81624 - Probability

Learning outcomes

As for probability, at the end of the course the student has good knowledge of probability theory of discrete and continuous random variables. Particular attention is paid to the theory of stochastic processes, both diffusive and with jumps. The student masters the main techniques of stochastic processes applied to finance, such as Ito's lemma, Girsanov theorem and change of measure methods for Lévy processes.

Course contents

Notation and basic set theory
Sets and functions
Outer measure
Lebesgue-measurable sets and Lebesgue measure
Basic properties of Lebesgue measure
Borel sets
Lebesgue-measurable functions
Random variables
Fields generated by random variables
Probability distributions
Independence of random variables
Integral
Definition of the integral
Monotone convergence theorems
Integrable functions
The dominated convergence theorem
Relation to the Riemann integral
Approximation of measurable functions
Integration with respect to probability distributions
Absolutely continuous measures:
examples of densities
Expectation of a random variable
Characteristic function
Spaces of integrable functions
The space L
The Hilbert space L
Properties of the L -norm
Inner product spaces
Orthogonality and projections
The LP spaces: completeness
Moments
Independence
Conditional expectation (first construction)
Product measures
Independence again
Conditional probability
Strong law of large numbers
Weak convergence
Central limit theorem

Brownian Motion

Elementary Stochastic Processes

Readings/Bibliography

https://www.amazon.it/Measure-Integral-Probability-Marek-Capinski/dp/1852337818/ref=sr_1_1?ie=UTF8&qid=1465801767&sr=8-1&keywords=capinski

ISBN 978-1-85233-781-0

https://www.amazon.it/Probability-Measure-3rd-BILLINGSLEY-PATRICK/dp/8126517719/ref=sr_1_1?ie=UTF8&qid=1465801791&sr=8-1&keywords=billingsley

 

https://www.amazon.it/Probability-Essentials-Jean-Jacod/dp/3540438718/ref=sr_1_1?ie=UTF8&qid=1497252809&sr=8-1&keywords=jacod+protter

ISBN 978-3-540-43871-7

 

https://www.amazon.it/Probability-1-Albert-N-Shiryaev/dp/1493979051/ref=tmm_other_meta_binding_swatch_0?_encoding=UTF8&qid=1528634616&sr=8-1

• ISBN-10: 1493979051

Teaching methods

Regular 2 or 3 hours lectures with occasional in-course tests.

Assessment methods

Written and oral exams. The written exam is articulated in a series of 6 exercises each with a maximum grade of 5 points. Every exercise attains to elements of the syllabus and the relevant Bibliography covered during the course lectures or otherwise hinted at during classes. The oral exam covers the same range of topics as well, with an eye to the candidate's perfomarnce in the previously held written test.

Teaching tools

Supplementary notes may be distributed during the course itself.

Office hours

See the website of Enrico Bernardi