Stochastic Processes (WS 2024)
Teacher: Prof. Dr. Peter Pfaffelhuber
Assistant and Tutor: Samuel Adeosun
Lecture: Mo 10-12, HS II, Albertstr. 23b
Exercises: Wed, 12-14, SR 218 Ernst-Zermelo-Str. 1 (starting on 23.10.24)
Examinations: Please check here what you need to do in order to obtain the ECTS points. In most cases, this is 50% of all credits of the exercise sheets, as well as presenting a solution two times during class for the pass/fail examination (Studienleistung). For the graded examination (Prüfungsleistung), there will be oral exams in both, BSc Mathematics and MSc programs.
General: The lecture contains (english) videos, which basically explain the content of the manuscript. During the lecture on Mondays, there will be a repetition of the videos. Here, you have the possibility to ask questions and interact with other people. There will be a new exercise sheet every Tuesday, which you have to hand in by Monday in then following week. It is best to do this in the mailbox in the basement of the Ernst-Zermelo-Str. 1. Two students can work together and hand in their solutions together.
Manuscript: The current version of the manuscript is here. There is also an older (German) version of the manuscript, which can be found here. Note that the manuscript builds upon the mansucripts on measure theory and probability theory.
Exercise sheets
| Sections | Handed out | to be handed in | |
|---|---|---|---|
| Full course on probability theory | 15.10.2024 | 21.10.2024 | |
| Section 13.1 | 22.10.2024 | 28.10.2024 | |
| Section 13.2, 13.3 | 29.10.2024 | 4.11.2024 | |
| Section 13.4, 13.5 | 5.11.2024 | 11.11.2024 | |
| Section 14.1 | 12.11.2024 | 18.11.2024 | |
| Section 14.2 | 19.11.2024 | 25.11.2024 | |
| Section 14.3 | 26.11.2024 | 2.12.2024 | |
| Section 14.4, 14.5 | 3.12.2024 | 9.12.2024 | |
| Section 15.1, 15.2 | 10.12.2024 | 16.12.2024 | |
| Section 15.3, 15.4 | 17.12.2024 | 7.1.2025 | |
| Section 15.4 | 7.1.2025 | 13.1.2025 | |
| Section 16.1, 16.2 | 14.1.2025 | 20.1.2025 | |
| Section 16.3, 16.4 | 21.1.2025 | 27.1.2025 | |
| Section 16.5 | 28.1.2025 | 3.2.2025 |
1. Introduction
Covers Section 13.1
2. The Poisson Process
Covers Sections 13.2
3. Brownian Motion
Covers Section 13.3
4. Filtrations and Stopping Times
Covers Section 13.4
5. Progressive Measurability
Covers Section 13.5
6. Introduction to Martingales
Covers Section 14.1
7. The stochastic integral as a martingale
Covers Section 14.2
8. Stopped Martingales
Covers Section 14.3
9. Martingale Convergence 1
Covers Section 14.4
10. Martingale Convergence 2
Covers Section 14.4
11. The Central Limit Theorem for Martingales
Covers Section 14.5
12. Martingales in Continuous Times
Covers Section 14.6
13. Markov Processes: Definition and Examples
Covers Section 15.1
14. The strong Markov property
Covers Section 15.2
15. The distribution of a Markov process
Covers Section 15.3
16. Semigroups and Generators
Covers Section 15.4
17. Examples of Markov processes
Covers Section 15.4
18. Brownian Motion: Quadratic Variation
Covers Section 16.1
19. Brownian Motion: Reflextion Pronciple
Covers Section 16.2
20. Brownian Motion: The Iterated Logarithm
Covers Section 16.3
21. Donsker's Theorem
Covers Section 16.4
22. The Skorohod embedding Theorem
Covers Section 16.4