Lecturer

Vincent Tassion

Coordinator

Laurin Köhler-Schindler

Content

Stochastic processes are a way to describe and study the behaviour of systems that evolve in some random way. In this course, the evolution will mostly be with respect to a scalar parameter interpreted as time, so that we discuss the temporal evolution of the system. We introduce several important classes of stochastic processes, analyse their properties and behaviour and show by some examples how they can be used. The main emphasis is on theory; in that sense, "applied" should be understood to mean "applicable".

Outline of the course:

• Discrete-time Markov chain
• Poisson process
• Poisson point process
• Renewal process
• Continuous-time Markov chain

Prerequisites

Prerequisites are familiarity with (measure-theoretic) probability theory as it is treated in the course "Probability Theory" (401-3601-00L).

Lecture Notes

Lecture notes will be regularly updated.

• Markov Chains (Chapters 1 & 2)
• Poisson Process (Chapter 3)
• Poisson Point Processes (Chapter 4)
• Renewal Processes (Chapter 5)

In addition, you can access the notes of lecture 12, the notes of lecture 13, and the notes of lecture 14. As mentioned in class, the lecture notes on continuous-time Markov Chains will possibly be made available at a later stage (for continuous-time Markov chains (Chapter 6), only the material from the notes of lectures 13-14 and the exercises are relevant for the exam).

You can also find the lecture notes by Prof. Sznitman (spring semester 2017) here.

Organization

Lectures take place online via Zoom each Tuesday from 09:15 to 12:00. Exercise classes also take place online via Zoom on Thursdays as indicated below. The Zoom links have been communicated by email to registered students. If you are a registered student and you need the link to be sent again, you can contact the coordinator of the course.

If you have any questions related to the content of the lectures or the exercises, you are very welcome to ask them on the forum.

Even though we recommend to attend the lectures live, recordings will be made available on the ETH video portal. The login details have been sent to all registered students by email. Please note that exercise classes will not be recorded to keep an interactive atmosphere and encourage spontaneous questions and discussions.

Exercises

The new exercises will be posted here every week on Wednesday. We recommend to look at the problems and prepare some questions before the exercise classes on Thursday.

Submission

You are welcome to submit your solutions, even though it is not oligatory. In case you decide to submit your solutions, this needs to be done before 18:00 of the corresponding due date (typically on Monday). Your solutions will then be corrected and returned by e-mail. Please follow the instructions below.

1. Scan your solution into a single PDF file. File format other than PDF will not be accepted. In case you do not have access to a scanner, scanning to a single PDF can be done efficiently using mobile applications such as Google Drive or Adobe Scan and many other ones. Please make sure that your scans are of good quality if you use these apps.
2. Name the created file as {email address}_{exercise sheet number}.pdf. The parts surrounded by brackets are placeholders that should be replaced by the email address at which you want to receive the corrected solution and the number of exercise sheet for which you are submitting a solution, respectively. A correctly named solution file for exercise sheet 1 is for instance given by markov.chain@probability.com_1.pdf.
3. Upload your solution using either the link A-L or the link M-Z in the table below (according to your surname).

Several comments are in order:

• Failing to follow the submission instructions will likely result in your submission not being corrected.
• Late submissions will not be considered.
• List of successfully uploaded files appears under the area used for uploading files. The font is, however, rather small so it might be easy to overlook.
• In case you submit your solution early and then decide to change something and submit again, we will only consider the most recent submission.

Exercise classes

First exercise class: Thursday, February 25

Enrollment and hand-in: The enrollment to the exercise classes by surname (see below) is binding for the submission of your solutions, but not for your attendance. This means that you can attend the exercise class that fits best with your schedule, but you need to submit your solutions according to your surname.

Time and location:

Literature

• R. N. Bhattacharya and E. C. Waymire, "Stochastic Processes with Applications", SIAM (2009), available online.
• R. Durrett, "Essentials of Stochastic Processes", Springer (2016), available online.
• M. Lefebvre, "Applied Stochastic Processes", Springer (2007), available online.
• S. I. Resnick, "Adventures in Stochastic Processes", Birkhäuser (2005).
• G. Last, M. Penrose, "Lectures on the Poisson Process", Cambridge University Press (2017), available online.