Advanced Probability
A.Y. 2025/2026
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.
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.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
1. INTRODUCTION TO THE PROBABILITY THEORY
2. CONVERGENCE OF PROBABILITY MEASURE
2.1 Weak convergence in R
2.2. Weak convergence in metric space
3. CONDITIONING
3.1. Conditional expectation:
3.2. Regular versione of a conditional probability
4. STOCHASTIC PROCESSES
4.1. Measurability and continuity
4.2. Costruction of path spaces and Kolmogorv Bochner Theorem
4.3. Gaussian
4.4. Levy's processes. Poisson and Wiener processes
4.5. Markov processes: Transition functions, semigroups, infinitesimal semigroup. Dynkin formula
4.6 Markov Chains
5. STOPPING TIMES
6. MARTINGALES
6.1. Discrete time martingales and their transformations
6.2. Convergence results and Doob's optional stopping theorem
6.3. UI Martingales
6.4 L1 and L2 Martingales
6.3. Continuous time martingales
6.4. Quadratic variation theory
7. MARKOV CHAINS
7.1. Discrete time Markov Chains
7.2. Continuous time Markov Chains
See the Lesson Diary at the official web page of the course
2. CONVERGENCE OF PROBABILITY MEASURE
2.1 Weak convergence in R
2.2. Weak convergence in metric space
3. CONDITIONING
3.1. Conditional expectation:
3.2. Regular versione of a conditional probability
4. STOCHASTIC PROCESSES
4.1. Measurability and continuity
4.2. Costruction of path spaces and Kolmogorv Bochner Theorem
4.3. Gaussian
4.4. Levy's processes. Poisson and Wiener processes
4.5. Markov processes: Transition functions, semigroups, infinitesimal semigroup. Dynkin formula
4.6 Markov Chains
5. STOPPING TIMES
6. MARTINGALES
6.1. Discrete time martingales and their transformations
6.2. Convergence results and Doob's optional stopping theorem
6.3. UI Martingales
6.4 L1 and L2 Martingales
6.3. Continuous time martingales
6.4. Quadratic variation theory
7. MARKOV CHAINS
7.1. Discrete time Markov Chains
7.2. Continuous time Markov Chains
See the Lesson Diary at the official web page of the course
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 be delivered at the blackboard. Occasionally, a computer may be used.
At times, students will be given points for reflection that they will be expected to explore independently.
At times, students will be given points for reflection that they will be expected to explore independently.
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
On MyAriel all the reference are specified in the Lectur Diary section.
No lecture notes are available.
- 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
On MyAriel all the reference are specified in the Lectur Diary section.
No lecture notes are available.
Assessment methods and Criteria
The exam consists of two parts:
A written part, which involves discussing and reasoning about certain results based on the topics covered in class.
An oral exam, which assesses the student's knowledge of the theoretical results presented during the course, their corresponding proofs, as well as the ability to apply and connect various results and to reason through the course topics.
The final grade takes both parts into account.
The grade is expressed on a scale of 30 and will be communicated immediately after the oral exam.
A written part, which involves discussing and reasoning about certain results based on the topics covered in class.
An oral exam, which assesses the student's knowledge of the theoretical results presented during the course, their corresponding proofs, as well as the ability to apply and connect various results and to reason through the course topics.
The final grade takes both parts into account.
The grade is expressed on a scale of 30 and will be communicated immediately after the oral exam.
MAT/06 - PROBABILITY AND STATISTICS - University credits: 9
Practicals: 36 hours
Lessons: 42 hours
Lessons: 42 hours
Professor(s)
Reception:
Monday, 10:30 am - 1:30 pm (upon appointment, possibly suppressed for academic duties)
Department of Mathematics, via Saldini 50, office 1017.
Reception:
Please write an email
Room of the teacher or online room