University of Bonn Institute for Applied Mathematics

Prof. Dr. Anton Bovier, Dr. Anna Kraut

Stochastic Processes
Sommersemester 2020



Start of the lecture: 21st of April 2020,
Time: Tuesday  8-10,
Friday  10 - 12.



Important announcement:

Due to the spread of the Coronavirus, the lecture will be held electronically via Zoom. According to instructions from the Vice Rector for Teaching, we may start only on April 21. 

In the meanwhile, we recommend to install ZOOM on your computer or tablet, to sign up for a free account, and to get familiar with it. Of course, you may also already look at the lecture notes. For the actual lectures, I will use a mixture of the notes and a whiteboard to write on. ZOOM allows interaction in several ways, so I think that you will be able to ask questions and participate nicely in the lectures. Of course this will be an experiment for all of us.

You can from now on sign up for the lecture in eCampus.

On eCampus, you can now find the links to join the lectures. There are two types: One will get you a calender entry that will remind you to join the lecture and contains the link to do so, the other is a direct link to the lectures. They are seperate for Tuesdays and Fridays. 



The topic of this lecture are stochastic processes, mainly in discrete time. We start we a review of measure theory. This partly complements the material from "Introduction to probability theory" but also helps to bring those who did not follow this course on the same footing.

The first important new topic are conditional expectations given a sigma-algebra. The study of stochastic processes then begins with the rigorous construction via the Daniel-Kolmogorov theorem. This is followed by the core of the course, the theory of discrete time martingales. This will include convergence theorems, martingale inequalities, and the theory of optional stopping, all of which will be crucial for the subsequence chapter on discrete time Markov processes with general state space. Here we will emphasise the connections between Markov processes and martingale and encounter the first version of the martingale problem. This will provide a new look at Dirichlet problems.

The course will conclude with a first look at a stochastic process in continuous time,  the celebrated Browning motion. A highlight will be the first functional limit theorem, Donsker's invariance principle, that establishes Brownian motion as a scaling limit of random walks. 


Lecture notes

The lecture notes for this course can be found here.



  • L.C.D. Rogers and D. Williamson, Diffusions, Markov processes and martingales. Vol. 1, Cambridge University Press

  • Y.S. Chow and H. Teicher, Probability theory. Independence, interchangeability, martingales. Springer, 1997
  • Achim Klenke, Wahrscheinlichkeitstheorie, Springer 2006

Exercise Classes:

The exercise classes are also held electronically via Zoom. 

Exercise Classes: 1.)  Monday 8-10,
2.)  Monday 10-12,
3.)  Monday 12-14,
4.)  Wednesday 12-14,
5.)  Wednesday 16-18

Collection of the exercise sheets:

Fridays before the lecture. Please discuss the exact way of collecting your submissions with your tutors. The first sheet will be uploaded on the 24th of April. Hence, the first "collection" of a sheet is on/until the 1st of May. 


Admission criteria

Due to the current situation, admission criteria will be waived.

Exercise Sheets

Sheet 1
Sheet 2 
Sheet 3
Sheet 4
Sheet 5
Sheet 6
Sheet 7
Sheet 8
Sheet 9
Sheet 10
Sheet 11
Mock exam




Second exam:

Date: Sept. 25, 9:00.

Place: Wolfgang-Paul-Hörsaal, Kreuzbergweg 28.


- Please arrive at the Wolfgang-Paul-Hörsaal no later than 8:40. We will let you in  at 8:45.

- Please try to keep the minimum distance of 1,50 m from your fellow students while you are waiting in front of the building.

- We then guide you into the building and to your seats in the lecture hall.

- Note that you have to wear a mask and disinfect your hands as soon as you enter the building. We provide some disinfectants at the entrance for you. 

- Mouth and nose covers may only be taken off once all candidates have taken their seats and the proctors have given a respective signal.

- Store your jacket and your bag immediately next to you at your seat. 

- Keep your student identity card and your identification card ready. 

- As soon as the preparation time begins, you have to fill and sign the contact tracing form. You will get two extra minutes of preparation time for that. 

- After the exam, we again guide you out of the building. Again, please make sure you that you keep the minimum distance of 1,50 m from your fellow students while you are in front of the building.


Further regulations: Please look at the cover sheet of the exam and the Vice Rector for Teaching and Learning’s handout for students, which you can find in the ecampus folder of this lecture. 


Post-exam review:  

Place: At the tables in the backyard of the Mathematik-Zentrum (Endenicher Allee 60). The following picture shows where we exactly meet. The red rectangle shows the place, where we are waiting, and where you can look inside your exams. The blue rectangle shows the place, where the waiting line should be. 


Time: The post-exam review will be held on Wednesday, Sept. 30, 11 a.m.., provided the weather is dry. Please chech this website for changes in advance.