Stochastische Differentialgleichungen (Winter Semester 2018/19)
- Lecturer: Prof. Dr. Lutz Weis
- Classes: Lecture (0105500), Problem class (0105510)
- Weekly hours: 4+2
PLEASE NOTE: There are are no lectures and tutorials on the following dates
- November 22: no tutorial - alternative date: December 20, 5:30 - 7pm in room 3.069
- November 23: no lecture - alternative date: 2019
- November 29: no tutorial - alternative date: February 7, 5:30 - 7pm in room 3.069
| Schedule | ||
|---|---|---|
| Lecture: | Tuesday 11:30-13:00 | SR 3.069 |
| Friday 9:45-11:15 | SR 3.069 | |
| Problem class: | Thursday 15:45-17:15 | SR 3.069 |
| Lecturers | ||
|---|---|---|
| Lecturer | Prof. Dr. Lutz Weis | |
| Office hours: | ||
| Room 2.047 Kollegiengebäude Mathematik (20.30) | ||
| Email: lutz.weis@kit.edu | Problem classes | Dr. Markus Antoni |
| Office hours: by appointment | ||
| Room 2.044 Kollegiengebäude Mathematik (20.30) | ||
| Email: markus.antoni@kit.edu | ||
In the mathematical approach to models appearing e.g. in nature sciences, engineering or finance which are described by an ordinary differential equation
we sometimes have to take random perturbations of the system into account. These 'errors' may originate from measurement errors, random environmental impacts, or incomplete information of the system's behaviour. These models can be described by stochastic differential equations of the form
where is a Brownian motion and
is a 'white noise'.
The purpose of this lecture is to study solutions of such stochastic differential equations. In order to do that we will start with a short recap of basic stochastic concepts (e.g. CDFs, independent random variables) and continue with an introduction to stochastic analysis (stochastic integrals, Ito formula, martingales, stopping times). With these results at hand we will prove fundamental existence, uniqueness, and stability results for stochastic differential equations as well as properties of their solutions (Markov property, path regularity).
This theory will be illustrated by examples appearing in mathematical finance, physics, biology, and engineering.
Prerequisites: Integration theory, basic stochastic concepts, Hilbert spaces
Exercise sheets
Sample exam
The sample exam will take place on Thursday, February 7, 2019, 3:45 - 7pm, in room 3.069 of the Kollegiengebäude Mathematik (20.30).
Exam
The final exam will take place on Thursday, February 21, 2019, 11am - 1pm, in room 1.067 of the Kollegiengebäude Mathematik (20.30).
The preliminary results of the exam can be found on the blue board next to room 2.041 or online in CAS.
The post-exam review will take place on Thursday, March 14, 2019 at 1pm in room 2.070 of the Kollegiengebäude Mathematik (20.30).
Literature
- J.M. Steele: Stochastic Calculus and Financial Applications, Springer, 2001
- L.C. Evans: An Introduction to Stochastic Differential Equations, AMS, 2013
- X. Mao: Stochastic Differential Equations and Applications, Woodhead Publishing, 2008