**Stochastic Processes Summer Semester 2022**

## Prof. Dr. Anton Bovier

### Florian Kreten

**Lecture**

Start of the lecture: | April 05, 2022 |

Place: | Kleiner Hörsaal (Wegelerstr. 10) |

Time: | Tuesday 8-10, Friday 10 - 12. |

To register for the course on ecampus, click here. To register for the exercise classes on ecampus, click here. If you plan to write the exam, please register also on BASIS for both the course and the exercise classes (later in the semester). If you have any questions regarding the organisation of the lecture, please write an email to florian.kreten@uni-bonn.de.

**Content**

The topic of this lecture are stochastic processes, mainly in discrete time. We start with 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.

**Exercise Sheets**

Sheet 1

Sheet 2

Sheet 3

Sheet 4

Sheet 5

Sheet 6

Sheet 7

Sheet 8

Sheet 9

Sheet 10

Sheet 11

Sheet 12

Sheet 13

Beginning with sheet 2, you should submit your solutions electronically via ecampus, *in groups of three*. For being admitted to the exam, you will need at least 50% of all points (relevant are sheets 2-11).

**Exercise Classes**

Start in the second week of the lecture period. All tutorials take place at the mathematical institute, Endenicher Allee 60. The assignment will we done in the first week of the lecture.

Group 1: Monday 8c.t, room 0.003 (Nicholas Zafiris)

Group 2: Monday 10c.t., room 0.003 (Soma Hansel)

Group 3: Thursday 8c.t., room 0.003 (Alexander Becker)

Group 4: Monday 12c.t., room N0.003 (Qijin Shi)

Group 5: Monday 10c.t., room N0.003 (Manuel Esser)

**References**

- 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

**Exams**

You have two attempts for passing the exam. You can participate in the second exam only if you fail the first one (or equivalently do not show up ...). As for now, both exams will be written exams. Please be there roughly 15 minutes before the exam starts:

**1st exam:** July 19, 2022, 09:00-11:00. Location: *Großer *Hörsaal, Wegelerstraße 10.

The post-exam review will be held on July 20, 10:00-12:00, in room 0.006 at the Mathematics Centre, Endenicher Allee 60.

**2nd exam:** September 20, 2022, 09:00-11:00. Location: *Kleiner *Hörsaal, Wegelerstraße 10

The post-exam review will be held on September 21, 10:00-11:00, in room 0.011 at the Mathematics Centre, Endenicher Allee 60.

*Regulations*:

- No external support is allowed such as smartphones, notes, calculators, etc...!
- Do not write with erasable pens.
- Keep your student identity card and your identification card ready.
- You can use every statement from the lecture and from the weekly problem sheets, unless the exercise there is to prove this statement. If you use a statement, please quote it appropriately and verify that the conditions to use the statement are fulfilled.