Teaching code

4S02482

Teacher

Marco Minozzo

Coordinator

Marco Minozzo

Credits

9

Language

Italian

Scientific Disciplinary Sector (SSD)

SECS-S/01 - STATISTICS

Period

Primo semestre (lauree magistrali) dal Oct 3, 2022 al Dec 23, 2022.

Lessons timetable Moodle Seminars 0

Learning objectives

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.

Prerequisites and basic notions

Students are supposed to have acquired all notions and basic concepts usually taught in a first undergraduate university course in probability and statistics: main discrete and continuous univariate distributions, main limit theorems such as the weak law of large numbers and the central limit theorem.

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; Chebyshev 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.

Bibliography

Didactic methods

In addition to the textbooks and the books in the reading list, other supporting material like written records of the lessons, handouts, exercises and past exam papers with solutions will be distributed during the course and will be made available on the E-learning platform of the University. Detailed indications, regarding the use of the textbooks, will be given during the course.
Course load is equal to 54 hours (equal to 9 ECTS). Exercises are an integral part of the course and, together with the classes, they are essential to a proper understanding of the topics of the course. The working language is Italian. In addition to lessons and exercise hours, there will be tutoring hours devoted to revision. More detailed information will be available in due course.
All lessons will be face-to-face and recorded. However, due to the COVID-19 health emergency, the way lessons will be delivered might change during the course of the semester.

Learning assessment procedures

The final exam consists of a written test (lasting about 2 hours and 30 minutes) made up of a selection of exercises. For the written test, the Moodle's QUIZ tool might be used. During the exam, only a calculator can be used and no other material (books, notes, etc.) will be allowed. The written test will be followed by an oral test (compulsory), which can only be accessed by students who have obtained at least a pass in the written test. To take the tests, students must present themselves with a university card or a suitable identification document.
For the 2022/2023 academic year, it is likely that students will have the possibility to request (according to the rules established by the Rector) a remote examination, on the basis of problems directly connected to the COVID-19 heath emergency. Regardless of the modality (face-to-face or remote), the exam will be calibrated to guarantee the same level of difficulty. Finally, we remind that, as far as the examination methods are concerned, there are no differences according to the number of lessons attended.

Exam language

Italiano