Bibliography
Examination Methods
To pass the exam, students must show:
- to have understood the theoretical notions, showing detailed knowledge of definitions and statements, as well as of some proofs;
- to be able to apply theory to problem solving.
The exam consists of a 180-minute written test which includes a theoretical part, with at least one proof required, and a part of exercises.
The assessment methods could change according to the academic rules