Advanced Probability

A.Y. 2020/2021
Overall hours
Learning objectives
Probability theory is now applied in a variety of fields including physics, engineering, informatics, biology, economics and social sciences. This course is an introduction to the rigorous theory of probability with perspective theme given by the Doob's theory of martingales and an introduction to the stochastic processes. Given the reference to the basic theory, the relevant concept of the conditional expectation is examined in depth. Stochastic processes and in particular their measurability properties and the construction of the path space are introduced. Two classes of processes are studied into details: the martingales at both discrete and continuous time, and the Markov processes, via its characterizations and the study of the discrete time Markov Chain.
Expected learning outcomes
Student learn how to treat and discuss the main properties of principal probabilistic objects.
He learn of the main mathematical properties of stochastic processes. He gain a knowledge of important classes of processes and of advanced techniques in stochastic analysis.
Course syllabus and organization

Single session

Lesson period
First semester
In the emergency phase, lectures will be erogated in synchronous via the platform MSTeams and/or Zoom. Asynchronous lectures will be considered (via pre-recorded video) either in the case of technical problems or in the case of request by the students. In the latter case, a video-recorded lectures will be available. In the case of asynchronous lectures, during the lecture time the lecturer will be available in chat or in video-conference for answering to possible questions by the students.

Possible meetings in presence will be proposed.

The final examination will consist only in a oral part, via MS Teams and Zoom.
Course syllabus
1. Introduction to the probability theory
2. Random variables and their integrability
3. Characteristic functions and their properties
4. Convergence of random variables and weak convergence of measures.
5. Conditional expectation: existence and properties.
6. Stochastic processes
6.1. Measurability and continuity
6.2. Costruction of path spaces and Kolmogorv Bochner Theorem
6.3. Gaussian processes
6.4. Markov processes
6.5. Levy's processes. Poisson and Wiener processes
7. Martingales
7.1. Discrete time martingales and their transformations
7.2. Convergence results and Doob's optional stopping theorem
7.3. UI Martingales
7.4 L^p Martingales
7.3. Continuous time martingales
7.4. Quadratic variation theory
8. Markov Chains
8.1. State classification: recurrence and transience
8.2. Random walk
8.5. Invariant distribution and asymptotic behavior
Prerequisites for admission
Notions related to a basic course of probability (construction of a probability space, random variables and vectors, induced probability measures induced by random elements, different kind of convergence in probability,...)
Teaching methods
The lessons will all be frontal lessons held on the blackboard.
Teaching Resources
- D. Williams, Probability with Martingales
- J. Jacod, P. Protter, Probability Essentials
- Billingsley , Probability and Measure, 1986
- V.Capasso-D.Bakstein, An Introduction to Continuous-Time Stochastic Processes: Theory, Models, and Applications to Finance, Biology, and Medicine, Birkhauser, 2015;
- Heinz Bauer, Probability Theory,1996
- P. Baldi, Stochastic Calculus: An Introduction Through Theory and Exercises, 2017
- AppleBaum, Levy Processes and Stochastic Calculus, 2004
- Norris, Markov Chains, 1999
Assessment methods and Criteria
The final examination consists of two parts: a written and an oral part.

- During the written part the student will be required to solve one or two exercizes and to prove one or two mathematical results analyzed during the course. The duration of the written exam is two hours.

- The oral exam can always be taken. In the oral part, the student will be required to illustrate results presented during the course in order to evaluate her/his profound comprehension of them as well as of their interactions.
- The final evaluation considers both parts even though it will be mainly based on the oral one.
- Grades are expressed in thirtieths.
MAT/06 - PROBABILITY AND STATISTICS - University credits: 9
Practicals: 36 hours
Lessons: 42 hours
upon appointment via email
Via Saldini 50, second floor
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Room of the teacher or online room