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Department of Mathematics "Giuseppe Peano"

# Laurea Magistrale (M.Sc.) in Stochastics and Data Science

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## Stochastic processes

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### Academic year 2020/2021

Course ID
MAT0037
Teachers
Prof. Laura Sacerdote (Lecturer)
Prof. Cristina Zucca (Lecturer)
Year
1st year
Teaching period
Second semester
Type
D.M. 270 TAF B - Distinctive
Credits/Recognition
6
Course disciplinary sector (SSD)
MAT/06 - probabilita' e statistica matematica
Delivery
Formal authority
Language
English
Attendance
Optional
Type of examination
Written and oral
Prerequisites
Good knowledge of Probability and Analysis
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## Course objectives

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.

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## 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.

Communication Skills

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.

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## Program

• 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.
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## Course delivery

ATTENTION: arrangement for next week lessons (March 3,4).

We will use e-learning facilities. We are looking for technical options for a web lesson (if possible) or uploading material. Please check your mail regularly

Lessons (48 hours, 6 CFU) are given in lecture rooms.

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## 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

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## 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.

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## Notes

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.

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## Class schedule

Enroll
• Open
Enrollment opening date
01/09/2019 at 00:00
Enrollment closing date
30/06/2020 at 00:00
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Last update: 14/05/2020 09:24
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