Academic year 2020/2021
- Course ID
- Prof. Laura Sacerdote (Lecturer)
Prof. Cristina Zucca (Lecturer)
- 1st year
- Teaching period
- Second semester
- D.M. 270 TAF B - Distinctive
- Course disciplinary sector (SSD)
- MAT/06 - probabilita' e statistica matematica
- Formal authority
- Type of examination
- Written and oral
- Good knowledge of Probability and Analysis
Sommario del corso
The course is aimed at giving the students the skills to use diffusion processes to represent different realities of practical interest. The student should use the different techniques for carrying out the analysis of the models. The student will demonstrate both the ability of self-study of advanced topics, connected to the content of the course, and the ability to collaborate. Students should also use the software Mathematica to perform some assigned simulations.
A module of the course, included in the overall courseload, will be taught by visiting professor Goran Peskir (University of Manchester, UK) on Diffusion processes and boundary classification (cf. International visiting professorsopen_in_new).
Results of learning outcomes
Knowledge and Understanding
Students will attain a knowledge of stochastic processes, in particular Brownian motion and diffusion processes.
Applying Knowledge and Understanding
Students will be able to study stochastic models of applied interest. They will know some of the important classes of stochastic processes and will be able to study their main functional and features.
Making Judgements and Learning Skills
Students will be able to apply theoretical or applied techiques to solve problems connected with the modeling of stochastic processes.
Students will be able to properly use English and to present their knowledge both in written form in the homeworks and in oral form in the final exam.
- Brownian Motion:
- Markov property,
- existence of the Brownian motion;
- maximum and first passage time distribution;
- arcosine laws;
- iterated logarithm law;
- Reflected Brownian motion;
- Heat equation and Brownian motion;
- multidimensional Brownian motion.
- Stationary Processes:
- mean square distance;
- autoregressive processes;
- ergodic theory and stationary processes;
- Gaussian processes
- Diffusion Processes:
- differential equations associated with some functionals of the process;
- backward and forward equations;
- stationary measures;
- boundary classification for regular diffusion processes;
- conditioned diffusion processes;
- spectral representation of the transition density for a diffusion;
- diffusion processes and stochastic differential equations;
- jump-diffusion processes;
- first passage time problems for diffusion processes.
- A 12 hours module, included in the courseload, will be taught by Visiting Professor Samuel Herrmann on Simulation of stochastic processes.
- Brownian Motion:
The course is composed of 48 hours of lectures which, for the AY 2020/2021, will be held remotely, either as live streaming or pre-recorded. All lectures will be recorded and made available on Moodle in due time, together with the related slides or notes and other course material.
Some additional activities to favour direct interaction between professors and students may be organised as online meetings and/or meetings in presence, under appropriate conditions of social distancing and compatibly and in compliance with future existing regulations. For meetings in presence, students who are not able to be physically present will have the chance to follow such activities through the online course material
Learning assessment methods
During the course homeworks are assigned. Solution of these exercises is part of the final exam. Teamwork is allowed for this part of the work. Exam is oral. Students that do not make homeworks will solve exercises immediately before the oral exam.
For the exercises there is no pubblic mark, just an evaluation which can be: passed, not passed. Only students that got a passed mark in the exercises can give the oral exam. The final mark will be given by the oral exam.
The evaluation of homeworks is valid only for the Summer exam session. From September session students are required to solve exercises immediately before the oral exam.
During the Covid-19 emergency the learning assessment method will consist in a written exam ( solution of exercises) followed by an oral exam via Webex video conference system. Those who solved the assigned exercises during the semester will be exempted from the written exam. The exemption holds until the September session, included
Suggested readings and bibliography
Schilling, Partzch, "Brownian Motion", De Gruyter
Karlin, Taylor. "A first Course in Stochastic Processes", Academic Press.
Karlin, Taylor. "A second Course in Stochastic Processes", Academic Press.
Mörters, Peres. "Brownian Motion", Cambridge University Press.
Kannan. "An introduction to stochastic processes", North Holland.
ATTENTION: arrangement for next week lessons (March 3,4).
Lesson of March, 3:
at 9:00 you find the material for the lesson on Moodle. At 10:45 I will be online to answer to your questions, using the chat of Moodle.
Lesson of March,4:
at 4:00 PM you will find the material for the lesson on Moodle. At 5:45 PM I will be online to answer to your questions, using the chat of Moodle.
Courses that borrow this teaching
- Enrollment opening date
- 01/09/2020 at 00:00
- Enrollment closing date
- 30/06/2021 at 00:00