- Lecturer
- Prof. Dr. Martin Schweizer
- Coordinator
- David Martins

This course gives an introduction to Brownian motion and stochastic calculus. The following topics are planned:

- Definition and construction of Brownian motion
- Some important properties of Brownian motion
- Basics of Markov processes in continuous time
- Stochastic calculus, including stochastic integration for continuous semimartingales, Itô's formula, Girsanov's theorem, stochastic differential equations and connections with partial differential equations
- Basics of Lévy processes

Familiarity with measure-theoretic probability as in the standard D-MATH course Probability Theory will be assumed. Textbook accounts can be found for example in

- J. Jacod, P. Protter, Probability Essentials, Springer (2004)
- R. Durrett, Probability: Theory and Examples, Cambridge University Press (2019)

The first lecture takes place on Tuesday, February 22. Lectures and classes will not take place during Easter week from Friday, April 15 until Sunday, April 24.

Teaching is currently planned to take place in person, although this may change at any point depending on the evolution of the pandemic and the measures taken by ETH Zürich.

Lecture time | Room |
---|---|

Tue 08-10 | HG E 3 |

Thu 08-10 | HG E 3 |

Lecture notes (that will be fairly closely followed during the lectures) as well as auxiliary notes on Probability Theory are available here. The lecture notes will be updated and extended throughout the semester. The required password will be distributed to the students enrolled in the course via email.

Note that the lecture notes are protected by copyright, and their dissemination in any form is strictly prohibited.

The first exercise class takes place on Friday, February 25. Lectures and classes will not take place during Easter week from Friday, April 15 until Sunday, April 24.

Teaching is currently planned to take place in person, although this may change at any point depending on the evolution of the pandemic and the measures taken by ETH Zürich. Classes will be conducted in English.

Time | Room | Assistant |
---|---|---|

Fri 08-09 | HG G 26.5 | David Martins |

Fri 09-10 | HG G 26.5 | Marco Rodrigues |

Fri 12-13 | HG G 26.3 | Emir Nairi |

Solutions should be submitted to your assistant's folder in the box dedicated to this course, next to HG G 53.2. The deadline is 2pm on Wednesday before the class in order to guarantee that they are marked in time. The marked exercise sheets will be returned in the next class or otherwise returned to the box for collection.

Alternatively, you can submit your scanned solutions online using SAMup. Please submit the solution to each sheet as a single pdf file. The correction will also be uploaded via SAMup.

New exercise sheets will be uploaded here on Tuesday before the corresponding Friday exercise class, along with a model solution to the exercise sheet from the previous week. Submitting solutions is not mandatory, but attempting to solve the sheets is very helpful with practicing the contents of the course and preparing for the exam.

Note that the exam questions will use all the material from the lecture as well as from the exercises.

Exercise sheet | Due by | Solutions |
---|---|---|

Exercise sheet 0 | - | Solutions 0 |

Exercise sheet 1 | March 2 | Solutions 1 |

Exercise sheet 2 | March 9 | Solutions 2 |

Exercise sheet 3 | March 16 | Solutions 3 |

Exercise sheet 4 | March 23 | Solutions 4 |

Exercise sheet 5 | March 30 | Solutions 5 |

Exercise sheet 6 | April 6 | Solutions 6 |

Exercise sheet 7 | April 13 | Solutions 7 |

Exercise sheet 8 | April 27 |
Solutions 8 |

Exercise sheet 9 | May 4 | Solutions 9 |

Exercise sheet 10 | May 11 | Solutions 10 |

Exercise sheet 11 | May 18 | Solutions 11 |

Exercise sheet 12 | May 25 | Solutions 12 |

Exercise sheet 13 | June 1 | Solutions 13 |

Präsenz sessions (office hours) with assistants from group 3 are available on Mondays and Thursdays starting from the fourth week of the semester. More information here.

The exam will be in person, oral and closed book. Each candidate will receive a question and have 20 minutes to prepare for the exam in room HG G 47.1. The preparation is also closed book, and the question studied by the candidate will then be the starting question in the exam, which lasts for 20 minutes and takes place in HG G 51.2. Exam dates are August 23, 24, 26.

Students should bring for the exam an identification document and some paper to write on during their preparation time. No other aids are allowed, and mobile phones must be put away during the preparation time. Also, to anticipate a potential question — there is no available list of the possible questions. The material for the exam comprises all the material covered in the lecture notes (with the exception of the Levy-Ciesielski construction of Brownian motion) and all the material covered in the exercise sheets. Students are expected to have a good overview of the material, understanding both the ideas and the proofs for results.

Präsenz sessions (office hours) with assistants from group 3 will be available on the last two weeks before the exam session. More information here.

Students are welcome to sign up and participate in the forum for any questions and discussions related to the course.

- R.F. Bass, Stochastic Processes, Cambridge University Press (2011).
- I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, Springer (1998).
- J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer (2016).
- D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer (1999).
- L.C.G. Rogers, D. Williams, Diffusions, Markov Processes and Martingales, vol. 1 and 2, Cambridge University Press (2000).