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.
TEXTBOOKS
- A. M. Mood, F. A. Graybill, D. C. Boes (1991). Introduzione alla Statistica. McGraw-Hill, Milano.
- B. V. Gnedenko (1979). Teoria della Probabilità. Editori Riuniti, Roma.
- R. V. Hogg, A. T. Craig (1994). Introduction to Mathematical Statistics, 5th Edition. Macmillan.
- A. N. Shiryaev (1996). Probability, 2nd Edition. Springer, New York.
- S. E. Shreve (2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer, New York.
- S. E. Shreve (2004). Stochastic Calculus for Finance I: The Binomial Asset Pricing Model. Springer, New York.
SUPPORTING MATERIAL
Other supporting material, 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.
READING LIST
- P. Baldi (2011). Calcolo delle Probabilità, 2a Edizione. Mc Graw-Hill, Milano.
- D. M. Cifarelli (1998). Introduzione al Calcolo delle Probabilità. Mc Graw-Hill, Milano.
- W. Feller (1968). An Introduction to Probability Theory and Its Applications, Volume 1, 3rd Edition. Wiley.
- B. V. Gnedenko (1979). Teoria della Probabilità. Editori Riuniti, Roma.
- G. R. Grimmett, D. R. Stirzaker (1991). Probability and Random Processes, Solved Problems, 2nd Edition. The Claredon Press, Oxford University Press.
- G. R. Grimmett, D. R. Stirzaker (2001). Probability and Random Processes, 3rd Edition. Oxford University Press.
- G. R. Grimmett, D. R. Stirzaker (2001). One Thousand Exercises in Probability. Oxford University Press.
- J. Jacod, P. Protter (2000). Probability Essentials. Springer, New York.
- S. Lipschutz (1975). Calcolo delle Probabilità, Collana Schaum. ETAS Libri.
- T. Mikosch (1999). Elementary Stochastic Calculus With Finance in View. World Scientific, Singapore.
STUDY GUIDE
Detailed indications, regarding the use of the textbooks, will be given during the course.
PREREQUISITES
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.
EXERCISES
Exercises are an integral part of the course and are necessary to an adequate understanding of the topics.
TEACHING METHODS
Course load is equal to 54 hours (equal to 9 ECTS). All classes are essential to a proper understanding of the topics of the course. The working language is Italian.
TUTORING ACTIVITIES
In addition to lessons and exercise hours, before each exam session there will be tutoring hours devoted to revision. More detailed information will be available during the course.
Reference books | |||||
Author | Title | Publisher | Year | ISBN | Note |
W. Feller | An Introduction to Probability Theory and Its Applications, Volume 1 (Edizione 3) | Wiley | 1968 | ||
P. Baldi | Calcolo delle Probabilità (Edizione 2) | McGraw-Hill | 2011 | 9788838666957 | |
S. Lipschutz | Calcolo delle Probabilità, Collana Schaum | ETAS Libri | 1975 | ||
T. Mikosch | Elementary Stochastic Calculus With Finance in View | World Scientific, Singapore | 1999 | ||
R. V. Hogg, A. T. Craig | Introduction to Mathematical Statistics (Edizione 5) | Macmillan | 1994 | ||
D. M. Cifarelli | Introduzione al Calcolo delle Probabilità | McGraw-Hill, Milano | 1998 | ||
A. M. Mood, F. A. Graybill, D. C. Boes | Introduzione alla Statistica | McGraw-Hill, Milano | 1991 | ||
G. R. Grimmett, D. R. Stirzaker | One Thousand Exercises in Probability | Oxford University Press | 2001 | 0198572212 | |
A. N. Shiryaev | Probability (Edizione 2) | Springer, New York | 1996 | ||
G. R. Grimmett, D. R. Stirzaker | Probability and Random Processes (Edizione 3) | Oxford University Press | 2001 | 0198572220 | |
G. R. Grimmett, D. R. Stirzaker | Probability and Random Processes: Solved Problems (Edizione 2) | The Clarendon Press, Oxford University Press, New York | 1991 | ||
J. Jacod, P. Protter | Probability Essentials | Springer, New York | 2000 | ||
G. Casella, R. L. Berger | Statistical Inference (Edizione 2) | Duxbury Thompson Learning | 2002 | ||
S. E. Shreve | Stochastic Calculus for Finance II: Continuous-Time Models | Springer, New York | 2004 | ||
S. E. Shreve | Stochastic Calculus for Finance I: The Binomial Asset Pricing Model | Springer, New York | 2004 | ||
B. V. Gnedenko | Teoria della Probabilità | Editori Riuniti Roma | 1979 |
The final exam consists of a written test (of two hours and 30 minutes) followed by an oral session (of approximately 30 minutes). Both written and oral sessions are mandatory. For the written test, students can use a scientific calculator; any other material (books, notes, etc.) is forbidden. To be admitted to the oral session, students must receive at least 15 out of 30 in the written test. Contents, assessment methods and criteria are the same for all students and do not depend on the number of classes attended.
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