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.
• 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.
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.
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.
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.
|W. Feller||An Introduction to Probability Theory and Its Applications, Volume 1 (Edizione 3)||Wiley||1968||Reading list|
|P. Baldi||Calcolo delle Probabilità (Edizione 2)||McGraw-Hill||2011||9788838666957||Reading list|
|S. Lipschutz||Calcolo delle Probabilità, Collana Schaum||ETAS Libri||1975||Reading list|
|T. Mikosch||Elementary Stochastic Calculus With Finance in View||World Scientific, Singapore||1999||Reading list|
|R. V. Hogg, A. T. Craig||Introduction to Mathematical Statistics (Edizione 5)||Macmillan||1994||Textbook|
|D. M. Cifarelli||Introduzione al Calcolo delle Probabilità||McGraw-Hill, Milano||1998||Reading list|
|A. M. Mood, F. A. Graybill, D. C. Boes||Introduzione alla Statistica||McGraw-Hill, Milano||1991||Textbook|
|G. R. Grimmett, D. R. Stirzaker||One Thousand Exercises in Probability||Oxford University Press||2001||0198572212||Reading list|
|A. N. Shiryaev||Probability (Edizione 2)||Springer, New York||1996||Textbook|
|G. R. Grimmett, D. R. Stirzaker||Probability and Random Processes (Edizione 3)||Oxford University Press||2001||0198572220||Reading list|
|G. R. Grimmett, D. R. Stirzaker||Probability and Random Processes: Solved Problems (Edizione 2)||The Clarendon Press, Oxford University Press, New York||1991||Reading list|
|J. Jacod, P. Protter||Probability Essentials||Springer, New York||2000||Reading list|
|G. Casella, R. L. Berger||Statistical Inference (Edizione 2)||Duxbury Thompson Learning||2002||Reading list|
|S. E. Shreve||Stochastic Calculus for Finance II: Continuous-Time Models||Springer, New York||2004||Textbook|
|S. E. Shreve||Stochastic Calculus for Finance I: The Binomial Asset Pricing Model||Springer, New York||2004||Textbook|
|B. V. Gnedenko||Teoria della Probabilità||Editori Riuniti Roma||1979||Textbook|
Due to the COVID-19 health emergency, at the moment it is not possible to predict whether the final examination will be face-to-face or remotely. If the examination will be held face-to-face, it will consist of a written test (lasting about 2 hours and 30 minutes) made up of a selection of exercises. For the written test, 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 a mark greater than or equal to 15/30 in the written test. To take the tests, students must present themselves with a university card or a suitable identification document.
In the event that the examination will be held remotely, it will consist of a written test through Moodle's QUIZ tool (lasting about 1 hour and 30 minutes) made up of a selection of numerical exercises and multiple choice questions. The written test will be followed by a compulsory oral test, which can only be accessed by students who have obtained a sufficiently adequate mark in the written test. The oral exam will also take place remotely through Zoom.
For the 2020/2021 academic year, remote examination is always guaranteed for all students who request it. Regardless of the modality (face-to-face or remotely), the exams will be calibrated to guarantee the same level of difficulty. Finally, we remind that the examination methods are the same for all students and there are no differences according to the number of lessons attended.