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

Sheet 1: Repetition Probabiliy Theory

Solutions

Full course on probability theory 15.10.2024 21.10.2024

Sheet 2: Introduction

Solutions

Section 13.1 22.10.2024 28.10.2024

Sheet 3: Poisson process and Brownian Motion

Solutions

Section 13.2, 13.3 29.10.2024 4.11.2024

Sheet 4: Filtrations, Stopping Times, Measurability

Solutions

Section 13.4, 13.5 5.11.2024 11.11.2024

Sheet 5: Martingales 1

Solutions

Section 14.1 12.11.2024 18.11.2024

Sheet 6: Martingales 2

Solutions

Section 14.2 19.11.2024 25.11.2024

Sheet 7: Martingales 2

Solutions

Section 14.3 26.11.2024 2.12.2024

Sheet 8: Martingales 3

Solutions

Section 14.4, 14.5 3.12.2024 9.12.2024

Sheet 9: Markov Processes 1

Solutions

Section 15.1 10.12.2024 16.12.2024

Sheet 10: Markov Processes 2

Solutions

Section 15.2 17.12.2024 7.1.2025

Sheet 11: Markov Processes 3

Solutions

Section 15.3, 15.4 7.1.2025 13.1.2025

Sheet 12: Brownian Motion 1

Solutions

Section 16.1, 16.2 14.1.2025 20.1.2025

Sheet 13: Brownian Motion 2

Solutions

Section 16.3, 16.4 21.1.2025 27.1.2025

Sheet 14: Brownian Motion 3

Solutions

Section 16.5 28.1.2025 3.2.2025

1. Introduction

Covers Section 13.1

Slides

2. The Poisson Process

Covers Sections 13.2

Slides

3. Brownian Motion

Covers Section 13.3

Slides

4. Filtrations and Stopping Times

Covers Section 13.4

Slides

5. Progrssive Measurability

Covers Section 13.5

Slides

6. Introduction to Martingales

Covers Section 14.1

Slides

7. The stochastic integral as a martingale

Covers Section 14.2

Slides

8. Stopped Martingales

Covers Section 14.3

Slides

9. Martingale Convergence

Covers Section 14.4

Slides

10. The Central Limit Theorem for Martingales

Covers Section 14.5

Slides

11. Martingales in Continuous Times

Covers Section 14.6

Slides

12. Markov Processes: Definition and Examples

Covers Section 15.1

Slides

13. Strong Markov Processes

Covers Section 15.2

Slides

14. The distribution of a Markov process

Covers Section 15.3

Slides

15. Semigroups and Generators

Covers Section 15.4

Slides

16. Brownian Motion: Quadratic Variation

Covers Section 16.1

Slides

17. Brownian Motion: Reflextion Pronciple

Covers Section 16.2

Slides

18. Brownian Motion: The Iterated Logarithm

Covers Section 16.3

Slides

19. Donsker's Theorem

Covers Section 16.4

Slides

20. The Skorohod embedding Theorem

Covers Section 16.4

Slides