81624 - Probability

Course Unit Page


This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

Quality education Industry, innovation and infrastructure

Academic Year 2021/2022

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
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
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


Recommended texts:

[1] A Probability Path

Sidney Resnick



Measures, Integrals and Martingales

Rene' Schilling


[3] Probability 1

Albert Shiryaev


Teaching methods

Traditional black-board based classes and every lecture will be made available through online platforms.

Assessment methods

Written ( and optional 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 (optional ) oral exam may cover the same range of topics as well, while keeping into accont the candidate's perfomance in the previously held written test.

Some more details may be in order:

[1] Grades are expressed on a scale from 0 to 30 cum Laude (30L), where 18
is the passing threshold.

[2] It is highly recommended that students attend to classes, where a
number of exam-like problems will be solved and presented completely.

[3] Written exams may last from 1 to 3 hours,
depending on the stage and the session they are attached to. Written
exams are open books.

[4] Written exams will be mainly made up by a
collection of exercises, some of them basic run of the mill stuff,
some more challenging.

[5] The method of execution, the precision of
presentation and, needless to say, the correctness and the accuracy of
the results will constitute the main factor in establishing the grade.

[6] The optional oral exam will be graded in a similar manner and an
arithmetic mean of the (i.e. written and oral) grades will yield the
final mark.

Teaching tools

Supplementary notes may be distributed during the course itself.

Office hours

See the website of Enrico Bernardi